systems with delays : analysis, control, and computations

Systems with Delays

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Publishers at ScrivenerMartin Scrivener([email protected])

Phillip Carmical ([email protected])

Systems with Delays

A.V. Kim and A.V. Ivanov

Analysis, Control, and Computations

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v

ContentsPreface ix

1 Linear time-delay systems 11.1 Introduction 1

1.1.1 Linear systems with delays 11.1.2 Wind tunnel model 21.1.3 Combustion stability in liquid propellant

rocket motors 31.2 Conditional representation of diff erential equations 5

1.2.1 Conditional representation of ODE and PDE 51.2.2 Conditional representation of DDE 6

1.3 Initial Value Problem. Notion of solution 91.3.1 Initial conditions (initial state) 91.3.2 Notion of a solution 10

1.4 Functional spaces 111.4.1 Space C[−τ,0] 121.4.2 Space Q[−τ,0] 121.4.3 Space Q[−τ,0) 131.4.4 Space H = Rη × Q[−τ,0) 14

1.5 Phase space H. State of time-delay system 151.6 Solution representation 16

1.6.1 Time-varying systems with delays 161.6.2 Time-invariant systems with delays 20

1.7 Characteristic equation and solution expansion into a series 241.7.1 Characteristic equation and eigenvalues 241.7.2 Expansion of solution into a series on

elementary solutions 26

vi Contents

2 Stability theory 392.1 Introduction 29

2.1.1 Statement of the stability problem 302.1.2 Eigenvalues criteria of asymptotic stability 312.1.3 Stability via the fundamental matrix 322.1.4 Stability with respect to a class of functions 33

2.2 Lyapunov-Krasovskii functionals 362.2.1 Structure of Lyapunov-Krasovskii quadratic

functionals 362.2.2 Elementary functionals and their properties 372.2.3 Total derivative of functionals with respect

to systems with delays 402.3 Positiveness of functionals 46

2.3.1 Defi nitions 462.3.2 Suffi cient conditions of positiveness 472.3.3 Positiveness of functionals 47

2.4 Stability via Lyapunov-Krasovskii functionals 492.4.1 Stability conditions in the norm || · || H 502.4.2 Stability conditions in the norm || · || 512.4.3 Converse theorem 522.4.4 Examples 53

2.5 Coeffi cient conditions of stability 542.5.1 Linear system with discrete delay 542.5.2 Linear system with distributed delays 56

3 Linear quadratic control 593.1 Introduction 593.2 Statement of the problem 603.3 Explicit solutions of generalized Riccati equations 67

3.3.1 Variant 1 673.3.2 Variant 2 683.3.3 Variant 3 69

3.4 Solution of Exponential Matrix Equation 733.4.1 Stationary solution method 733.4.2 Gradient methods 74

3.5 Design procedure 753.5.1 Variants 1 and 2 753.5.2 Variant 3 76

3.6 Design case studies 763.6.1 Example 1 763.6.2 Example 2 78

Contents vii

3.6.3 Example 3 783.6.4 Example 4 803.6.5 Example 5: Wind tunnel model 823.6.6 Example 6: Combustion stability in liquid

propellant rocketmotors 84

4 Numerical methods 894.1 Introduction 894.2 Elementary one-step methods 91

4.2.1 Euler’smethod 924.2.2 Implicit methods (extrapolation) 954.2.3 Improved Euler’smethod 964.2.4 Runge-Kutta-like methods 97

4.3 Interpolation and extrapolation of the model pre-history 984.3.1 Interpolational operators 984.3.2 Extrapolational operators 1004.3.3 Interpolation-Extrapolation operator 101

4.4 Explicit Runge-Kutta-like methods 1024.5 Approximation orders of ERK-like methods 1044.6 Automatic step size control 106

4.6.1 Richardson extrapolation 1064.6.2 Automatic step size control 1074.6.3 Embedded formulas 108

5 Appendix 1115.1 i-Smooth calculus of functionals 111

5.1.1 Invariant derivative of functionals 1115.1.2 Examples 116

5.2 Derivation of generalized Riccati equations 1245.3 Explicit solutions of GREs (proofs of theorems) 134

5.3.1 Proof of Th eorem 3.2 1345.3.2 Proof of Th eorem 3.3 1375.3.3 Proof of Th eorem 3.4 139

5.4 Proof of Th eorem 1.1. (Solution representation) 139

Bibliography 143

Index 164

ix

Preface

At present there are elaborated eff ective control and numerical methods and corresponding soft ware for analysis and simulating diff erent classes of ordinary diff erential equations (ODE) and partial diff erential equations (PDE). Th e progress in this direction results in wide application of these types of equations in practice. Another class of diff erential equations is represented by delay diff erential equations (DDE), also called systems with delays, time-delay systems, hereditary systems, functional diff erential equations.

Delay diff erential equations are widely used for describing and mathe-matical modeling of various processes and systems in diff erent applied prob-lems [3, 5, 1, 27, 32, 33, 34, 40, 50, 62, 63, 183, 91, 107, 108, 111, 127, 183].

Delay in dynamical systems can have several causes, for example: tech-nological lag, signal transmission and information delay, incubational period (infection diseases), time of mixing reactants (chemical kinetics), time of spreading drugs in a body (pharmaceutical kinetics), latent period (population dynamics), etc.

Th ough at present diff erent theoretical aspects of time-delay theory (see, for example, [3, 1, 27, 32, 34, 50, 62, 63, 67, 72, 73, 183, 91, 107, 111, 127] and references therein) are developed with almost the same completeness as the corresponding parts of ODE theory, practical implementation of many methods is very diffi cult because of infi nite dimensional nature of systems with delays.

Also it is necessary to note that, unlike ODE, even for linear DDE there are no methods of fi nding solutions in explicit forms, and the absence of generally available general-purpose soft ware packages for simulating such systems cause a big obstacle for analysis and application of time-delay systems.

In this book we try to fi ll up this gap.

x Preface

Th e main aim of the book is to present new constructive methods of DDE theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays1.

Th e main outstanding features of this book are the following:

1. on the basis of i-smooth analysis we give a complete descrip-tion of the structure and properties of quadratic Lyapunov-Krasovskii functionals2;

2. we describe a new control design technique for systems with delays, based on an explicit form of solutions of linear qua-dratic control problems;

3. we present new numerical algorithms for simulating DDE.

Acknowledgements

N.N.Krasovskii, A. B. Lozhnikov, Yu.F.Dolgii, A. I. Korotkii, O. V. Onegova, M. V. Zyryanov, Young Soo Moon, Soo Hee Han.

Research was supported by the Russian Foundation for Basic Research (projects 08-01-00141, 14-01-00065, 14-01-00477, 13-01-00110), the pro-gram “Fundamental Sciences for Medicine” of the Presidium of the Russian Academy of Sciences, the Ural-Siberia interdisciplinary project.

1 Th e present volume is devoted to linear time-delay system theory. We plan to prepare a special volume devoted to analysis of nonlinear systems with delays.2 Including properties of positiveness, and constructive presentation of the total derivative of functionals with respect to time-delay systems.

1

Chapter 1

Linear time-delaysystems

1.1 Introduction

1.1.1 Linear systems with delays

In this book we consider methods of analysis, control andcomputer simulation of linear systems with delays

x(t) = A(t) x(t)+Aτ (t) x(t−τ(t))+

0∫−τ(t)

G(t, s) x(t+s) ds+u ,

(1.1)where A(t), Aτ (t) are n×n matrices with piece-wise contin-uous elements, G(t, s) is n×n matrix with piece-wise con-tinuous elements on R× [−τ, 0], u is a given n–dimensionalvector-function, τ(t) : R → [−τ, 0] is a continuous func-tion, τ is a positive constant.

Much attention will be paid to the special class of lineartime-invariant systems

x(t) = A x(t)+Aτ x(t− τ)+

0∫−τ

G(s) x(t+ s) ds+u , (1.2)

2 Systems with Delays

where A, Aτ are n × n constant matrices, G(s) is n × nmatrix with piece-wise continuous elements on [−τ, 0], τ isa positive constant1.

Usually we will consider u as the vector of control para-meters. There are two possible variants:

1) u = u(t) is the function of time t;2) u depend on the current and previous state of the

system, for example,

u = C x(t) +

0∫−τ

D(s) x(t + s) ds . (1.3)

Consider some models of control systems with delays.

1.1.2 Wind tunnel model

A linearized model of the high-speed closed-air unit windtunnel is [134, 135]

x1(t) = −a x1(t) + a k x2(t− τ) ,

x2(t) = x3(t) , (1.4)

x3(t) = −ω2 x2(t)− 2 ξ ω x3(t) + ω2u3(t) ,

with a =1

1.964, k = −0.117, ω = 6, ξ = 0.8, τ = 0.33 s.

The state variable x1, x2, x3 represent deviations froma chosen operating point (equilibrium point) of the follow-ing quantities: x1 = Mach number, x2 = actuator positionguide vane angle in a driving fan, x3 = actuator rate. Thedelay represents the time of the transport between the fanand the test section.

The system can be written in matrix form

x(t) = A0x(t) + Aτx(t− τ) + B u(t) , (1.5)1I.e. in this case τ(t) ≡ τ .

Linear Time-delay Systems 3

where

A0 =

⎡⎢⎣−a 0 0

0 0 1

0 −ω2 −2 ξ ω

⎤⎥⎦ ,

Aτ =

⎡⎢⎣

0 a k 0

0 0 0

0 0 0

⎤⎥⎦ ,

B =

⎡⎢⎣

0

0

ω2

⎤⎥⎦ .

1.1.3 Combustion stability in liquid propellantrocket motors

A linearized version of the feed system and combustionchamber equations, assuming nonsteady flow, is given by2

φ(t) = (γ − 1) φ(t)− γ φ(t− δ) + μ(t− δ)

μ1(t) =1

ξJ

[−ψ(t) +

p0 − p1

2Δp

]

μ(t) =1

(1− ξ)J[−μ(t) + ψ(t)− P φ(t)]

ψ(t) =1

E[μ1(t)− μ(t)] . (1.6)

Hereφ(t) = fractional variation of pressure in the combustionchamber,t is the unit of time normalized with gas residence time,θg, in steady operation,τ = value of time lag in steady operation,p = pressure in combustion chamber in steady operation,

2The example is adapted from [36, 58].


τpγ = const for some number γ,

δ =τ

θg,

μ(t) = fractional variation of injection and burning rate,ψ(t) = relative variation of p1,

p1 = instantaneous pressure at that place in the feedingline where the capacitance representing the elasticityis located,

ξ = fractional length for the constant pressure supply,J = inertial parameter of the line,P = pressure drop parameter,μ1(t) = fractional variation of instantaneous mass flow up-stream of the capacitance,Δp = injector pressure drop in steady operation,p0 = regulated gas pressure for constant pressure supply,E = elasticity parameter of the line.

For our purpose we have taken

u =p0 − p1

2Δp

to be a control variable and guided by [36] have adoptedthe following representative numerical values:γ = 0.8, ξ = 0.5, δ = 1, J = 2, P = 1, E = 1.

This gives, for x(t) = (φ(t), μ1(t), μ(t), ψ(t))′,

x(t) = A0x(t) + Aτx(t− 1) + Bu(t) , (1.7)

where

A0 =

⎡⎢⎢⎢⎢⎣

0.2 0 0 0

0 0 0 −1

−1 0 −1 1

0 1 −1 0

⎤⎥⎥⎥⎥⎦ ,


Aτ =

⎡⎢⎢⎢⎢⎣−0.8 0 1 0

0 0 0 0

0 0 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎦ ,

B =

⎡⎢⎢⎢⎢⎣

0

1

0

0

⎤⎥⎥⎥⎥⎦ .

The system (1.7) has two roots with positive real part:λ1,2 = 0.11255± 1.52015 i.

1.2 Conditional representation of differ-ential equations

1.2.1 Conditional representation of ODE andPDE

Let us remember that for ODE

x(t) = g(t, x(t)) , (1.8)

the conditional representation is

x = g(t, x) , (1.9)

i.e. the argument t is not pointed out in state variable x(t).The conditional representation of the partial differential

equation∂y(t, x)

∂t= a

∂2y(t, x)

∂x2,

is∂y

∂t= a

∂2y

∂x2, (1.10)

i.e. the arguments t and x are not pointed out in the func-tion y(t, x).


Thus in order to obtain the conditional representationof an ODE it is necessary to make in this equation thefollowing substitutions{

x(t) to replace by x,x′(t) to replace by x′. (1.11)

Example 1.1. The linear control ODE

x′(t) = a(t)x(t) + u(t) ,

can be written in the conditional form as

x′ = a(t)x + u(t) ,

note, we omit variable t only in the state variable x(t) butnot in the coefficients a(t) and u(t). One can omit t alsoin the control variable u(t), in this case the conditionalrepresentation will be

x′ = a(t)x + u .

�

Remark 1.1 It is necessary to emphasize, conditionalrepresentation is very useful for describing local propertiesof differential equations, for application of geometrical lan-guage and methods. �

1.2.2 Conditional representation of DDE

Let us introduce the conditional representation of systemswith delays (1.1). First of all it necessary to note, differ-ential equations with time lags differ from ODE by pres-ence (involving) point x(t − τ) and/or segment x(t + s),−τ ≤ s < 0, which characterize previous history (pre-history) of the solution x(t).

The conditional representation of time-delay systems(1.1) can be introduced in the following way. In H an


element of trajectory of the system is written as a pairxt ≡ {x(t); x(t + s),−τ ≤ s < 0} ∈ H . Then, using thenotation

xt ≡ {x(t); x(t + s),−τ ≤ s < 0} ≡≡ {x(t); x(t + ·)} ≡ {x, y(·)}t (1.12)

we obtain the conditional representation

x = A(t) x + Aτ (t) y(−τ(t)) +

0∫−τ(t)

G(t, s) y(s) ds + u ,

(1.13)for system (1.1) in the space H .

Correspondingly, the conditional representation of time-invariant system (1.2) is

x = A x + Aτ y(−τ) +

0∫−τ

G(s) y(s) ds + u . (1.14)

Conditional representations (1.9), (1.13) and (1.14) haveno “physical sense”, and formulas (1.13) and (1.14) areunderstood as systems (1.1) and (1.2) considered in thephase space H . It is convenient to use representations(1.9), (1.13) and (1.14) for investigating local propertiesof differential equations.

Remark 1.2. It is necessary to emphasize that we usein {x, y(·)}t (see (1.13) ) different letters for denoting cur-rent vector x(t) = x and the function-delay x(t + ·) = y(·),because they play different roles in the dynamic of time-delay systems. One can use a fruitful analogy betweena train and the presentation {x, y(·)}t of the element oftrajectory of the system. The current point x plays therole of a locomotive and the function-delay y(·) presentsthe trucks which follow the locomotive. This is not only an


imaginary analogy. In many examples the solutions havesome kind of “inertness” because of the presence of delayterms. �

So, in order to obtain conditional representation of DDEwe make in this equation the following replacements:1) substitution (1.11) for current point x(t),2) substitution:

x(t + s) to replace by y(s),−τ ≤ s < 0, (1.15)

for pre-history x(t + s), −τ ≤ s < 0.

In particular,

x(t + τ(t)) is replaced by y(−τ(t)) ,

x(t + τ) is replaced by y(−τ) ,

x(t + s) is replaced by y(s) for − τ ≤ s < 0 .

Remark 1.3. Employment of conditional representa-tions (1.13) and (1.14) allows clearly to separate in thestructure of time delay systems the finite dimensional com-ponents x and infinite dimensional components y(·) and toformulate results in such a way that if function-delay y(·)disappears then the results turn into the corresponding re-sults of ODE theory (with exactness in notation). It allowsto carry out a methodological analysis of results and meth-ods of the theory of differential equations with deviatingarguments. �


1.3 Initial Value Problem. Notion of so-lution

1.3.1 Initial conditions (initial state)

In the present section we consider the statement of theinitial value problem for time-delay systems. Remember,for ODE (1.9) the initial condition has the form: (t0, x

0),where t0 is the initial time moment and x0 is the initialstate.

In order to define the solution x(t) of time-delay system(1.13) (or (1.14) ) it is necessary to know an initial point x0

and an initial function y0(·), i.e. at the initial time momentt0 the solution x(·) should satisfy the initial conditions:

x(t0) = x0 , (1.16)

x(t0 + s) = y0(s), −τ ≤ s < 0. (1.17)

So an initial state of a system with delays we will con-sider as a pair h0 = {x0, y0(·)}.

Remark 1.4. In general case an initial point x0 andan initial function y0(·) are not related, i.e. can be chosenindependently. �

Thus, we can formulate the initial value problem forsystem (1.13): for a given initial state (position) h0 ={x0, y0(·)} and an initial time moment t0 to find the so-lution x(t), t ≥ t0 − τ, of system (1.13) which satisfies theinitial conditions (1.16), (1.17).

Now we can give the definition of a solution of an ini-tial value problem, but first it is necessary to discuss whatfunctions y0(·) we will consider as initial functions.

In many papers and books the classes of continuous andmeasurable initial functions are considered. These classes of


functions are very useful for investigating different aspectsof time-delay systems, however

1) in many applied problems initial conditions are discon-tinuous, so in this case the class of continuous initialfunctions is insufficient;

2) the consideration of systems with delays with respectto measurable initial functions requires application ofmathematical methods which could be complicatedfor engineers and applied mathematicians, and besidesthat, measurable initial conditions are very rare casesin practical problems for time-delay systems.

So in the present book we develop different aspects ofthe theory of time-delay systems for initial conditions h0 ={x0, y0(·)} with piece-wise continuous functions y0(·) ={y0(s), −τ ≤ s < 0}, because this class of initial functionscover almost all admissible initial conditions.

It is necessary to emphasize, in the first place, piece-wisecontinuous functions include the class of continuous func-tions and, in the second place, measurable functions can beapproximated by piece-wise continuous functions.

1.3.2 Notion of a solution

Now let us define what we will understand under the solu-tions of time-delay systems.

Definition 1.1. The solution of system (1.13) corre-sponding to an initial time moment t0 and an initial stateh0 = {x0, y0(·)} is the function x(t) = x(t; t0, h

0) whichsatisfies the following conditions:

1) x(t) is defined on some interval [t0 − τ, t0 + κ), κ > 0,

2) x(t) satisfies the initial conditions (1.16)–(1.17),


3) x(t) is continuous on [t0, t0 + κ) and has on this timeinterval piece-wise continuous derivative,

4) x(t) satisfies the equation (1.13) on [t0, t0 + κ)3.

�

It is necessary to make some comments in respect of thisdefinition.

1) Initial state h0 = {x0, y0(·)} can be discontinuous, sothe corresponding solution can be discontinuous oninitial time interval [t0 − τ, t0]. However the solutionshould be continuous for t ∈ [t0, t0 + κ).

2) The derivative x′(t) of the solution can have discon-tinuities on [t0, t0 + κ) but, it is very important, werequire that at points of discontinuities of the solutionof equation (1.13) is satisfied for the right-hand sidederivative x′(t + 0).

3) It is necessary to note, at initial time moment t0 thederivative of the solution x(t) can be discontinuous,in this case it is supposed that at initial time momentt0 equations (1.13) are satisfied for the right-hand sidederivative x′(t0 + 0).

1.4 Functional spaces

The contemporary time-delay system theory is developedon the basis of functional approach to description and in-vestigation of such equations. That is, segments of solu-tions are considered as elements of some functional space.This approach will be discussed in the following sections.

In the present section we describe functional spacesC[−τ, 0], Q[−τ, 0] and H = Rn × Q[−τ, 0) which will beused for the realization of this approach.

3At points of discontinuity of the derivative x′(t) the equation (1.13) shouldbe satisfied for the right-hand side derivative x′(t + 0).


1.4.1 Space C[−τ, 0]

C[−τ, 0] is the set of n-dimensional continuous on [−τ, 0]functions. For any two functions φ(·), ψ(·) ∈ C[−τ, 0] thereis defined the distance

‖φ(·)− ψ(·)‖C = max−τ≤s≤0

‖φ(s)− ψ(s)‖ .

1.4.2 Space Q[−τ, 0]

Q[−τ, 0] is the set of n-dimensional functions q(s), −τ ≤s ≤ 0, with the properties:1) q(·) is continuous on the interval [−τ, 0] except, may be,

a finite set of points of discontinuity of the first kind (atwhich q(·) is continuous on the right) ;

2) q(·) is bounded on [−τ, 0].

Let us make some remarks:1) the term discontinuity of the first kind at a point s∗ ∈

(−τ, 0) means that at this point the function q(·) hasfinite unequal left-side and right-side limits4;

2) the term continuous on the right means that at the points∗ of discontinuity we set

q(s∗) = lims→s∗+0

q(s) ,

i.e. at this point the function takes value q(s∗) equal tothe right-side limits;

3) different functions of Q[−τ, 0] can have different pointsof discontinuity.

The distance between two elements q(1)(·), q(2)(·) of thisspace is defined as

‖q(1)(·)− q(2)(·)‖Q = sup−τ≤s≤0

‖q(1)(s)− q(2)(s)‖ .

4These points are called points of discontinuity.


Every continuous on [−τ, 0] function belongs to Q[−τ, 0],hence C[−τ, 0] ⊂ Q[−τ, 0].

1.4.3 Space Q[−τ, 0)

The set Q[−τ, 0) consists of n-dimensional functions y(s),−τ ≤ s < 0, with the properties:1) y(·) is continuous on the half-interval [−τ, 0) except, may

be, a finite set of points of discontinuity of the first kind(at which q(·) is continuous on the right) ;

2) y(·) is bounded on [−τ, 0);3) there exists finite left-side limit at zero lim

s→0−y(s).

Remark 1.5. For example, the function y∗(s) =

sin

(1

s

), −τ ≤ s < 0, does not belong to Q[−τ, 0) be-

cause left-side limit lims→0−

y(s) does not exist. The function

y∗(s) =1

s, −τ ≤ s < 0, also does not belong to Q[−τ, 0)

because is unbounded at zero. �

The distance between two elements y(1)(·), y(2)(·) ∈Q[−τ, 0) is defined as

‖y(1)(·)− y(2)(·)‖τ = sup−τ≤s<0

‖y(1)(s)− y(2)(s)‖ .

It is necessary to note, functions of the space Q[−τ, 0)are constrictions of functions of the space Q[−τ, 0] on thehalf-interval [−τ, 0), i.e. can be obtained from functions ofthe space Q[−τ, 0] by removing their values at zero. Forexample, for a function q(·) ∈ Q[−τ, 0] the correspond-ing constriction on the half-interval [−τ, 0) is the functiony(·) = {q(s),−τ ≤ s < 0} which satisfies all above men-tioned properties of elements Q[−τ, 0).

However, there is no one-to-one correspondence betweenspaces Q[−τ, 0] and Q[−τ, 0) ! The space Q[−τ, 0] broader


than Q[−τ, 0). For example, to one element y(·) ∈ Q[−τ, 0)correspond infinite numbers of functions

qx(·) =

{x for s = 0

y(s) for − τ ≤ s < 0

of the space Q[−τ, 0], because for every x ∈ Rn the functionqx(·) is the element of Q[−τ, 0].

1.4.4 Space H = Rη ×Q[−τ, 0)

The space Rn × Q[−τ, 0) is the Cartesian product of thefinite dimensional space Rn and infinite dimensional spaceQ[−τ, 0), i.e. elements of this space are pairs {x, y(·)}where x is the vector from Rn and y(·) is the functionfrom Q[−τ, 0).

For convenience we denote the space Rn×Q[−τ, 0) usingone letter H , i.e. H = Rn×Q[−τ, 0), and correspondently,its elements we denote also by one letter h = {x, y(·)}.

The distance between two elements

h(1) = {x(1), y(1)(·)} , h(2) = {x(2), y(2)(·)} ∈ H

is defined as

‖h(1) − h(2)‖H = max{‖x(1) − x(2)‖, ‖y(1) − y(2)‖τ}.One can see the spaces H and Q[−τ, 0] are isometric,

that is there exists one-to-one mapping A : Q[−τ, 0] → Hwhich preserves the distance between elements. For ex-ample, to a function q(·) ∈ Q[−τ, 0] corresponds pair{q(0); q(s), τ ≤ s < 0} ∈ H , and vice versa, to an element{x, y(·)} ∈ H corresponds the function

q(·) =

{x for s = 0

y(s) for − τ ≤ s < 0

which is the element of Q[−τ, 0]. Hence, from the pointof view of the mathematical properties two spaces H andQ[−τ, 0] are identical, i.e. they are one and the same space!


So sometimes we will use term “function” and for pair{x, y(·)} ∈ H .

However, for the description of properties of systemswith delays it will be more convenient to use the presenta-tion H = Rn ×Q[−τ, 0).

1.5 Phase space H. State of time-delaysystem

Phase space of a dynamical system is called the set of allpossible instantaneous states of the system.

In case of ODE attention, usually, is not paid to thenotion of the phase space because it is almost always Rn.

But infinite dimensional systems (functional differentialequations, partial differential equations) one can considerin different phase spaces, so the choice of suitable phasespace plays in this case a very important role.

For time-delay system (1.13) phase space is usuallytaken as the set of definition the right-hand sides f of thesystem and is defined in its turn by the class of admissibleinitial functions.

In the present book we consider systems with respectto piece-wise continuous initial conditions. So it will beconvenient to take the set H = Rn×Q[−τ, 0) as the phasespace of systems with delays. In the phase space H thestate at time moment t of a system with delays is the pair{x(t); x(t+ ·)} of the current point x(t) and function-delayx(t + ·) = {x(t + s),−τ ≤ s < 0}. We will denote this pairas

xt = {x(t); x(t + ·)}. (1.18)

Note, (1.18) is the element of the space H = Rn×Q[−τ, 0).

Remark 1.6. As was noted in the previous section thespaces H and Q[−τ, 0] are isometric. Nevertheless for ourgoals it is convenient to use the phase space in the form


H = Rn × Q[−τ, 0) in order to explicitly distinguish fi-nite dimensional and infinite dimensional components inthe structure of systems and functionals. This structureof the space H (as the Cartesian product of finite dimen-sional and infinite dimensional spaces) allows to clean upthe structure of systems with delays, to give more precisedescriptions of some constructions and properties. �

Remark 1.7. Though we chose H as the basic phasespace for describing and investigating time-delay systems,nevertheless in many cases it is convenient to use (and wewill do it!) the space C[−τ, 0] for describing some proper-ties of nonlinear systems. The space C[−τ, 0] is convenientfor investigating and describing asymptotic properties oftime-delay systems, and the space H is convenient for in-vestigating local properties of systems. �

Remark 1.8. Linear time-delay systems often are con-sidered in the phase space Rn × L2[−τ, 0), because thisspace is the Hilbert space and has many constructions con-venient for describing and investigation linear systems. �

1.6 Solution representation

In this section we give special representations of solutions oflinear time-varying and time-invariant systems with delays.

1.6.1 Time-varying systems with delays

Consider a linear system

x = A(t) x+Aτ (t) y(−τ)+

0∫−τ

G(t, s) y(s) ds+u(t) (1.19)


where A(t), Aτ (t) and u(t) are n×n, n×n and n×1 matriceswith piece-wise continuous on R coefficients; G(t, s) is n×nmatrix with piece-wise continuous on R× [−τ, 0] elements.

Similar to ODE’s the solutions of system (1.19) can beexpressed in terms of fundamental matrix5

F [t, ξ] =

⎡⎢⎢⎢⎢⎣

f11(t, ξ) f12(t, ξ) . . . f1n(t, ξ)

f21(t, ξ) f22(t, ξ) . . . f2n(t, ξ)

. . . . . .

fn1(t, ξ) fn2(t, ξ) . . . fnn(t, ξ)

⎤⎥⎥⎥⎥⎦ (1.20)

which is the solution of the matrix delay differential equa-tion

∂F [t, ξ]

∂t= A(t) F [t, ξ] +

+Aτ (t) F [t−τ, ξ]+

0∫−τ

G(t, s) F [t+s, ξ] ds, t > ξ , (1.21)

under the condition{F [ξ, ξ] = I ,

F [t, ξ] = 0 for t < ξ .(1.22)

Theorem 1.1. The solution x(t) = x(t; t0, h0) of sys-

tem (1.21) corresponding to an initial condition{t0 ∈ R ,

h0 = {x0, y0(·)} ∈ H = Rn ×Q[−τ, 0)(1.23)

has the form

x(t) = F [t, t0] x0+

0∫−τ

F [t, t0+τ +s] Aτ(t0+τ +s) y0(s) ds+

5Also called the state transition matrix.


+

0∫−τ

⎡⎣ s∫−τ

F [t, t0 + s− ν] G(t0 + s− ν, ν) dν

⎤⎦ y0(s) ds+

+

t∫t0

F [t, ρ] u(ρ) dρ . (1.24)

Proof of the theorem is given in Appendix. �

Example 1.2. Let us find a fundamental matrix of thesystem {

x1 = aτ (t) y2(−τ) ,

x2 = 0 ,(1.25)

where aτ (t) is a continuous on R function.The corresponding matrices A(t) and Aτ (t) are

A =

(0 0

0 0

), Aτ =

(0 aτ (t)

0 0

).

The fundamental matrix

F [t, ξ] =

(f11(t, ξ) f12(t, ξ)

f21(t, ξ) f22(t, ξ)

),

should satisfy the system of differential equations

∂F [t, ξ]

∂t≡(

f11(t, ξ) f12(t, ξ)

f21(t, ξ) f22(t, ξ)

)=

=

(0 aτ (t)

0 0

) (f11(t− τ, ξ) f12(t− τ, ξ)

f21(t− τ, ξ) f22(t− τ, ξ)

),

where dot “·” denotes the derivative with respect to thevariable t.


Thus we have the system of 4 differential equations⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

f11(t, ξ) = aτ (t) f21(t− τ, ξ) ,

f12(t, ξ) = aτ (t) f22(t− τ, ξ) ,

f21(t, ξ) = 0 ,

f22(t, ξ) = 0 ,

with the initial conditions (see (1.22))⎧⎪⎪⎨⎪⎪⎩

f11(ξ, ξ) = f22(ξ, ξ) = 1 ,

f21(ξ, ξ) = f12(ξ, ξ) = 0 ,

fij(t, ξ) = 0 for t < ξ , i, j = 1, 2 .

Subsequently calculating we can find

f21(t, ξ) = 0 ,

f22(t, ξ) =

{1 for t ≥ ξ ,

0 for t < ξ ,

f11(t, ξ) =

{1 for t ≥ ξ ,

0 for t < ξ ,

f12(t, ξ) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

t∫ξ+τ

aτ (ν)dν for t ≥ ξ + τ ,

0 for t < ξ + τ ,

hence

F [t, ξ] =

(0 0

0 0

)for t < ξ ,

F [t, ξ] =

(1 0

0 1

)for t ∈ [ξ, ξ + τ ] ,


F [t, ξ] =

⎛⎜⎜⎝ 1

t∫ξ+τ

aτ (ν)dν

0 1

⎞⎟⎟⎠ for t > ξ + τ .

�

1.6.2 Time-invariant systems with delays

For linear time-invariant system

x = A x + Aτ y(−τ) +

0∫−τ

G(s) y(s) ds + u(t) (1.26)

the fundamental matrix F [t, ξ] has the property

F [t, ξ] = F [t− ξ] ,

and the solution of the initial value problem (1.26), (1.23)has the form

x(t) = F [t− t0] x0 +

0∫−τ

F [t− t0 − τ − s] Aτy0(s) ds +

+

0∫−τ

⎡⎣ s∫−τ

F [t− t0 − s + ν] G(ν) dν

⎤⎦ y0(s) ds+

+

t∫t0

F [t− ξ] u(ξ) dξ . (1.27)

Taking into account that for time-invariant systems wecan always take the initial moment t0 = 0, hence in this


case the fundamental matrix

F [t] =

⎡⎢⎢⎢⎢⎣

f11(t) f12(t) . . . f1n(t)

f21(t) f22(t) . . . f2n(t)

. . . . . .

fn1(t) fn2(t) . . . fnn(t)

⎤⎥⎥⎥⎥⎦ (1.28)

is the solution of the matrix delay differential equation

dF [t]

dt= A F [t] + Aτ F [t− τ ] +

0∫−τ

G(s) F [t + s] ds , t > 0 ,

(1.29)under the condition{

F [0] = I ,

F [t] = 0 for t < 0 .(1.30)

Hence the solution corresponding to the initial momentt0 and an initial pair {x0, y0(s)} can be presented in theform

x(t) = F [t] x0 +

0∫−τ

F [t− τ − s] Aτy0(s) ds +

+

0∫−τ

⎡⎣ s∫−τ

F [t− s + ν] G(ν) dν

⎤⎦ y0(s) ds+

t∫0

F [t−ξ] u(ξ) dξ .

(1.31)Example 1.3. Let us find a fundamental matrix of the

system {x1 = x2 ,

x2 = −y1(−τ) .(1.32)

For this system the corresponding constant matrices A


and Aτ have the forms

A =

(0 1

0 0

), Aτ =

(0 0

−1 0

).

The fundamental matrix

F [t− ξ] =

(f11(t− ξ) f12(t− ξ)

f21(t− ξ) f22(t− ξ)

),

is the solution of the following system of differential equa-tions

∂F [t− ξ]

∂t=

(f11(t− ξ) f12(t− ξ)

f21(t− ξ) f22(t− ξ)

)=

=

(0 1

0 0

) (f11(t− ξ) f12(t− ξ)

f21(t− ξ) f22(t− ξ)

)+

+

(0 0

−1 0

) (f11(t− ξ − τ) f12(t− ξ − τ)

f21(t− ξ − τ) f22(t− ξ − τ)

).

So, in order to find the elements of the fundamental ma-trix F it is necessary to solve the system of 4 differentialequations with delays⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

f11(t− ξ) = f21(t− ξ) ,

f12(t− ξ) = f22(t− ξ) ,

f21(t− ξ) = −f11(t− ξ − τ) ,

f22(t− ξ) = −f12(t− ξ − τ) ,

(1.33)

with respect to the initial conditions (see (1.22)){f11(0) = f22(0) = 1 ,

f12(0) = f21(0) = 0 ,(1.34)


fij(t− ξ) = 0 for t < ξ , i, j = 1, 2 . (1.35)

Let us solve this system using the step method.

STEP 1. Because of the condition (1.35) the system (1.33)has on the time interval [ξ, ξ + τ ] the form

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

f11(t− ξ) = f21(t− ξ) ,

f12(t− ξ) = f22(t− ξ) ,

f21(t− ξ) = 0 ,

f22(t− ξ) = 0 .

Taking into account the initial conditions (1.34) we ob-tain

f21(t− ξ) = 0 , f22(t− ξ) = 1 , for t ∈ [ξ, ξ + τ ] ,

and

f11(t− ξ) = 1 , f12(t− ξ) = t− ξ , for t ∈ [ξ, ξ + τ ] .

Thus

F [t− ξ] =

(1 t− ξ

0 1

)for t ∈ [ξ, ξ + τ ] .

STEP 2. On the next time interval [ξ + τ, ξ + 2τ ] system(1.33) has the form

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

f11(t− ξ) = f21(t− ξ) ,

f12(t− ξ) = f22(t− ξ) ,

f21(t− ξ) = −1 ,

f22(t− ξ) = t− ξ − τ


with the initial conditions⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

f11(τ) = 1 ,

f12(τ) = τ ,

f21(τ) = 0 ,

f22(τ) = 1 .

In general case the coefficients of the fundamental ma-trix F have the form

f11(t− ξ) = f22(t− ξ) =

κ[ t−ξτ

]∑m=0

(−1)m (t− ξ −mτ)2m

(2m)!,

where κ[t] denotes the integer part of t,

f12(t− ξ) = f21(t− ξ) =

κ[ t−ξτ

]∑m=0

(−1)m (t− ξ −mτ)2m+1

(2m + 1)!.

�

1.7 Characteristic equation and solution

expansion into a series

1.7.1 Characteristic equation and eigenvalues

Consider a linear time-invariant system with delays

x = A x + Aτ y(−τ) +

0∫−τ

G(s) y(s) dt (1.36)

where A, Aτ and constant n × n matrices, G(s) is n × nmatrix with piece-wise continuous on [−τ, 0] elements.

Similar to ODE case let us look for solutions of (1.36)in an exponential form

x(t) = eλtC (1.37)


where λ is a complex number and C ∈ Rn. Substituting(1.37) into (1.36) gives

λeλtC = AeλtC + Aτeλ(t−τ)C +

0∫−τ

G(s) eλ(t+s)C ds .

Canceling the factor eλt and rearranging terms we obtain

[A− λIn×n + Aτe

−λτ +

0∫−τ

G(s) eλs ds]C = 0

orχ(λ) C = 0 , (1.38)

where

χ(λ) = A− λIn×n + Aτe−λτ +

0∫−τ

G(s) eλs ds . (1.39)

A nonzero vector C satysfying (1.37) exists if and onlyif the matrix χ(λ) is singular, i.e.

det χ(λ) = 0 . (1.40)

this is the so-called characteristic equation. The determi-nant Δ(λ) = det χ(λ) is called the characteristic quasipoly-nomial (characteristic function).

The complex number λ = α + iβ, which is a solution ofcheracteristic equation (1.40), is called an eigenvalue. Thecorresponding vector C ∈ Rn, satisfying (1.38), is calledan eigenvector.

The dynamics of system (1.36) is completely defined bythe roots6 of equation (1.39). However, unlike ODE theseroots can be found in explicit forms only in some rare cases.

Nevertheless there are qualitative results concerning dis-tribution of eigenvalues.

6Characteristic roots.


Theorem 1.2. Either χ(λ) is a polynomial7 or χ(λ) hasinfinitely many roots λ1, λ2, . . . such that Reλk → −∞ ask →∞. �

Theorem 1.3. Let λ is an eigenvalue, then

1) |λ| ≤ V ar[−τ,0]A if Re λ ≥ 0,

2) |λ| ≤ e−τ Re λ V ar[−τ,0]A if Re λ ≤ 0.

Corollary 1.1. For every specific system (1.36) thereexists a real number γ such that system (1.36) has no zerosin the right half-plane Reλ > γ. �

1.7.2 Expansion of solution into a series on ele-mentary solutions

We already have shown that for every characteristic root λthere exists a vector C ∈ Rn such that the function eλtCwill satisfy system (1.36) for every t ∈ R. If the root hasa multiplicity m > 1 then, in general case, there can besolutions of the form

φ(t) = eλtp(t) (1.41)

where p(t) : R→ Rn is a polynomial of a degree less thanm.

The maximal number of linear independent solutions ofthe form (1.41) corresponding to a characteristic root λ isequal to its multiplicity. We will call these solutions ele-mentary solutions of system (1.36).

Every solution of system (3.41) can be connected witha series [192, 193]

x(t) ∼∞∑

k=1

pk(t) ezkt , (1.42)

7Hence it has a finite number of roots.


where zk, k = 1,∞, are poles of system (3.41), pk(t) is apolynomial (the degree of the polynomial is 1 less than themultiplicity of the root zk). If for some Δ > 0 and α thereare no poles of system (3.41) in the strip α −Δ < Re z <α + Δ, then the asymptotic formula holds:

x(t) =∑

Re zk>α

pk(t) ezkt + O(eαt) . (1.43)

From this result follows an important corollary. If realparts of all eigenvalues are negative, i.e. Re zk < 0, thenevery solution of system (1.36) tends to zero.

29

Chapter 2

Stability theory

2.1 Introduction

The method of the Lyapunov function1 is one of the mosteffective methods for investigation of ODE dynamics. Effi-ciency of the Lyapunov function method for ODE is basedon the fact that application of Lyapunov’s function allowsus to investigate stability of solutions without solving cor-responding ODE.

In case of DDE the direct Lyapunov method was elab-orated in [111, 112] in terms of the infinite-dimensionalLyapunov-Krasovskii functionals.

In this chapter we

1) describe general structure of the quadratic Lyapunov-Krasovskii functionals;

2) derive the constructive formula of total derivative ofthe functionals with respect to systems with delays;

3) present basic theorems of the Lyapunov-Krasovskiifunctional methods for investigating stability of sys-tems with delays.

1This method is also called the direct or the second Lyapunov method.


2.1.1 Statement of the stability problem

In this chapter we consider linear time-invariant systemswith delays

x = A x + Aτy(−τ) +

0∫−τ

G(s) y(s) ds , (2.1)

h = {x, y(·)} ∈ H .Obviously, system (2.1) has the zero solution x(t) = 0.

Further we will investigate stability of this solution.The origin (the zero element) of space H is the station-

ary point of system (2.1), hence, generally speaking, wecan identify the zero solution x(t) ≡ 0 and the origin ofH . So further, the terms “stability of the zero solution”and “stability of the origin” will be used as synonyms.

Further we will use the following definitions.

Definition 2.1. The zero solution x(t) ≡ 0 of system(2.1) is stable if for any positive ε there exists a positive δsuch that if ‖h‖ < δ then ‖x(t; t0, h)‖ ≤ ε for all t ≥ t0. �

Definition 2.2. The zero solution x(t) ≡ 0 of system(2.1) is asymptotically stable if it is stable and

‖x(t; t∗, h)‖ → 0 as t →∞ .

�

Definition 2.3. The zero solution x(t) ≡ 0 of system(2.1) is exponentially stable if there exist positive constantsa and b such that for any (t∗, h) ∈ R×H

‖x(t; t∗, h)‖ ≤ a ‖h‖He−b (t−t∗) for t ≥ t∗ .

�

Remark 2.1. The interval [−τ, 0] is compact, so in allabove definitions one can use the functional norm ‖xt‖H of

Stability Theory 31

the solutions instead of the finite dimensional norm ‖x(t)‖.�

Note that, using suitable substitution, we can reduceinvestigating stability of arbitrary solutions of specific DDEsystem to investigating stability of the zero solution of some“perturbed” DDE.

Moreover, if a solution of system (2.1) corresponding tosome initial function is (asymptotically) stable then a so-lution corresponding to any other initial function also willbe (asymptotically) stable. hence for linear DDE we cansay about (asymptotic) stability of DDE system, but notonly a specific solution.

Also note the following useful proposition (A.Zverkin).

Theorem 2.1.

1) System (2.1) is stable if and only if for every(t0, h) ∈ R ×H the corresponding solution x(t; t0, h)is bounded;

2) If for every (t0, h) ∈ R×H the corresponding solutionx(t; t0, h) of system (2.1) tends to zero then the systemis asymptotically stable.

�

2.1.2 Eigenvalues criteria of asymptotic stability

As we already mentioned in subsection “Expansion of solu-tion into a series on elementary solutions”, every solutionof system (2.1) tends to zero if all eigenvalues have negativereal parts. In other words, the condition

Re zk < 0


for all eigenvalues, is necessary and sufficient condition ofasymptotic stability of system (2.1).

Also now we can note that for linear DDE asymptoticstability and exponential stability are equivalent.

2.1.3 Stability via the fundamental matrix

At present there are no effective algorithms of computingthe eigenvalues for linear systems with distributed delaysin order to check stability.

In this subsection we discuss another method of practi-cal verification of stability of the closed-loop system. Themethod is very simple for implementation and consists ofcomputing the fundamental matrix of the system. Thefundamental matrix can be numerically calculated usingTime-delay system toolbox [4].

Consider the homogeneous time-invariant system

x = A x + Aτ y(−τ) +

0∫−τ

G(s) y(s) ds . (2.2)

We can fomulate (see, for example, [76, 32]) the follow-ing stability conditions in terms of the fundamental matrix.

Theorem 2.2. System (2.2) is

1) stable if and only if there exists a constant k > 0 suchthat ∥∥∥F [t]

∥∥∥n×n

≤ k , t ≥ 0 ; (2.3)

2) asymptotically stable if and only if there exist con-stants k > 0 and α > 0 such that∥∥∥F [t]

∥∥∥n×n

≤ k e−α t , t ≥ 0 . (2.4)

Stability Theory 33

�

The fundamental matrix can be found numerically us-ing Time-delay system toolbox [4] as the solution ofsystem (1.29), (1.30).

Remark 2.2. Thus, one can easily check stability (orinstability) of system (1.26) solving numerically system(1.29), (1.30) and verifying the corresponding properties(2.3) or (2.4) of the matrix F [t].

Note, if at least one of the coefficients of the matrix F [t]is not uniformly bounded then system (1.26) is unstable. �

2.1.4 Stability with respect to a class of functions

First of all it is necessary to note that, as emphasized bymany authors (see, for example [111, 107]), complete cor-rect statement of a stability problem for a concrete systemwith delays should include description of a class of admis-sible initial functions (initial disturbances).

In this case it is sufficient to consider stability of so-lution of specific time-delay system only with respect toadmissible initial disturbances.

Remark 2.3. In [172] one can find an example ofa time-delay system which is unstable with respect to theclass of all continuous disturbances, but is stable with re-spect to more narrow class of admissible initial functions.�

Of course, classes of admissible initial disturbances aredifferent in different problems, so in general stability theoryusually the class of all continuous or piece-wise continuousinitial functions (disturbances) is considered. Nevertheless,in some problems such class of initial functions can be su-perfluous.


Let L be a subset (a system of functions) of the space H .

Definition 2.4. System (2.1) is stable with respect toa class of functions L if for any h ∈ L the correspondingsolution is bounded. �

Definition 2.5. System (2.1) is asymptotically stablewith respect to a class of functions L if

limt→∞

‖x(t; h)‖ = 0 (2.5)

for any h ∈ L. �

Note, from linearity of system (2.1) it follows that if thesystem is (asymptotically) stable with respect to a class Lthen the system will be also (asymptotically) stable withrespect to the space L∗ = span{L} spanned on L, and,moreover, the system will be (asymptotically) stable withrespect to the class

L = span

{⋃h∈L

⋃t≥0

xt(h)

}.

As we already mentioned, in many cases it is difficultto prove stability of a system with respect to the class allcontinuous initial functions. In this case one can check sta-bility of the system with respect to a class of test initialfunctions L by computer simulation.

The corresponding class of functions can be chosen, forexample, in the following way.

It is well known that there exist orthogonal systems ofcontinuous on the interval [−τ, 0] functions {φi(·)}∞i=0 suchthat every function ψ(·) ∈ C[−τ, 0] can be expanded inseries2

ψ(s) =∞∑i=0

γi φi(s) , −τ ≤ s ≤ 0 , (2.6)

2For example, trigonometrical system.

Stability Theory 35

with some coefficients {γi}∞i=0 ⊂ R.One can consider first k functions

φ1(·), φ2(·), . . . , φk(·) ∈ C[−τ, 0] (2.7)

as basic (test) functions, and investigate stability of system(2.1) with respect to this finite class of functions. In thiscase the system will be stable with respect to subspace offunctions, which are linear combinations of “basic” func-tions (2.7).

From the linearity of system (2.1) it follows that if forevery basic function φ1,. . .,φk the corresponding solutionx(t, φi) tends to zero as t → ∞, then for arbitrary con-stants γ1,. . .,γk the solution x(t, φ), corresponding to aninitial function (2.9), also tends to zero. So, it is sufficientto check convergence to zero only for functions φ1,. . .,φk.

Definition 2.6. System (2.1) is asymptotic stable withrespect to a class of functions (2.7) if

limt→∞

‖x(t; φi)‖ = 0 (2.8)

i = 1, . . . , k. �

Also it is necessary to note, though the series (2.6) con-tains an infinite number of terms, nevertheless taking intoaccount the presence of some uncertainties at every specific(applied) problem one can consider a class of admissibleinitial disturbances as a finite sum

ψ(s) =k∑

i=0

γi φi(s) , −τ ≤ s ≤ 0 , (2.9)

γ1,. . .,γk ∈ R, assuming that remainder part∞∑

i=k+1

γi φi(·)

of the series (2.6) corresponds to uncertainties.

Depending on the concrete problem one can choose hisown system of (linear independent) test functions.


2.2 Lyapunov-Krasovskii functionals

2.2.1 Structure of Lyapunov-Krasovskii quadraticfunctionals

For investigating linear finite dimensional systems

x = A x (2.10)

the quadratic Lyapunov functions

v(x) = x′P x (2.11)

are usually used (here P is n× n symmetric matrix).For linear DDE (2.1) similar role play the quadratic

Lyapunov-Krasovskii functionals, which have in generalcase the following presentation

V [x, y(·)] = x′P x + 2x′0∫

−τ

D(s) y(s) ds +

+

0∫−τ

y′(s) Q(s) y(s) ds +

0∫−τ

0∫−τ

y′(s) R(s, ν) y(ν) ds dν +

+

0∫−τ

0∫ν

y′(s) Π(s) y(s) ds dν +

+

0∫−τ

⎡⎣⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠⎤⎦ dν (2.12)

where P , D(s), Q(s), R(s, ν), Π(s), Γ(ν, s) are n × n ma-trices and s, ν ∈ [−τ, 0].

One can see that general quadratic functional (2.12) iscomposed by the system of more simple elementary func-tionals

V [x, y(·)] = W1[x] + W2[x, y(·)] +

+ W3[y(·)] + W4[y(·)] + W5[y(·)] + W6[y(·)]where

Stability Theory 37

W1[x] = x′P x , (2.13)

W2[x, y(·)] = 2 x′0∫

−τ

D(s) y(s) ds , (2.14)

W3[y(·)] =

0∫−τ

y′(s) Q(s) y(s) ds , (2.15)

W4[y(·)] =

0∫−τ

0∫−τ

y′(s) R(s, ν) y(ν) ds dν , (2.16)

W5[y(·)] =

0∫−τ

0∫ν

y′(s) Π(s) y(s) ds dν , (2.17)

W6[y(·)] =

0∫−τ

⎡⎣⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠⎤⎦ dν .

(2.18)So the properties of functional (2.12) are defined by the

properties of these elementary functionals.In the next subsection we describe some properties of

these functionals.

2.2.2 Elementary functionals and their properties

In Appendix we presented basic constructions of i-smoothcalculus and examples of calculating invariant derivativesof some general classes of functionals.

In this subsection we present the corresponding formu-las for described above elementary functionals.

The following formulas are valid:


For functional (2.13) :

∂W1[x]

∂x= 2 P x , ∂yW1[x] = 0

(P is symmetric matrix).


∂W2[x]

∂x= 2

0∫−τ

D(s) y(s) ds ,

∂yW2[x] = 2 x′[D(0) x−D(−τ) y(−τ)−

0∫−τ

dD(s)

dsy(s) ds

].

The formulas follow from Example 5.4 with

ω[x, s, y(s)] = 2 x′D(s) y(s) .


∂W3[y(·)]∂x

= 0 ,

∂yW3[x, y(·)] = x′Q(0) x− y′(−τ) Q(−τ) y(−τ)−

−0∫

−τ

y′(s)dQ(s)

dsy(s) ds .


ω[s, y(s)] = y′(s) Q(s) y(s) .

Stability Theory 39


∂W4[y(·)]∂x

= 0 ,

∂yW4[x, y(·)] =

= x′0∫

−τ

[R(0, s) + R′(s, 0)

]y(s) ds−

− y′(−τ)

0∫−τ

[R(−τ, s) + R′(s,−τ)

]y(s) ds−

−0∫

−τ

0∫−τ

y′(s)[∂R(s, ν)

∂s+

∂R(s, ν)

∂ν

]y(ν) ds dν .


γ[s, y(s); ν, y(ν)] = y′(s) R(s, ν) y(ν) .


∂W5[y(·)]∂x

= 0 ,

∂yW5[x, y(·)] =

= τ x′Π(s) x−0∫

−τ

y′(s) Π(s) y(s) ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν .


ω[s, y(s)] = y′(s) Π(s) y(ν) .



∂W6[y(·)]∂x

= 0 .

∂yW6[x, y(·)] =

= 2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν−⎛⎝ 0∫−τ

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫−τ

y(s) ds

⎞⎠ .

The formulas follow from Example 5.6.

2.2.3 Total derivative of functionals with respectto systems with delays

Total derivative of the quadratic function (2.11) with re-spect to system (2.10) has the simple form

v(2.10) = x′(A′P + P A

)x . (2.19)

Analysing total derivative (2.19) one can check variousproperties of original system (2.10) without calculating itssolutions.

Let us derive a formula of total derivative of thequadratic Lyapunov-Krasovskii functional (2.12) with re-spect to system (2.1).

The constructive formula of total derivative of Lyapunov-Krasovskii functional V [h] with respect to system (2.1) hasthe form

V(2.1)[x, y(·)] = ∂yV [x, y(·)] +

+∂V ′[x, y(·)]

∂x

[A x + Aτy(−τ) +

0∫−τ

G(s) y(s) ds

](2.20)

Stability Theory 41

where ∂yV [x, y(·)] is the invariant derivative of the func-tional V [x, y(·)].

Taking into account that

∂V [x, y(·)]∂x

=∂W1[x]

∂x+

∂W2[x, y(·)]∂x

+

+∂W3[y(·)]

∂x+

∂W4[y(·)]∂x

+∂W5[y(·)]

∂x+

∂W6[y(·)]∂x

and∂yV [x, y(·)] = ∂yW1[x] + ∂yW2[x, y(·)] +

+∂yW3[x, y(·)]+∂yW4[x, y(·)]+∂yW5[x, y(·)]+∂yW6[x, y(·)] ,we obtain

V(2.1)[x, y(·)] = 2

⎡⎣x′P +

0∫−τ

y′(s) D′(s) ds

⎤⎦×

×[A x + Aτy(−τ) +

0∫−τ

G(s) y(s) ds

]+

+ 2 x′[D(0) x−D(−τ) y(−τ)−

0∫−τ

dD(s)

dsy(s) ds

]+

+x′Q(0) x−y′(−τ) Q(−τ) y(−τ)−0∫

−τ

y′(s)dQ(s)

dsy(s) ds+

+ x′0∫

−τ

[R(0, s) + R′(s, 0)

]y(s) ds−

− y′(−τ)

0∫−τ

[R(−τ, s) + R′(s,−τ)

]y(s) ds−


−0∫

−τ

0∫−τ

y′(s)[∂R(s, ν)

∂s+

∂R(s, ν)

∂ν

]y(ν) ds dν +

+ τ x′Π(0) x−0∫

−τ


−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν +

+2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν−⎛⎝ 0∫−τ

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫−τ

y(s) ds

⎞⎠ =

= 2 x′P ′A x + 2 x′P ′Aτy(−τ) + 2 x′P ′0∫

−τ

G(s) y(s) ds +

+ 2 x′A′0∫

−τ

D(s) y(s) ds + 2 y′(−τ)A′τ

0∫−τ

D(s) y(s) ds +

+ 2

0∫−τ

0∫−τ

y′(s) D(s) G(ν) y(ν) ds dν +

+ 2 x′[D(0) x−D(−τ) y(−τ)−

0∫−τ

dD(s)

dsy(s) ds

]+

+x′Q(0) x−y′(−τ) Q(−τ) y(−τ)−0∫

−τ

y′(s)dQ(s)

dsy(s) ds+

+ x′0∫

−τ

[R(0, s) + R′(s, 0)

]y(s) ds−

Stability Theory 43

− y′(−τ)

0∫−τ

[R(−τ, s) + R′(s,−τ)

]y(s) ds−

−0∫

−τ

0∫−τ

y′(s)[∂R(s, ν)

∂s+

∂R(s, ν)

∂ν

]y(ν) ds dν +

+ τ x′Π(0) x−0∫

−τ


−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν +

+2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν−⎛⎝ 0∫−τ

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫−τ

y(s) ds

⎞⎠ ,

thus finally we obtain the following formula of totalderivative:

V(2.1)[x, y(·)] =

= x′[2 P ′A + 2 D(0) + Q(0) + τ Π(0)

]x +

+ x′[2 P ′Aτ − 2 D(−τ)

]y(−τ) +

+x′0∫

−τ

[2 P ′G(s)− 2

dD(s)ds

+ R(0, s) + R′(s, 0) + 2 A′D(s)]

y(s) ds+

+ y′(−τ)

0∫−τ

[2 A′τD(s)− R(−τ, s) + R′(s,−τ)

]y(s) ds +

+

0∫−τ

0∫−τ

y′(s)[2 D(s)G(ν) − ∂R(s, ν)

∂s− ∂R(s, ν)

∂ν

]y(ν) ds dν −


− y′(−τ) Q(−τ) y(−τ) −

−0∫

−τ

y′(s)[Π(s) +

dQ(s)

ds

]y(s) ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν +

+2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν−⎛⎝ 0∫−τ

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫−τ

y(s) ds

⎞⎠ ,

(2.21){x, y(·)} ∈ H .

Note, total derivative (2.21) is defined on elements ofH , but not at solutions.

The relation between total derivative (2.21) and thederivative of Lyapunov-Krasovskii functionals along solu-tions gives us the following

Theorem 2.3. Functional (2.12) has right-hand deriv-atives along solutions of system (2.1), and for any (t0, h) ∈R×H

d

dtV [xt(t0, h)] = V(2.1)[xt(t0, h)] , for t ≥ t0 . (2.22)

�

In many cases it is sufficient to consider DDE in spacesof smoother functions than H , for example, C[−τ, 0],C1[−τ, 0], Lipk[−τ, 0]. This is connected with the factthat solutions xt(t0, h) of DDE belong to these spaces whent ≥ t0 + τ . So often we can require the invariant differen-tiability of Lyapunov functionals not on the whole space

Stability Theory 45

H , but only on its subset.

Remark 2.4. The most general formula of total deriva-tive is defined as right-hand Dini-derivative along solutions[111, 112]

V +(2.1)[h] = lim

Δt→0

1

Δt

(V [xt∗+Δt(t∗, h)]− V [t, h]

). (2.23)

From the mathematical point of view the application of for-mula (2.23) in theorems of the Lyapunov functional methodis natural and allows us to prove the corresponding con-verse theorems. Nevertheless direct implementation of for-mula (2.23) is difficult because it requires, at least formally,calculating the solution xt of system (1.18).

R.Driver [49] proposed to calculate the total derivativeas

V ∗(2.1)[h] = lim

Δt→+0

1

Δt

(V [hΔt]− V [h]

), (2.24)

where h = {x, y(·)}, hΔt = {x + f(h)Δt, y(Δt)(·)},

y(Δt) =

{x + f(h)s for 0 ≤ s < Δt ,

y(s) for − τ ≤ s < 0 ,

where

f(h) = A x + Aτy(−τ) +

0∫−τ

G(s) y(s) ds .

Though the formula (2.24) does not require calculating so-lutions, nevertheless utilization of this formula is also com-plicated because of computation of right-hand Dini deriv-atives. �


2.3 Positiveness of functionals

In applications usually positive and non-negative Lyapunov-Krasovskii functionals are used. In this section we discussthese properties of functionals.

2.3.1 Definitions

Definition 2.7. Functional

V [x, y(·)] : Rn ×Q[−τ, 0) → R

is

1) positive definite on H (on C[−τ, 0]) if there exists afunction a ∈ K such that for any h ∈ H (h ∈ C[−τ, 0])the following inequality is hold

V [h] ≥ a(‖h‖H) ;

2) positive on H (on C[−τ, 0]) if

V [h] > 0

for h �= 0, h ∈ H (h ∈ C[−τ, 0]);

3) non-negative on H (on C[−τ, 0]) if for any h ∈ H(h ∈ C[−τ, 0])

V [h] ≥ 0 .

�

Note that positiveness and positive definiteness are notequivalent for functionals on H (and C[−τ, 0]), i.e. notevery positive on C[−τ, 0] functional will be also positivedefinite on C[−τ, 0] (see further Remark 2.5).

Stability Theory 47

2.3.2 Sufficient conditions of positiveness

Lyapunov-Krasovskii functional (2.12) is positive if, for ex-ample, ⎡

⎣ 1

τP D(s)

D′(s) Q(s)

⎤⎦ > 0 , (2.25)

Π(s) ≥ 0 , R(s, ν) ≥ 0 , Γ(s, ν) ≥ 0 (2.26)

for s, ν ∈ [−τ, 0]. Condition (2.25) guarantees positivenessof the sum

W1[x] + W2[x, y(·)] + W3[y(·)] ,and conditions (2.26) guarantee non-negativeness on H offunctionals W4[y(·)], W5[y(·)] and W6[y(·)], respectively.

In general case analysis of positiveness of the quadraticfunctionals is a very difficult task. It requires special in-vestigation in every concrete case.

2.3.3 Positiveness of functionals

Remember that for continuous finite dimensional functionsv(x) : Rn → R there are two equivalent definitions of pos-itive definiteness. Let us formulate them as a proposition.

Lemma 2.1. Let a function v(x) be continuous in theregion {x ∈ Rn : ‖x‖ < L}. Then the following conditionsare equivalent:

1. v(x) > 0 for 0 < ‖x‖ < L,3

2. there exists a function a ∈ K such that v(x) ≥ a(‖x‖)for ‖x‖ < L. �

3If L = ∞ then it is also supposed that lim‖x‖→∞

inf v(x) = 0.


For a continuous functional V [h] a similar propositioncan be proved in space SLk[−τ, 0] consisting of functionsz(·) : [−τ, 0] → Rn which satisfy the Lipschitz condition‖z(s1) − z(s2)‖ ≤ k|s1 − s2|, for s1, s2 ∈ [−τ, 0], whereconstant k > 0. In this space we consider the metricρ ( z(1)(·), z(2)(·) )= ‖z(1)(·) − z(2)(·)‖C . Functions from SLk[−τ, 0] arenot supposed to be differentiable, hence Lipk[−τ, 0] ⊂SLk[−τ, 0]. This space has the following properties.

Lemma 2.2.

1. SLk[−τ, 0] is the nonlinear complete metric space,

2. Sε ={z(·) ∈ SLk[−τ, 0] : ρ(z(·), 0) = ε

}(ε > 0) is

the compact set.

�

Theorem 2.5. For a continuous functional V [·]:SLk[−τ, 0] → R the following conditions are equivalent:

1. functional V is positive on SLk[−τ, 0], i.e. V [z(·)] > 0for 0 �= z(·) ∈ SLk[−τ, 0] and lim

‖z(·)‖C→∞

inf V [z(·)] �=0;

2. there exists a function a ∈ K such that V [z(·)] ≥a(‖z(·)‖

C) for z(·) ∈ SLk[−τ, 0].

Proof. It is evident that from condition 2 of the theo-rem follows validity of condition 1. Let us prove the con-verse implication. Consider the function

w(r) = minz(·)∈Sr

V [z(·)], r ≥ 0 , (2.27)

which is continuous and w(r) > 0 for r > 0. The functionalV is continuous and positive definite, and for any r > 0 thesphere Sr is compact, hence lim

r→∞w(r) > 0, and therefore

Stability Theory 49

there exists a function a ∈ K such that w(r) > a(r) forr > 0. �

Theorem 2.6. If V : SLk[−τ, 0] → R is a continuousfunctional, then there exists b ∈ K such that V [z(·)] ≤b(‖z(·)‖

C) for z(·) ∈ SLk[−τ, 0].

Proof. One can easily check that the function b(r) =max

z(·)∈Sr

V [z(·)] satisfies the terms of the theorem. �

Remark 2.5. In the space C[−τ, 0] the proposition,similar to Theorem 2.5, is not valid, because the spherein C[−τ, 0] is not a compact set. Consider, for example,

the functional V [z(·)] =

0∫−τ

z2(s)ds. Obviously V [z(·)] > 0

for z(·) �= 0. Let us fix arbitrary ε > 0 and constructa sequence {z(i)}∞i=1 ⊂ C[−τ, 0] by the rule: z(i)(s) =

εsi

τ i, −τ ≤ s ≤ 0. Calculate V [z(i)(·)] =

ε2

τ 2i

0∫−τ

s2ids =

τε2

2i + 1. Hence V [z(i)(·)] → 0 as i → ∞, meanwhile

‖z(i)(·)‖C

= max−τ≤s≤0

ε|si|τ i

= ε. �

2.4 Stability via Lyapunov-Krasovskii func-tionals

As we already mentioned, necessary and sufficient con-ditions for asymptotic stability of a linear time-invariantsystem consist in negativeness of real parts of all roots ofthe corresponding characteristic equation. However, unlikeODE, for DDE at present there are no effective methods ofdirect verification of this property of eigenvalues.


Application of Lyapunov-Krasovskii functionals allowsone to avoid these difficulties and investigate stability ofDDE without calculating eigenvalues or DDE solutions.

Though, using this approach we can obtain, as a rule,only sufficient conditions of stability, utilization of differ-ent types of Lyapunov-Krasovskii functionals enables usto obtain various forms of stability conditions in terms ofparameters of systems.

In this section we present basic theorems of Lyapunov-Krasovskii functional method for linear systems with de-lays. The results of this section are based on [111, 112, 49].

Let us define K as the set of continuous strictly increas-ing functions a(·) : [0, +∞)→ [0, +∞), a(0) = 0.

2.4.1 Stability conditions in the norm ‖ · ‖H

Theorem 2.7. If there exist quadratic Lyapunov-Krasovskiifunctional V [x, y(·)] and a function a ∈ K such that forall h = {x, y(·)} ∈ H

1. V [h] ≥ a(‖h‖H) ,

2. V(2.1)[h] ≤ 0 ,

then system (2.1) is stable. �

Theorem 2.8. If there exist quadratic Lyapunov-Krasovskii functional V [x, y(·)] and functions a, b, c ∈ Ksuch that for any h = {x, y(·)} ∈ H the following condi-tions are satisfied

1. a(‖h‖H) ≤ V [h] ≤ b(‖h‖H) ,

2. V(2.1)[h] ≤ −c(‖h‖H) ,

then system (2.1) is asymptotically stable. �

Stability Theory 51

In many cases we can construct the Lyapunov function-als, for which total derivatives are only non-positive (butnot negative definite). Nevertheless under some additionalconditions it can be sufficient for asymptotic stability ofDDE [111].

Theorem 2.9. If there exists quadratic Lyapunov-Krasovskii functional V [x, y(·)] and a function a ∈ Ksuch that for all h = {x, y(·)} ∈ H \ {0}

1. V [h] ≥ a(‖h‖H) ,

2. V(2.1)[h] < 0 ,


2.4.2 Stability conditions in the norm ‖ · ‖Theorem 2.10. If there exist quadratic Lyapunov-Krasovskii functional V [x, y(·)] and a function a ∈ Ksuch that for all {x, y(·)} ∈ H

1. V [x, y(·)] ≥ a(‖x‖) ,

2. V(2.1)[x, y(·)] ≤ 0 ,

then system (2.1) is stable. �

Theorem 2.11. If there exist quadratic Lyapunov-Krasovskii functional V [x, y(·)] and functions a, b, c ∈ Ksuch that for any h = {x, y(·)} ∈ H the following condi-tions are satisfied

1. a(‖x‖) ≤ V [x, y(·)] ≤ b(‖h‖H) ,

2. V(2.1)[x, y(·)] ≤ −c(‖x‖) ,



Remark 2.6. Taking into account that ‖h‖H ≥ ‖x‖ forany h = {x, y(·)} ∈ H , hence we can substitute the firstcondition of Theorem 11.3 by

1. V [h] ≥ a(‖x‖) .

�

Remark 2.7. Because of the smoothing of DDE solu-tions one can substitute in the above theorems the spaceH by the spaces of more smooth functions, for example,C[−τ, 0] or Lipk[−τ, 0]. �

2.4.3 Converse theorem

The following converse theorem is valid [111, 76].

Theorem 2.12. If system (2.1) is asymptotically stablethen for any positive definite n×n matrix W there exist apositive definite quadratic Lyapunov-Krasovski functionalV [x, y(·)] and a constant k > 0 such that

V [x, y(·)] ≤ k ‖h‖2H (2.28)

V(2.1)[x, y(·)] ≤ −x′W x . (2.29)

Moreover for any L > 0 there exists a constant cL suchthat

cL ‖x‖3 ≤ V [x, y(·)] (2.30)

for ‖h‖H ≤ L, h = {x, y(·)}. �

Unfortunately the theorem does not give us rules offunding parameters of Lyapunov-Krasovskii functionals,nevertheless the theorem guarantees that we can follow thisway and our attempts can be successful.

Stability Theory 53

2.4.4 Examples

Consider two examples.

Example 2.1. For equation

ax = −bx −0∫

−τ

(τ + s)y(s)ds (a > 0, b ≥ 0) (2.31)

one can consider the Lyapunov functional

V [x, y(·)] = ax2 +

0∫−τ

( 0∫s

y(u)du)2

ds.

The functional V is invariantly differentiable and its totalderivative with respect to equation (2.31) has the form4

V(2.31)[x, y(·)] = 2x(− bx−

0∫−τ

(τ + s)y(s)ds)

+2x

0∫−τ

0∫s

y(u)duds−(0∫

−τ

y(s)ds)2

= −2bx2−(0∫

−τ

y(s)ds)2

.

Thus, if b = 0 then the zero solution of (2.31) is uniformlystable, and if b > 0 then the zero solution is globally uni-formly asymptotically stable. �

Example 2.2. [111]. Let us apply invariantly differen-tiable Lyapunov functional

V [x, y(·)] =x2

2α+ μ

0∫−τ

y2(s)ds (α, μ > 0)

for investigating stability of the origin of the linear equation

x = −α x + β y(−τ) , (2.32)

4Here we also use the equality

0∫−τ

(τ + s)y(s)ds =

0∫−τ

0∫s

y(u)duds .


where α and β are constants. The total derivative of Vwith respect to (2.32) is the quadratic form of the variablesx and y(−τ)

V(2.32)[t, x, y(·)] = −x2 +β

αxy(−τ) + μx2 − μy2(−τ) .

This quadratic form is negative definite if

4(1− μ)μ >β2

α2, (2.33)

hence, if there exists μ > 0 that satisfies condition (2.33),then the zero solution of (2.32) is uniformly stable. Forμ = 0.5 the left-hand side of (2.33) achieves a maximum,and in this case inequality (2.33) takes the form β2 < α2

or |β| < α. �

2.5 Coefficient conditions of stability

In this section we present some stability conditions(in terms of system coefficients) obtained using specificLyapunov-Krasovskii functionals. More complicated con-ditions and further references one can find, for example, in[4, 107, 108, 51].

2.5.1 Linear system with discrete delay

Consider a system

x = A x + Aτy(−τ) (2.34)

where A and Aτ are constant n × n matrices. Supposethat the eigenvalues of the matrix A have negative realparts. Then there exists a symmetric matrix C such thatthe matrix

D = A′C + C A

is negative definite.

Stability Theory 55

Let us consider Lyapunov-Krasovskii functional

V [x, y(·)] = x′P x +

0∫−τ

y′(s) Q y(s) ds (2.35)

where Q is n× n constant positive definite matrix.Obviously there exist positive constants a and b such

thata ‖x‖ ≤ V [x, y(·)] ≤ b ‖h‖H .

One can easily calculate total derivative

V(2.34)[x, y(·)] = −x′D x + 2 x′P Aτ y(−τ)+

+x′Q x− y′(−τ) Q y(−τ) . (2.36)

The right-side of (2.36) is the quadratic form of the vari-ables x and y(−τ). Let us estimate this quadratic form.

Let matrices Q and (P −Q) be positive definite. Thenthere exist positive constants λ and μ such that

x′ (P −Q) x ≥ λ ‖x‖2 , (2.37)

x′Q x ≥ μ ‖x‖2 . (2.38)

Let us suppose that√λ μ− ‖P Aτ‖n×n > 0 . (2.39)

Note, if this inequality is valid then there exists a constantα ∈ (0, min{λ, μ}) such that√

(λ− α) (μ− α)− ‖P Aτ‖n×n > 0 ,

then one can estimate

V(2.34)[x, y(·)] ≤≤ −λ ‖x‖2 + 2 ‖P Aτ‖n×n ‖x‖ ‖y(−τ)‖ − μ ‖y(−τ)‖2 =

= −(λ− α) ‖x‖2 − (μ− α) ‖y(−τ)‖2 +


+ 2 ‖P Aτ‖n×n ‖x‖ ‖y(−τ)‖ −− α

(‖x‖2 + ‖y(−τ)‖2

)≤ 5

≤ −2√

(λ− α) (μ− α) ‖x‖ ‖y(−τ)‖+

+ 2 ‖P Aτ‖n×n ‖x‖ ‖y(−τ)‖ −− α

(‖x‖2 + ‖y(−τ)‖2

)≤

≤ −2(√

(λ− α) (μ− α)− ‖P Aτ‖n×n

)‖x‖ ‖y(−τ)‖ −

− α ‖x‖2 − α ‖y(−τ)‖2 ≤≤ −α ‖x‖2 .

Thus all conditions of Theorem 2.11 are satisfied and wecan formulate the following proposition.

Theorem 2.13 [76]. Let conditions (2.37) – (2.39) besatisfied, then system (2.34) is asymptotically stable. �

2.5.2 Linear system with distributed delays

Consider a system

x = A x +

0∫−τ

G(s) y(s) ds (2.40)

where A is constant n × n matrix, G(s) is n × n matrixwith continuous elements on [−τ, 0].

Consider n×n nonsingular matrix C(s) with continuouselements on [−τ, 0], and define the matix

Q(s) =

s∫−τ

C ′(ν) C(ν) dν , s ∈ [−τ, 0] . (2.41)

5Further we use inequality −(a + b) ≤ −2√

a b for a, b ≥ 0.

Stability Theory 57

Let P be a symmetric positive definite n × n matrix.Consider the Lyapunov-Krasovskii functional of the form

V [x, y(·)] = x′P x +

0∫−τ

y′(s) Q(s) y(s) ds . (2.42)

One can easily prove that there exist positive constantsa, b, c such that

a ‖x‖ ≤ V [x, y(·)] ≤ b ‖x‖+ c ‖y(·)‖τ .

The total derivative of functional (2.42) with respect tosystem (2.40) can be presented in the following form

V(2.40)[x, y(·)] =

= x′[A′ P + P A + Q(0)

]x + 2 x′ P

0∫−τ

G(s) y(s) ds−

− y′(−τ) Q(−τ) y(−τ)−0∫

−τ

y′(s)dQ(s)

dsy(s) ds =

= x′M x−y′(−τ) Q(−τ) y(−τ)−0∫

−τ

ξ{x,y(·)}(s) ξ′{x,y(·)}(s) ds

(2.43)where

M = A′P + P A + Q(0) +

+ P

⎡⎣ 0∫−τ

G(s) C−1(s) (C−1(s))′G′(s) ds

⎤⎦ P ,

ξ{x,y(·)}(s) = y′(s) C ′(s)− x′ P G(s) C−1(s) .

Taking into account that Q(−τ) = 0 and that the lastterm in (2.43) is non-negative, we obtain

V(2.40)[x, y(·)] ≤ x′M x . (2.44)


Hence we can formulate the following proposition.

Theorem 2.14 [51]. If there exist symmetric positivedefinite matrix P , symmetric negative definite matrix M ,and nonsingular matrix C(s), −τ ≤ s ≤ 0, such that

−M + A′P + P A +

0∫−τ

C(s) C ′(s) ds +

+ P

⎛⎝ 0∫−τ

G(s) C−1(s) (C−1(s))′G′(s) ds

⎞⎠ P = 0 , (2.45)


Note, if we fix some matrices M and C(s) then equation(2.45) is the classic matrix Riccati equation with respectto the matrix P .

Chapter 3

Linear quadratic control

3.1 Introduction

In this chapter we discuss a problem of designing stabilizingcontroller for linear systems with delays.

Further we will be interested mainly in investigating as-ymptotic stability of systems, so, for brevity, in the sequelthe word “stability” will be often used instead of “asymp-totic stability”1.

For linear finite-dimensional systems, linear quadraticregulator (LQR) theory plays a special role among variousapproaches because an optimal gain can be easily calcu-lated by solving an Algebraic Riccati Equation (ARE) andthe corresponding control stabilizes the closed-loop systemunder mild conditions.

For systems with delays, theoretical aspects of LQRproblem have been also well developed in different direc-tions [14, 45, 46, 52, 66, 91, 104, 114, 121, 129, 165, 179,182], and it was shown that the optimal control (which isa linear operator on a space of functions) is given by so-lutions of some specific differential equations, the so-calledgeneralized Riccati equations (GREs) [52, 165, 166]. But,unfortunately, for systems with delays the above mentioned

1I.e. in this chapter under stability we understand the asymptotic stability ofsystems.


advantages of finite-dimensional systems are not preservedbecause there are no effective methods of solving GREs.Approximate numerical methods [52, 105, 165, 166, 174] forthe system of GREs (which consists of the algebraic matrixequation, ordinary and partial differential equations) arevery complicated and their practical realization is far moredifficult than that for the corresponding algebraic Riccatiequation.

Among various papers devoted to LQR problems an ex-plicit solution was obtained in [180, 181] under some specialconditions for generalized quadratic cost functional.

However, in order to find an explicit solution of GREsit is necessary to calculate unstable poles of an open-loopsystem and to compute a set of special functions, which arestill difficult tasks.

In this chapter we describe methods of finding explicitsolutions of GREs using special choices of the parametersof the generalized quadratic functional.

The approach is based on the principles that general-ized quadratic cost functional and its coefficients are againdesign parameters.

3.2 Statement of the problem

In this chapter we consider an LQR problem for systemswith delays

x = A x + Aτ y(−τ) +

0∫−τ

G(s)y(s)ds + B u (3.1)

where A, Aτ , B are n× n, n× n, n× r constant matrices,G(s) is n × n matrix with continuous elements on [−τ, 0],x ∈ Rn and u ∈ Rr.

We consider the system (3.2) in the phase space H =Rn ×Q[−τ, 0).

Linear Quadratic Control 61

Remember that (3.1) is the conditional representationof system

x(t) = A x(t) + Aτ x(t− τ) +

0∫−τ

G(s)x(t + s)ds + B u(t) .

(3.2)We consider a state of time-delay systems as a pair

{x, y(·)} ∈ H , hence the corresponding representation of alinear state feedback control is

u(x, y(·)) = C x +

0∫−τ

D(s) y(s) ds , (3.3)

where C is r × n constant matrices, D(s) is r × n matrixwith continuous on [−τ, 0] elements.

Calculated along specific trajectory xt the closed-loopcontrol (3.3) has the presentation

u(xt) = C x(t) +

0∫−τ

D(s) x(t + s) ds . (3.4)

Closed-loop system, corresponding to system (3.1) andstate feedback control (3.3), can be easily constructed as

x = (A + B C) x + Aτ y(−τ) +

0∫−τ

(G(s) + B D(s)

)y(s) ds

(3.5)that corresponds to the conventional representation

x(t) = (A + B C) x(t) + Aτ x(t− τ) +

+

0∫−τ

(G(s) + B D(s)

)x(t + s) ds .


We investigate a problem of constructing stabilizingfeedback control (3.3) on the basis of minimization of thegeneralized quadratic cost functional

J =

∞∫0

{x′(t)Φ0 x(t) + 2 x′(t)

0∫−τ

Φ1(s) x(t + s) ds +

+

0∫−τ

0∫−τ

x′(t + s) Φ2(s, ν) x(t + ν) ds dν +

+

0∫−τ

x′(t + s) Φ3(s) x(t + s) ds +

+

0∫−τ

0∫ν

x′(t + s) Φ4(s) x(t + s) ds dν +

+ x′(t− τ) Φ5 x(t− τ) +

+ u′(t) N u(t)}

dt (3.6)

on trajectories of system (3.1).Here Φ0 and Φ5 are constant n × n matrices; Φ1(s),

Φ3(s) and Φ4(s) are n×n matrix with piece-wise continuouselements on [−τ, 0], Φ2(s, ν) is n × n matrix with piece-wise continuous elements on [−τ, 0] × [−τ, 0], N is r × rsymmetric positive definite matrix.

The state weight functional in (3.6) is the quadraticfunctional

Z[x, y(·)] = x′Φ0 x + 2x′0∫

−τ

Φ1(s) y(s) ds +

+

0∫−τ

0∫−τ

y′(s) Φ2(s, ν) y(ν) ds dν +


+

0∫−τ

y′(s) Φ3(s) y(s) ds +

+

0∫−τ

0∫ν

y′(s) Φ4(s) y(s) ds dν +

+ y′(−τ) Φ5 y(−τ) (3.7)

on space H = Rn×Q[−τ, 0), so we can write cost functional(3.6) in the compact form

J =

∞∫0

{Z[xt] + u′(t) N u(t)

}dt . (3.8)

Remark 3.1. It is noted that most papers consider thequadratic functional

J∗ =

∞∫0

{x′(t)Φ0 x(t) + u′(t) N u(t)

}dt , (3.9)

however, taking into account that the matrices Φ0, Φ1(s),Φ2(s, ν), Φ3(s), Φ4(s) and Φ5 are, generally speaking, de-sign parameters, the problem (3.1), (3.6) has more degreeof freedom. �

To the generalized LQR problem (3.1), (3.6) corre-sponds the following system of matrix generalized Riccatiequations2

P A + A′ P + D(0) + D′(0) + F (0) + τ Π(0) + Φ0 = P K P ,(3.10)

dD(s)

ds+[P K − A′

]D(s)− P ′G(s) = R(0, s) + Φ1(s) ,

(3.11)

2Derivation of generalized Riccati equations is given in Appendix.


∂R(s, ν)

∂s+

∂R(s, ν)

∂ν=

= D′(s) G(ν) + G′(s) D(ν)−D′(s) K D(ν) + Φ2(s, ν) ,(3.12)

dF (s)

ds+ Π(s) = Φ3(s) , (3.13)

dΠ(s)

ds= Φ4(s) , (3.14)

with the boundary conditions

D(−τ) = P Aτ , (3.15)

R(−τ, s) = A′τD(s) , (3.16)

F (−τ) = Φ5 , (3.17)

and the symmetry conditions

P = P ′ , R(s, ν) = R′(ν, s) ,

for −τ ≤ s ≤ 0, −τ ≤ ν ≤ 0.Here

K = B N−1 B′ . (3.18)

In the next section we show that on the basis of suitablechoices of matrices Φ0, Φ1(s), Φ2(s, ν), Φ3(s, ν), Φ4(s, ν)and Φ5 we can simplify equations (3.10) – (3.16) and findsolutions in explicit forms.

Theorem 3.1. If:

1) state weight quadratic functional (3.7) is positive def-inite on H = Rn ×Q[−τ, 0);

2) GREs (3.10) – (3.16) have a solution P , D(s), R(s, ν),F (s) and Π(s) such that the quadratic functional

W [x, y(·)] = x′P x + 2 x′0∫

−τ

D(s) y′(s) ds +


+

0∫−τ

0∫−τ

y′(s) R(s, ν) y(ν) ds dν +

+

0∫−τ

y′(s) F (s) y(s) ds +

+

0∫−τ

0∫ν

y′(s) Π(s) y(s) ds dν (3.19)

is positive definite on H = Rn ×Q[−τ, 0),

then system (3.2) is stabilizable and the feedback control

u∗(xt) = −N−1B′[P x(t) +

0∫−τ

D(s) x(t + s) ds]

(3.20)

provides the optimal solution of generalized LQR problem(3.1), (3.6) in the stabilizing class of controls and the op-timal value of the cost functional J for an initial position{x, y(·)} is given by (3.19).

Proof. To prove the theorem we show that the closed-loop system (3.5), corresponding to the control (3.20), isasymptotically stable.

Let us consider positive definite functional (3.19) asLyapunov-Krasovskii functional for closed-loop system.

Taking into account that matrices P , D(s), R(s, ν),F (s) and Π(s) satisfy the system of GREs (3.10) – (3.16),one find that total derivative of functional (3.19) with re-spect to closed-loop system (3.5) has the form

W(3.5)[x, y(·)] = −Z[x, y(·)] . (3.21)

The weight quadratic functional Z[x, y(·)] is positivedefinite, hence the functional (3.21) will be negative defi-nite on H .


Thus the closed-loop system is asymptotically stable. �

Remarks 3.2.

1) to prove theorem it is sufficient to check positive def-initeness of functionals (3.7) and (3.19) not on wholeH = Rn ×Q[−τ, 0), but only on SLk[−τ, 0];

2) the theorem is valid if instead of the positive defi-niteness of functionals (3.7) and (3.19) the followingconditions are satisfied:

Z[x, y(·)] ≥ a(‖x‖) ,

W [x, y(·)] ≥ b(‖x‖) ,

for a(·), b(·) ∈ K;

3) positiveness of quadratic functionals (3.7) and (3.19)can be verified using matrix inequality methods.

�

Remarks 3.3. Note that

1) functional Z[x, y(·)] is positive definite if, for example,⎡⎣ 1

τΦ0 Φ1(s)

Φ′1(s) Φ3(s)

⎤⎦ > 0 ,

Φ2(s, ν) ≥ 0 , Φ4(s) ≥ 0 , Φ5 ≥ 0 .

for s, ν[−τ, 0];

2) functional W [x, y(·)] is positive definite if, for exam-ple, ⎡

⎣ 1

τP D(s)

D′(s) F (s)

⎤⎦ > 0 ,

Π(s) ≥ 0 , R(s, ν) ≥ 0

for s, ν[−τ, 0]. �


3.3 Explicit solutions of generalized Ric-cati equations

Now we present an approach to finding explicit solutionsof GREs (3.10) – (3.16). The approach is based on anappropriate choice of matrices Φ0, Φ1 and Φ2 in the costfunctional (3.7) or (3.8).

3.3.1 Variant 1

Theorem 3.2. Let

1) matrix P be the solution of the matrix equation

P A + A′P + M = P K P , (3.22)

where M is a symmetric n× n matrix;

2) matrices D(s) and R(s, ν) are defined by

D(s) = e−[P K−A′](s+τ)PAτ , (3.23)

R(s, ν) =

{Q(s) D(ν) for (s, ν) ∈ Ω1 ,D′(s) Q′(ν) for (s, ν) ∈ Ω2 ,

(3.24)

where

Ω1 ={

(s, ν) ∈ [−τ, 0]× [−τ, 0] : s− ν < 0}

,

Ω2 ={

(s, ν) ∈ [−τ, 0]× [−τ, 0] : s− ν > 0}

,

andQ(s) = A′τe

[P K−A′](s+τ) , (3.25)

3) F (s) and Π(s) are n × n matrices with continuousdifferentiable elements on [−τ, 0].

Then the matrices P , D(s), R(s, ν), F (s) and Π(s) aresolutions of GREs (3.10) – (3.16) with matrices

Φ0 = M −(D(0) + D′(0)

)− F (0)− τ Π(0) ,


Φ1(s) = P ′G(s)− R(0, s) ,

Φ2(s, ν) = D′(s) K D(ν)−D′(s) G(ν)−G′(s) D(ν) ,

Φ3(s) =dF (s)

ds+ Π(s) ,

Φ4(s) =dΠ(s)

ds,

Φ5 = F (−τ) . (3.26)

Proof. The statement of the theorem can be verifiedby the direct substitution (detailed proof of the theorem isgiven in Appendix). �

Remark 3.4. From Theorem 3.2 it follows that

Φ1(s) = −D′(0) Q′(s) . (3.27)

�

3.3.2 Variant 2

Theorem 3.3. Let

1) matrix P be the solution of the exponential matrixequation (EME)

P A+A′P +e−[P K−A′] τP Aτ +A′τP e−[P K−A′]′ τ +M =

= PKP , (3.28)

where M is a symmetric n× n matrix;

2) matrices D(s) and R(s, ν) have the forms (3.23) –(3.25)

3) F (s) and Π(s) are n × n matrices with continuousdifferentiable elements on [−τ, 0].



Φ0 = M − F (0)− τ Π(0) ,

Φ1(s) = P ′G(s)− R(0, s) ,

Φ2(s, ν) = D′(s) K D(ν)−D′(s) G(ν)−G′(s) D(ν) .

Φ3(s) =dF (s)

ds+ Π(s) ,

Φ4(s) =dΠ(s)

ds,

Φ5 = F (−τ) . (3.29)


3.3.3 Variant 3

Theorem 3.4. Let

1) F (s) and Π(s) be n × n matrices with continuousdifferentiable elements on [−τ, 0], M be a symmetricn× n matrix;

2) matrix P be the solution of the Riccati matrix equa-tion

P (A+Aτ)+(A′+A′τ )P +F (0)+τ Π(0)+M = P K P ,(3.30)

3) matrices D(s) and R(s, ν) have the form

D(s) ≡ P Aτ , (3.31)

R(s, ν) ≡ A′τ P Aτ . (3.32)



Φ0 = M

Φ1(s) =(PK −A′ −A′τ

)P Aτ − P G(s),

Φ2(s, ν) = A′τP K P Aτ − A′τP G(ν)−G′(s) PAτ ,

Φ3(s) =dF (s)

ds+ Π(s) ,

Φ4(s) =dΠ(s)

ds,

Φ5 = F (−τ) . (3.33)


Remark 3.5. If we take, for instance,

Φ0 = q2K − q (A + Aτ )− q (A′ + A′τ )− F (0)− τ Π(0) ,

where q is arbitrary number, then the matrix equation(3.30) has the solution

P = q E ,

which is the positive definite matrix. �

The simple form of the solution allows us to exam-ine non-negativeness of the corresponding quadratic func-tional. The substitution of D(s) and R(s, ν) into (3.19)yields

W [x, y(·)] =

= x′P x + 2 x′0∫

−τ

P Aτ y(s) ds +


+

0∫−τ

0∫−τ

y′(s) A′τ P Aτ y(ν) ds dν =

= x′P x +(2 x′)

P

0∫−τ

Aτ y(s) ds +

+( 0∫−τ

y′(s) A′τ ds)

P( 0∫−τ

Aτ y(ν) dν)

=

=(x + Aτ

0∫−τ

y(s) ds)′

P(x + Aτ

0∫−τ

y(s) ds)

.

Hence, if matrix P is positive definite then the functionalW [x, y(·)] is non-negative on H = Rn ×Q[−τ, 0).

In case of system (3.1) only with discrete delay (i.e.G(s) ≡ 0) can we obtain sufficient conditions of positive-ness of the weight functional Z[x, y(·)] (3.7).

First of all we can see that in this case the matrix Φ2

has the form

Φ2 = A′τP B N−1B′ P Aτ ,

hence the corresponding term in the functional (3.7) canbe presented in the following form

0∫−τ

0∫−τ

y′(s) Φ2 y(ν) ds dν =

=

0∫−τ

0∫−τ

y′(s) A′τP B N−1B′ P Aτ y(ν) ds dν =

=

⎛⎝ 0∫−τ

y′(s) ds

⎞⎠′

A′τP B N−1B′ P Aτ

⎛⎝ 0∫−τ

y′(ν) dν

⎞⎠ =


=

⎛⎝B′ P Aτ

0∫−τ

y′(s) ds

⎞⎠′

N−1

⎛⎝B′ P Aτ

0∫−τ

y′(s) ds

⎞⎠ ,

(3.34)and, obviously, this term is non-negative on H , because thematrix N−1 is positive definite.

From presentations (3.33) it follows that the fifth andsixth terms of functional (3.7) are non-negative on H if

Φ4(s) =dΠ(s)

ds≥ 0 for s ∈ [−τ, 0] ,

Φ5 = F (−τ) ≥ 0 .

Note that the quadratic functional

x′Φ0 x + 2x′0∫

−τ

Φ1(s) y(s) ds +

0∫−τ

y′(s) Φ3(s) y(s) ds

in (3.7) is positive if, for example,⎡⎣ 1

τΦ0 Φ1(s)

Φ′1(s) Φ3(s)

⎤⎦ > 0 for s ∈ [−τ, 0] .

Thus we obtain the following sufficient conditions forpositiveness of the weight functional (3.7) with coefficients(3.33) :

M ≥ 0 ,

dΠ(s)

ds≥ 0 for s ∈ [−τ, 0] ,

F (−τ) ≥ 0 ,⎡⎢⎢⎣

1

τM

(P K −A′ − A′τ

)P Aτ

A′τP(P K − A′ − A′τ

)′ dΠ(s)

ds+ Π(s)

⎤⎥⎥⎦ > 0

for s ∈ [−τ, 0].


We emphasize once more that we set matrices M , F (s)and Π(s) by ourself. Also remember, that we obtainedformula (3.34) under assumption G(s) ≡ 0.

3.4 Solution of Exponential Matrix Equa-

tion

To construct explicit solutions of GREs on the basis ofthe described approach it is necessary to solve ARE (3.22)or specific EME (3.28). ARE appears in various controlproblems and methods of its solving are well-developed,including effective software realizations.

We will not discuss theoretical aspects of solvability ofEME. Probably it is connected with controllability and ob-servability of system (2.1). The aim of this section is todiscuss approximate methods of solving EME.

Approximate solutions of EME can be found on the ba-sis of general methods of solution matrix equations

F (P ) = 0 , (3.35)

where P is n× n matrix and F (P ) is given by

F (P ) ≡ P A + A′P + M − PKP +

+ e−[P K−A′] τP Aτ + A′τP e−[P K−A′]′ τ . (3.36)

In this section we describe two of such methods.

3.4.1 Stationary solution method

The method consists of the following procedure. Fix n×nmatrix P0, which is considered as the initial approximation,and solve the initial value problem⎧⎨

⎩P (t) + F (P (t)) = 0 , t > 0 ,

P (0) = P0 .(3.37)


If there exists a finite limit P∗ = limt→∞

P (t) of the solution

P (t) of problem (3.37), then we can consider the limit ma-trix P∗ as an approximate solution of (3.35).

Initial value problem (3.37) can be solved using stan-dard numerical procedures. The described stationary so-lution method is realized in Time-delay System Toolbox [99].

3.4.2 Gradient methods

To solve matrix equation (3.35) one can also use gradientmethods. Consider, for example, application of the Newtonmethod.

Denote by Γ(P ) a one-to-one operator that maps n× nmatrix P into a n2–dimensional vector P according therule:

Γ

⎛⎜⎜⎝

p11 p12 · · · p1n

p21 p22 · · · p2n

· · · · · · · · · · · ·pn1 pn2 · · · pnn

⎞⎟⎟⎠ =

= (p11, p12, · · · , p1n, p21, p22, · · · , p2n, · · · , pn1, pn2, · · · , pnn)′ .

Then we can rewrite equation (3.35) in the form

Γ(F (P )) = F (P ) = 0 . (3.38)

Obviously, a matrix P is the solution of (3.35) if andonly if the corresponding vector P is the solution of (3.38).Then one can realize the following iteration procedure:

dF (Pk)

dP

(Pk+1 − Pk

)= −F (Pk), (3.39)

wheredF (P )

dPis the Jacobean (which determinant should

be non-zero at every iteration).


3.5 Design procedure

3.5.1 Variants 1 and 2

To construct the feedback control u∗(x, y(·)) according tothe approach described above, it is necessary only to findthe matrix P which is the solution of ARE (3.22) or EME(3.28). Then, taking into account the explicit form of thematrix D(s) (3.23), we obtain the following explicit formof the feedback controller in case of Variants 1 and 2

u∗(x, y(·)) = −N−1B′[P x+

0∫−τ

e−[P K−A′](s+τ)PAτ y(s) ds].

(3.40)Stabilizing properties of this controller one can check byTheorem 3.1.

Note to prove stabilizing properties of the feedback con-trol (3.40) it is sufficient to check asymptotic stability ofthe corresponding closed-loop system

x = (A − B N−1B′P ) x + Aτ y(−τ) +

+

0∫−τ

(G(s)−B N−1B′e−[P K−A′](s+τ)PAτ

)y(s) ds . (3.41)

Stability of closed-loop system can be checked usingsome sufficient conditions.

It is necessary to note, verification of asymptotic stabil-ity of closed-loop system (3.41) with respect to all initialfunctions of H is a very laborious and difficult task.

However, using computer simulation and special func-tions of Time-delay system toolbox one can check sta-bility of system (3.41) with respect to a special classes func-tions L ⊂ H .


3.5.2 Variant 3

In this case D(s) = P Aτ , hence the explicit form of thefeedback control is

u∗(x, y(·)) = −N−1B′[P x +

0∫−τ

PAτ y(s) ds]

(3.42)

and the corresponding closed-loop system has the form

x = (A − B N−1B′P ) x + Aτ y(−τ) +

+

0∫−τ

(G(s)−B N−1B′PAτ

)y(s) ds . (3.43)

3.6 Design case studies

In this chapter we apply the proposed approach to design-ing feedback controllers for linear time-delay systems. Inall examples the simulation was realized using the softwarepackage [99].

3.6.1 Example 1

Consider the system [58]

x =

[0 1

0 0

]x +

[1 0

0 0

]y(−1) +

[1

1

]u . (3.44)

Note, the open-loop system has two roots with nonneg-ative real parts: λ1 = 0.56714 and λ2 = 0.0. To constructthe controller according to the proposed method let us takethe weighting matrices as

M =

[1 0

0 1

], N = 1 .


The matrix P , which is the solution of the correspondingARE (3.22), has the form

P =

[1 0

0 1

]

and the closed-loop control is

u0(x, y(·)) =[ −1 −1

]x+

+[ −1 −1

] 0∫−1

{e

⎡⎣ −1 −1

0 −1

⎤⎦×S [

0.3679 00 0

]y(s)

}ds .

(3.45)Using special functions of Time-delay System Toolbox [99]one can check that solutions of the closed-loop systems tendto zero (see Figure 3.1).

0 5 10 15 20 25 30 35−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t

x

x2

x1

Fig. 3.1


3.6.2 Example 2


x =

[0 1

0 0

]x +

[−0.3 −0.1

−0.2 −0.4

]y(−5) +

[0

0.333

]u .

(3.46)The open-loop system has two roots with positive realparts. Let us take the following weight matrices:

M =

[1 0

0 1

], N = 1 .

Solution P of the corresponding ARE (3.22) has the form

P =

[2.6469 3.0030

3.0030 7.9486

]

and the corresponding closed-loop control is

u0(x, y(·)) =[ −1 −2.6469

]x +

[0 −0.333

]I ,(3.47)

where

I =

0∫−1

{e

⎡⎣ 0 −0333

1 −0.8814

⎤⎦×S [

0.1053 0.1921−0.0052 0.1297

]y(s)

}ds .

Using special functions of Time-delay System Toolbox[99] one can check that solutions of the closed-loop systemstend to zero (see Figure 3.2).

3.6.3 Example 3


x =

[0 1

0 0

]x +

[0.3 0.6

0.2 0.4

]y(−5) +

[0

1

]u . (3.48)


0 5 10 15 20 25 30 35−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

t

x

x2

x1

Fig. 3.2

Open-loop system is unstable. Let us take the weightingmatrices as

M =

[1 0

0 1

], N = 1 .

The matrix P , which is the solution of the correspondingARE (3.22), has the form

P =

[1.7321 1.0000

1.0000 1.7321

]

and the closed-loop control is

u0(x, y(·)) =[ −1 −1.7321

]x +

[0 −1

]I , (3.49)

where

I =

0∫−5

{e

⎡⎣ 0 −1

1 −1.7321

⎤⎦×S [ −0.0080 −0.0159

−0.0043 −0.0086

]y(s)

}ds .


Using special functions of Time-delay System Toolbox[99] one can check that solutions of the closed-loop systemstend to zero (see Figure 3.3).

0 5 10 15 20 25 30 35−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

x

x1

x2

Fig. 3.3

3.6.4 Example 4


x = Ax + Aτy(−0.5) + Bu , (3.50)

where

A =

[0.1 11 0.1

], Aτ =

[0.2 −0.20 −0.2

],

B =

[0.1 00.1 −0.1

],

and

G(s) ≡[

0 00 0

], τ = 0.5 .


Let the weight matrices are

M =

[1 00 1

], N =

[2 00 2

].

and matrices Φ0, Φ1(s), Φ2(s, ν) has the form (3.29).To find the solution P of the corresponding EME (3.28)

using the stationary solution method it is necessary to solveon the interval [0, 10] the matrix differential equation

dP (t)

dt= P (t) A + A′P (t) + e−[P (t) K−A′] τP (t) Aτ +

+ A′τP (t) e−[P (t)K−A′]′ τ + M − P (t)KP (t) (3.51)

with the initial condition

P (0) =

[0 1020 30

].

Each component of the matrix P (t) tends to a constant.The limit matrix

P (10) =

[102.2789 87.882987.8829 69.7475

](3.52)

can be considered as the approximate solution of EME(3.28).

The corresponding closed-loop control is

u0(x, y(·)) =

[ −8.8312 −8.81904.3987 4.4203

]x +

+

[ −0.0500 −0.05000 0.0500

]I , (3.53)

where

I =

0∫−0.5

{e

⎡⎣ −0.7831 −0.3230

0.1181 −1.2239

⎤⎦×S [

10.2113 −20.323010.1265 −20.2949

]y(s)

}ds .


0 1 2 3 4 5 6 7 8 9 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

x

x1

x2

Fig. 3.4

Using functions of Time-delay System Toolbox [99] onecan check that solutions of the closed-loop systems tend tozero (see Figure 3.4).

Also using special functions of Time-delay System Tool-box one can find optimal value of the cost functionalJuopt ≈ 3, 2189.

For example, values of the cost functional correspond-ing to the scaled optimal control are: J0.8uopt ≈ 3, 3971,J1.2uopt ≈ 3, 3312.

3.6.5 Example 5: Wind tunnel model

A linearized model of the high-speed closed-air unit windtunnel was described in Chapter 1 (see (1.5).

Let us design for this model the closed-loop control usingLQR algorithms.

Let us take the weight matrices as


M =

⎡⎢⎣

1 0 0

0 1 0

0 0 1

⎤⎥⎦ , N = 1 .

Solution P of the corresponding ARE (3.22) has theform

P =

⎡⎣ 0.9820 0 0

0 1.0837 0.01150 0.0115 0.0169

⎤⎦ .

Thus the corresponding closed-loop control is

u0(x, y(·)) =[

0 −0.4142 −0.6101]x+[

0 0 −36]I ,

where

I =

0∫−τ

{e

⎡⎢⎢⎢⎣−0.5092 0 0

0 0 −50.91170 1.0000 −41.1639

⎤⎥⎥⎥⎦×S ⎡

⎣ 0 0.0495 00 0 00 0 0

⎤⎦ y(s)

}ds .

The corresponding closed-loop system has the form

x =

⎡⎢⎣−0.5092 0 0

0 0 1.0000

0 −50.9117 −41.1639

⎤⎥⎦x +

+

⎡⎢⎣

0 0.0596 0

0 0 0

0 0 0

⎤⎥⎦ y(−τ) +

⎡⎢⎣

0 0 0

0 0 0

0 0 −1296

⎤⎥⎦ I (3.54)

Using Time-delay System Toolbox one can check thatsolutions of the closed-loop systems tend to zero (see Figure3.5).


0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

t

x x2

x3

x1

Fig. 3.5

3.6.6 Example 6: Combustion stability in liquidpropellant rocket motors

A linearized version of the feed system and combustionchamber equations was described in Chapter 1 (see (1.6).

Let us design for this model the closed-loop control usingLQR algorithms.

Let us take the weight matrices as

M =

⎡⎢⎢⎢⎢⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤⎥⎥⎥⎥⎦ , N = 1 . (3.55)

Using function lqdelay we can find the matrices

C =[

0.0398 −1.1134 0.2332 −0.1198]

,


D0 =[

0 −1 0 0]

,

D1 =

⎡⎢⎢⎣−0.2 0.0398 −1 0

0 −1.1134 0 10 0.2332 −1 −10 −1.1198 1 0

⎤⎥⎥⎦ ,

D2 =

⎡⎢⎢⎣−3.3101 0 4.1376 00.1794 0 −0.2243 0−0.0180 0 0.0225 00.2386 0 −0.2983 0

⎤⎥⎥⎦ .

Thus to system (1.7) with the weight matrices (3.55) cor-responds LQR control

u0(x, y(·)) =[

0.0398 −1.1134 0.2332 −0.1198]x +

+[

0 −1 0 0]×

0∫−5

{eD1×SD2y(s)

}ds .

The corresponding closed-loop system has the form

x(t) =

⎡⎢⎢⎢⎢⎣

γ − 1 0 0 0

0.0398 −1.1134 0.2332 −1.1198

−1 0 −1 1

0 1 −1 0

⎤⎥⎥⎥⎥⎦x(t) +

+

⎡⎢⎢⎢⎢⎣−γ 0 1 0

0 0 0 0

0 0 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎦x(t− δ) +

⎡⎢⎢⎢⎢⎣

0 0 0 0

0 −1 0 0

0 0 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎦×

×0∫

−5

{eD1×SD2y(s)

}ds . (3.56)


Using functions of Toolbox one can check that solutionsof the closed-loop systems tend to zero (see Figure 3.6).

0 5 10 15 20 25 30 35 40 45 50−0.5

0

0.5

1

t

x

x1x2x3x4

Fig. 3.6

Note that for γ = 0.95 and δ = 0.87 one can find solu-tions of GREs. However, the corresponding controller doesnot stabilize the system (see Figure 3.7).


0 10 20 30 40 50 60 70 80 90 100−150

−100

−50

0

50

100

150

200

t

x

x1x2x3x4

Fig. 3.7

89

Chapter 4

Numerical methods

4.1 Introduction

In this chapter we describe an approach to constructing nu-merical methods for linear time-varying systems with de-lays

x = A(t) x + Aτ (t) y(−τ(t)) +

0∫−τ(t)

G(t, s) y(s) ds + v(t)

(4.1)with the initial conditions

x(t0) = x0 , (4.2)

x(t0 + s) = y0(s) , −τ ≤ s < 0 . (4.3)

Here A(t), Aτ (t) are n×n matrices with piece-wise continu-ous elements, G(t, s) is n×n matrix with piece-wise contin-uous elements on R × [−τ, 0], u is a given n–dimensionalvector-function, τ(t) : R → [−τ, 0] is a continuous func-tion, τ is a positive constant; {x0, y0(·)} ∈ Rl ×Q[−τ, 0).

For convenience, we will use the following notation forsystem (4.1)

x = f(t, x, y(·)) (4.4)


where

f(t, x, y(·)) ≡ A(t) x + Aτ (t) y(−τ(t)) +

+

0∫−τ(t)

G(t, s) y(s) ds + v(t) .

Note, unlike ODE, even for linear DDE there are nogeneral methods of finding solutions in explicit forms. Soelaboration of numerical algorithms is the only way to findtrajectories of the corresponding systems.

At present various specific numerical methods are con-structed for solving specific delay differential equations.Most investigations are devoted to numerical methodsfor systems with discrete delays and Volterra integro-differential equations.

An exhaustive review of papers published until 1972 onDDE numerical methods is given in [38]. Consequent de-velopment of DDE numerical analysis and the correspond-ing bibliography is reflected in [63, 8, 9, 10, 11] and thecorresponding chapters of the books [77, 72].

For specific classes of DDE there were elaborated specialcodes: [12, 35, 56, 83, 149, 156, 186].

Unfortunately, most of these algorithms are laboriuosfor practical implementation even for simple DDE initialvalue problems, because the algorithms are based on com-plicated schemes of handling the discontinuities of DDEsolutions.

In this chapter we follow the approach [65, 22, 102] toconstructing numerical DDE methods. The approach isbased on the assumption of smoothness of DDE solutions.

The distinguishing feature of the approach is that thenumerical methods for DDE are direct analogies of the cor-responding classical numerical methods of ODE theory, i.e.,if delays disappear, then the methods coincide with ODEmethods.

Numerical Methods 91

Of course, exact (analytical) solutions of DDE have, asa rule, discontinuities of derivatives which can affect thenumerical algorithms used for their approximate solving.

However

• for a specific DDE, an initial function can be approx-imated, as a rule, by a sequence of (initial) functionswhich generate smooth solutions1,

• our numerical experiments showed that described inthe book algorithms are robast with respect to discon-tinuities of derivatives of DDE solutions.

4.2 Elementary one-step methods

The aim of this section is to demonstrate the basic ideaof the general approach (to constructing numerical meth-ods) on a simple one-step numerical scheme for initial valueproblem (4.4) – (4.3).

For the sake of simplicity we consider a uniform (reg-ular) grid tn = t0 + nΔ, n = 0, 1, . . . , N , of the interval

[t0, t0 + θ](

here Δ =θ

N

); and suppose that the ratio

τ

Δ= m is a positive integer.

Our aim is to obtain on the interval [t0, θ] approxima-tions un ∈ Rl, n = 0, 1, . . . , N , to the solution x(t) of theinitial value problem (4.4) – (4.3) at points t0,. . .,tN ; thatis

un ≈ x(tn) , n = 0, 1, . . . , N .

Definition 4.1. A sequence {un}, that approximatesthe solution x(t), is called the discrete model2 of system(4.4). �

1Hence, taking into account continuous dependence of DDE solution on initialdata and the approximate character of numerical procedures, we can assume thatthe given initial function generates the smooth solution.

2Numerical model, approximate model.


4.2.1 Euler’s method

General scheme

The method is very simple but not practical. However, anunderstanding of this method builds the way for the con-struction of the more practical (but also more complicated)numerical methods for DDE.

The discrete model

u0 = x0 , (4.5)

un+1 = un + Δf(tn, un, utn(·)) (4.6)

is called Euler’s method.

Interpolation

To find at time tn the next approximation un+1 usingEuler’s scheme (4.6) it is necessary to calculate the right-part f(t, x, y(·)) of system (4.4) on the pre-history

{ui, n−m ≤ i ≤ n} (4.7)

of the discrete model. Pre-history (4.7) of the discretemodel is a finite set of vectors un−m,. . .,un, meanwhile thefunctionals f in the right part of system (4.4) is defined, ingeneral case, on functions of H . Hence, to calculate a valueof the functional f on the pre-history of the discrete modelit is necessary to make an interpolation of the approximatesolution un.

Thus under utn(·) in (4.6) it is necessary to understanda function

utn(·) ≡ {u(s) , tn − τ ≤ s < 0 } , (4.8)

constructed by the finite set of points (4.7) using an inter-polational procedure.

Note, because of the interpolational error, an order ofaccuracy of method (4.6) should also depend on interpola-tional error.


One can use a simple piece-wise constant interpolation

u(t) =

{ui, t ∈ [ti, ti+1) ,y0(t0 − t), t ∈ [t0 − τ, t0) ,

(4.9)

to construct utn(·).The method (4.5), (4.6), (4.9) is Euler’s method with

piece-wise constant interpolation of the discrete pre-history(of the model).

Convergence of Euler’s method

Let us investigate convergence of the method.

Definition 4.2. Numerical method1) converges, if ‖un − x(tn)‖ → 0 as Δ → 0 for all n =1, . . . , N ;2) has a convergence order p, if there exists a constant C

such that‖un − x(tn)‖ ≤ CΔp for all n = 1, . . . , N . �

Euler’s method (4.6) – (4.9) converges and has the con-vergence order p = 1.

Theorem 4.1. Let the solution x(t) of the initial valueproblem (4.4) – (4.3) be twice continuous differentiablefunction. Then Euler’s method (4.6) – (4.9) converges andhas the convergence order p = 1. �

The described Euler’s method with the piece-wise con-stant interpolation is the simplest of converging methods.To obtain more accurate methods it is necessary to use highorder interpolational procedures and more complicated dis-crete models. Such methods will be discussed in the nextchapter.


Now let us consider the realization of Euler’s scheme forspecific systems with constant, time-varying and distrib-uted delays.

Constant discrete delay

Consider a system with the discrete delay

x(t) = A(t) x(t) + Aτ x(t− τ) . (4.10)

If the ratioτ

Δ= m is a positive integer, then the numerical

model (4.6) has the simple form

un+1 = un + Δ[

A un + Aτ un−m

].

Note that in this case an interpolation is not necessary!However, if we use τ–incommensurable mesh of the time

interval then it is also necessary to make an interpolationfor approximation of the delay term.

Time-varying discrete delay

Consider a system with time-varying delay

x(t) = A x(t) + Aτ x(t− τ(t)) (4.11)

0 < τ(t) ≤ τ . In this case the corresponding Euler’s schemeis

un+1 = un + Δ[

A un + Aτ u(tn − τ(tn))],

where u(t) : [tn−m, tn] → Rl is an interpolation of the dis-crete values un−m, . . . , un.

Distributed delay

Consider a system with distributed delay

x(t) = A x(t) + Aτ x(t− τ) +

0∫−τ

G(s) x(t + s) ds . (4.12)


According to Euler’s scheme we calculate only discretevalues un−m, . . . , un. So in order to compute the inte-gral it is necessary, similar to time-varying case, to con-struct an interpolational function u(t) of the discrete valuesun−m, . . . , un. Then the corresponding numerical model is

un+1 = un +Δ[A un +Aτ u(tn−τ)+

0∫−τ

G(s) u(tn +s) ds].

4.2.2 Implicit methods (extrapolation)

In Euler’s method we use the presentation

x(tn+1) = x(tn) +

tn+1∫tn

f(t, x(t), xt(·)) dt , (4.13)

and approximate the integral by the formula

tn+1∫tn

f(t, x(t), xt(·)) dt ≈ Δ f(tn, x(tn), xtn(·)) .

Similar to ODE case, it seems reasonable that a moreaccurate value would be obtained if we were to approximatethe integral in (4.13) by the trapezoidal rule

tn+1∫tn

f(t, x(t), xt(·)) dt ≈

≈ Δ

2

[f(tn, x(tn), xtn(·)) + f(tn+1, x(tn+1), xtn+1(·))

],

that leads to the numerical scheme

un+1 = un +Δ

2

[f(tn, un, utn(·)) + f(tn+1, un+1, utn+1(·))

].

(4.14)


Equation (4.14) gives us only the implicit formula forun+1 (because un+1 is also involved in the right-hand sideof (4.14)), so this scheme is the implicit numerical method.

In order to use the implicit method (4.14) it is neces-sary to calculate values of the functional f(t, x, y(·)) onfunctions

utn+1(·) = {u(tn+1 + s) , −τ ≤ s < 0} . (4.15)

In case of discrete delays, i.e. (4.11), we can use aninterpolation u(t) : [tn−m, tn] → Rl in order to calculate

utn+1(·) = u(tn+1 − τ(tn+1)) (4.16)

if τ(tn+1) ≥ Δ.However, if τ(tn+1) < Δ, then, in order to calculate

(4.16), it is necessary to make an extrapolation of thepre-history utn(·) on the interval [tn, tn + Δ].

Remark 4.1. In case of distributed delays it is also nec-essary to make an extrapolation. �

This method has accuracy O(Δ2) if the second order in-terpolation is used.

4.2.3 Improved Euler’s method

One can modify implicit method (4.14) in order to obtainan explicit method.

We can predict un+1 by Euler’s formula

un+1 = un + Δ f(tn, x(tn), xtn(·))and, in order to obtain a more accurate approximation,substitute this value into the right-hand side of (4.14) in-stead of un+1

un+1 = un +Δ

2

[f(tn, un, utn(·)) +


+ f(tn+1, un + Δ f(tn, un, utn(·)), utn+1(·))]. (4.17)

This explicit scheme is the improved Euler’s method.

This method has accuracy O(Δ2) if the second orderinterpolation is used.

4.2.4 Runge-Kutta-like methods

In this section we describe for DDE numerical methodswhich are direct generalization of the classic Runge-Kuttamethods of ODE’s. Note that parameters of these methodsare the same as in ODE case, i.e., if delays disappear thenwe obtain the classic Runge-Kutta method for ODE.

Runge-Kutta-like methods of the second order

The Runge-Kutta-like method (of order 2) has the form

f(tn + a Δ, un + b Δ f(tn, un, utn), utn+aΔ) , (4.18)

where constants a and b are to be selected. For example,if we take a = b = 1

2, then we obtain the midpoint method

un+1 = un + Δ f(tn +Δ

2, un +

Δ

2f(tn, un, utn), utn+Δ

2(·)) .

(4.19)We emphasize that the coefficients of the method are

the same as in ODE case.

Runge-Kutta-like method of the fourth order

Runge-Kutta method of order 4 is the classic and one ofthe most popular numerical method for ODE, because itsrate of convergence is O(Δ4) and it is easy to code.

For DDE this method has the following form

un+1 = un +1

6Δ (h1 + 2h2 + 2h3 + h4) ,


h1 = f(tn, un, utn(·)) ,

h2 = f(tn +Δ

2, un +

Δ

2h1, utn+Δ

2(·)) ,

h3 = f(tn +Δ

2, un +

Δ

2h2, utn+Δ

2(·)) ,

h4 = f(tn + Δ, un + Δh3, utn+Δ(·)) .

The method has the fourth order of convergence (underan appropriate smoothness of solutions) if we use the pre-history interpolation utn by piece-wise cubic splines andthe continued extrapolation utn+Δ

2(·).


4.3 Interpolation and extrapolation ofthe model pre-history

4.3.1 Interpolational operators

In this section we describe methods of interpolation andextrapolation of the pre-history of the discrete model un

using functions composed by polynomials of p-th degree.

Let us consider the same partition of the time interval[t0, t0 + θ] as in the previous section. Remember, this par-tition is uniform only for the sake of simplicity.

Also remember that the pre-history {ui}n of the discretemodel {ui}N

−m at time tn is the set of m + 1 vectors:

{ui}n = {ui ∈ Rl, n−m ≤ i ≤ n} .

This set of vectors defines at time tn the future dynamicsof the discrete model.

Definition 4.3. Interpolational operator I of the dis-crete model pre-history is a mapping I : {ui}n → u(·) ∈Q[tn − τ, tn]. �

Definition 4.4. We say that an interpolational operatorI has an approximation order p at a solution x(t) if thereexist constants C1, C2 such that

‖x(t)− u(t)‖ ≤ C1 maxi≥0,n−Nτ≤i≤n

‖ui − xi‖+ C2Δp (4.20)

for all n = 0, 1, . . . , N and t ∈ [tn − τ, tn]. �

Example 4.1. The following mapping uses the piece-wise linear interpolation and is the interpolational operatorof the second order:

I : {ui}n → u(t) =


=

⎧⎨⎩((t− ti)ui+1 + (ti+1 − t)ui

) 1

Δ, t ∈ [ti, ti+1] ,

y0(t0 − t), t ∈ [t0 − τ, t0) .(4.21)

�

General interpolational operators I can be constructedusing splines of a degree p. Without loss of generality we

can suppose thatm

p= k is a natural, otherwise one can

take m divisible p. Let us divide the interval [tn − τ, tn] =[tn−m − τ, tn] by k subintervals [tni−1

, tni], i = 0, 1, . . . , k −

1, of the length pΔ in such a way that tn0 = tn, tn1 =tn−p,. . . . At every subinterval [tni−1

, tni] we construct an

interpolational polynomial Lp(t) = Lip(t) according to the

values uni−p, uni−p+1, . . .,uni:

Lip(t) =

p∑l=0

uni−l

ni∏j=ni−p; j =ni−l

t− tjtni−l − tj

. (4.22)

Then we can define the following interpolational operatorI (of the discrete pre-history)

I : {ui}n → u(t) =

{Li

p(t), tni−1≤ t < tni

, t ≥ t0,

y0(t0 − t), t ∈ [t0 − τ, t0) .

(4.23)Theorem 4.2. Let the solution x(t) of the initial value

problem (4.4) – (4.3) be (p + 1)–times continuous differen-tiable on the interval [t0 − τ, t0 + θ]. Then interpolationaloperator (4.23) has an approximation order p + 1. �

One can use other types of interpolation for DDE nu-merical methods.

4.3.2 Extrapolational operators

Some DDE numerical methods require to calculate a pre-history utn+a(·) of the discrete model for a > 0. In this


case it is necessary to use an extrapolation of the model onthe interval [tn, tn + a].

Definition 4.5. Extrapolational operator Ea ( a > 0 )of the discrete model pre-history is a mapping E : {ui}n →u(·) ∈ Q[tn, tn + aΔ]. �

Definition 4.6. We say that an extrapolational opera-tor Ea has an approximation order p at a solution x(t) ifthere exist constants C3, C4 such that

‖x(t)− u(t)‖ ≤ C3 maxn−m≤i≤n

‖ui − xi‖+ C4(Δ)p (4.24)

for all n = 0, 1, . . . , N − 1, and t ∈ [tn, tn + aΔ]. �

One of the extrapolation methods, is an extrapolationby continuity of an interpolational polynomial

E : {ui}n → u(t) = L0p(t), t ∈ [tn, tn + aΔ] , (4.25)

over the right side of the point tn; here L0p(t) is the interpo-

lational polynomial of a degree p constructed by the valuesuj at the interval [tn−p, tn]:

L0p(t) =

p∑l=0

un−l

n∏j=n−p;j =n−l

t− tjtn−l − tj

.

Definition 4.7. An extrapolation constructed by:- spline interpolation on the interval [tn, tn − τ ],- continuation of the last polynomial on [tn, tn + Δ],is called an extrapolation by continuation or continued ex-trapolation. �

Theorem 4.3. Let a solution x(t) of initial value prob-lem (4.4) – (4.3) be (p + 1)-times continuous differentiableon [t0− τ, t0 + θ]. Then the continued extrapolation opera-tor, corresponding to an interpolational spline of a degreep, has an approximation order of the degree p + 1. �


4.3.3 Interpolation-Extrapolation operator

In some cases it is convenient to unify interpolational op-erator and extrapolation operator into the one operator ofinterpolation-extrapolation.

Definition 4.8. Interpolation-extrapolation operatorIE of the pre-history of a discrete model is a mapping

IE : {ui}n → u(·) ∈ Q[tn − τ, tn + aΔ] ,

a > 0 is a constant. �

Definition 4.9. An interpolation-extrapolation opera-tor IE has an approximation order p at a solution x(t) ifthere exist constants C5, C6 such that

‖x(t)− u(t)‖ ≤ C5 maxn−m≤i≤n

‖ui − xi‖+ C6(Δ)p (4.26)

for all n = 0, 1, . . . , N − 1, and t ∈ [tn − τ, tn + aΔ]. �

Definition 4.10. An operator IE is consistent if

u(ti) = ui , i = n−m, . . . , n .

�

Definition 4.11. An operator IE satisfies the Lipschitzcondition if there exists a constant LI such that for any

discrete pre-histories {u(1)i }n and {u(2)

i }n

max[tn−τ≤t≤tn+aΔ

‖u(1)(t)− u(2)(t)‖ ≤ LI maxn−m≤i≤n

‖u(1)i − u

(2)i ‖,

where u(1)(·) = IE({u(1)

i }n

), u(2)(·) = IE

({u(2)

i }n

). �

The methods of interpolation and extrapolation de-scribed in this section are consistent and satisfy the Lip-schitz condition.


4.4 Explicit Runge-Kutta-like methods

Let some interpolation operator I and extrapolation oper-ator E be fixed.

Explicit k–stage3 Runge-Kutta-like method (further weuse the abbreviation ERK) with the interpolation I andthe extrapolation E is the numerical model

u0 = x0 ; (4.27)

un+1 = un + Δk∑

i=1

σi hi(un, utn(·)), n = 1, . . . , N − 1 ,

(4.28)

h1(un, utn(·)) = f(tn, un, utn(·)) , (4.29)

hi(un, utn(·)) =

= f(tn + aiΔ, un + Δi−1∑j=1

bijhj(un, utn(·)), utn+aiΔ(·)) .

(4.30)The pre-history of the discrete model is defined as

ut(s) =

⎧⎪⎨⎪⎩

y0(t + s− t0) for t + s < t0,

I({ui}n) for tn − τ ≤ t + s < tn,

E({ui}n) for tn ≤ t + s ≤ tn + aΔ,(4.31)

a = max1≤i≤k

|ai| .

Numbers ai, σi, bij are called the coefficients of themethod. We denote σ = max

1≤i≤k|σi|, b = max

1≤i≤k; 1≤j≤k−1|bij|.

Let us investigate a convergence order (in the sense ofDefinition 4.2) of ERK-like methods.

3k is a natural number.


Definition 4.12. Residual ψ(tn) of ERK-like methodis the function

ψ(tn) =xn+1 − xn

Δ−

k∑i=1

σi hi(xn, xtn(·)) .

�

Note that a residual is defined on an exact solution x(t)and does not depend on an interpolation and an extrapo-lation.

Definition 4.13. A residual ψ(tn) has an order p ifthere exists a constant C such that ‖ψ(tn)‖ ≤ CΔp for alln = 0, 1, . . . , N − 1. �

Theorem 4.4. Let numerical method (4.27) – (4.31)have1) an approximation order p1 > 0,2) error of pre-history interpolation of an order p2 > 0,3) error of pre-history extrapolation of an order p3 > 0.Then the method converges and has the convergence orderp = min {p1, p2, p3}. �

4.5 Approximation orders of ERK-like

methods

For ODE an approximation order of an explicit numericalRunge-Kutta method is defined using the expansion of anexact solution and a right part of ODE into the Taylor se-ries.

Example 4.2. It is known that for ODE

x = f(t, x)


the improved Euler method

un+1 = un +Δ

2

(f(tn, un) + f(tn + Δ, un + Δf(tn, un))

)has the second approximation order at a sufficiently smoothsolution. Consider the procedure of estimation of the ap-proximation order of this method. The residual of themethod is

ψ(tn) =xn+1 − xn

Δ− 1

2

(f(tn, xn)+f(tn, xn +Δf(tn, xn))

).

Expanding an exact solution x(t) into Taylor’s series weobtain

xn+1 = xn + x(tn)Δ + x(tn)Δ2

2+ O(Δ3) =

= xn + f(tn, xn)Δ +Δ2

2

[∂f(tn, xn)

∂t+

∂f ′(tn, xn)

∂xf(tn, xn)

]+ O(Δ3) .

Also we have

f(tn + Δ, xn + Δf(tn, xn)) =

= f(tn, xn)+

[∂f(tn, xn)

∂t+

∂f ′(tn, xn)

∂xf(tn, xn)

]Δ+O(Δ2) .

Substituting these formulas into the residual we obtainψ(tn) = O(Δ2). �

For DDE an approximation order of a numerical methodalso can be found using expansion of a solution and a rightpart of DDE into Taylor’s series. However, in this case it isnecessary to use the techniques of the i–smooth analysis.

We emphasize that coefficients of Taylor’s series expan-sion of a solution and a right part of DDE are the same asfor ODE. Thus the following proposition is valid.

Theorem 4.5. If an ERK-method for ODE has an ap-proximation order p then an ERK-like method for DDE


with the same coefficients also has an approximation or-der p. �

This theorem together with Theorem 4.4 (on a conver-gence order) allow us to construct for DDE analogies of allknown ERK-methods of ODE theory. Of course, in DDEcase it is necessary to use the suitable operators of inter-polation and extrapolation.

For example, the improved Euler method for DDE (withthe same coefficients as in Example 4.2) with piece-wiselinear interpolation (4.21) and extrapolation (4.25) has thesecond convergence order.

The 4–stage ERK-like method for DDE has the follow-ing form

un+1 = un +1

6Δ (h1 + 2h2 + 2h3 + h4) ,

h1 = f(tn, un, utn(·)) ,

h2 = f(tn +Δ

2, un +

Δ

2h1, utn+Δ

2(·)) ,

h3 = f(tn +Δ

2, un +

Δ

2h2, utn+Δ

2(·)) ,

h4 = f(tn + Δ, un + Δh3, utn+Δ(·)) .

This method has the fourth order of convergence (underan appropriate smoothness of solutions) if we use the pre-history interpolation by piece-wise cubic splines and thecontinued extrapolation.

For an approximation order p ≥ 5 there is no p-stageERK-methods; this fact is called the Butcher barriers [72].Further we describe 6-stage ERK-method of order p = 5 –the so-called Runge-Kutta-Fehlberg method.


4.6 Automatic step size control

4.6.1 Richardson extrapolation

In case of DDE the Richardson extrapolation can be ob-tained in the same way as for ODE. This procedure al-lows us to derive a practical error estimate of a numericalmethod.

Consider for the initial value problem (4.4) – (4.3) anumerical method of an order p. Fix Δ > 0 and calculatetwo values u1 and u2 of the corresponding numerical model.Denote x1 = x(t0 + Δ) and x2 = x(t0 + 2Δ), then

ε1 = x1 − u1 = CΔp+1 + O(Δp+2) ,

ε2 = x2 − u2 =

= CΔp+1 + CΔp+1(1 + O(Δ)) + O(Δp+2) =

= 2CΔp+1 + O(Δp+2) . (4.32)

Factor 2 arises in ε2 because it consists of the transferrederror of the first step and the local error of the second step.

Let w be the value of the numerical model correspondingto one step of the double length 2Δ. Then

x2 − w = C(2Δ)p+1 + O(Δp+2) . (4.33)

From (4.32) and (4.33) we obtain

ε2 =u2 − w

2p − 1+ O(Δp+2) . (4.34)

Hence the value

u2 = u2 +u2 − w

2p − 1

approximates x2 = x(t0 + 2Δ) with the order p + 1.This procedure is called Richardson extrapolation and

allows one to elaborate a class of extrapolational methodsfor ODE, among which the most powerful is, apparently,the Gragg-Bulirsch-Stoer algorithm [72].


4.6.2 Automatic step size control

On the basis of estimate (4.34) one can organize a proce-dure of an automatic step size control that guarantees agiven accuracy tol. Below we describe the correspondingalgorithm using notation err for the error.

Let Δold be an initial value of the step. We calculate twovalues u1 and u2 of the discrete model corresponding to thisstep, and the value w of the discrete model correspondingto the double step 2Δold. Calculate the error

err =1

2p − 1max

i=1,...,l

|u2,i − wi|di

,

where the index i denotes the corresponding coordinate ofthe vectors, di is a scale factor. If di = 1 then we have anabsolute error, if di = |u2,i| then we have a relative error.One can use other norms and scales.

From relations

err = C(2Δold)p+1 ,

tol = C(2Δnew)p+1

we obtain the formula for a new step size

Δnew =

(tol

err

) 1p+1

Δold .

There are possible two variants:1) Δnew < Δold, then we accept the new step size Δnew;2) Δnew > Δold, then we accept two previous model values

u1 and u2, and to calculate u3 we use Δold, or even canit enlarge.

For practical realization of the algorithm for ODE thefollowing more complicated procedure

Δnew = min

{facmax,max

{facmin, fac

(tol

err

) 1p+1}}

Δold


is usually used. It allows one to avoid big increasing ordecreasing of a step size. In many programs fac = 0.8,facmax ∈ [1.5, 5].

4.6.3 Embedded formulas

In the previous subsection we described the algorithm of astep size control on the basis of one numerical method fortwo different step sizes Δ and 2Δ.

However, to obtain an error estimate and to organize anautomatic step size control procedure one can also use val-ues of two numerical models of different orders with respectto one step size.

This approach is especially effective if coefficients ai,bij of Butcher’s tableau of the lower-order method coincidewith the part of the coefficients of the higher-order method,because, in this case, for the high order method one canuse some of the already calculated values of the low ordermethod. Such methods are called embedded methods.

A method of an order p

un+1 = un + Δ

k∑i=1

σihi(utn(·))

is considered as the basic method, and a method of theorder p + 1

un+1 = un + Δk∑

i=1

σihi(utn(·))

is used for estimation of an error.

An example of the embedded methods is the pair ofimproved Euler method and Runge-Kutta method of the

third order0

1 1

1

2

1

4

1

4

un+11

2

1

20

un+11

6

1

6

4

6

This method is called the Runge-Kutta-Fehlberg methodof the order 2–3 (RKF 2(3)) (for DDE it is necessary to usethe discrete model pre-history interpolation and extrapo-lation of the second order).

More accurate is Runge-Kutta-Fehlberg method of theorder 4 – 5 (RKF 4(5))

0

1

4

1

43

8

3

32

9

3212

13

1932

2197−7200

2197

7296

2197

1439

216−8

3680

513− 845

41041

2− 8

272 −3544

2565

1859

4104−11

40

un+125

2160

1408

2565

2197

4104−1

50

un+116

1350

6656

12825

28561

56430− 9

50

2

55

This method is usually used in most of the softwarepackages for ODE (in DDE case it is necessary to use aninterpolation-extrapolation operator of the fourth order).

111

Chapter 5

Appendix

5.1 i-Smooth calculus of functionals

In functional

V [x, y(·)] : Rn ×Q[−τ, 0) → R (5.1)

x is the finite dimensional variable, so we can calculate the

gradient∂V

∂x(of course, if these derivatives exist).

In this section we describe basic constructions of the in-variant derivative of a functional with respect to the func-tional variable y(·).

5.1.1 Invariant derivative of functionals

In the sequel, for {x, y(·)} ∈ H and Δ > 0 we denote byEΔ[x, y(·)] the set of functions Y (·) : [−τ, Δ] → Rn suchthat:

1. Y (0) = x ,

2. Y (s) = y(s) , −τ ≤ s < 0 ,

3. Y (·) is continuous on [0, Δ] .


That is, EΔ[x, y(·)] is the set of all continuous contin-uations of {x, y(·)} on the interval [0, Δ]. Also we let

E[h] =⋃Δ>0

EΔ[h].

For functional (5.1) and a function Y (·) ∈ E[h] we canconstruct the function

ψY(ξ) = V [x, yξ(·)] , (5.2)

where yξ(·) = {Y (ξ + s),−τ ≤ s < 0} ∈ Q[−τ, 0) andξ ∈ [0, Δ]. Note, function (5.2) and the interval [0, Δ] de-pend on the choice of Y (·) ∈ E[h].

Definition 5.1. Functional (5.1) has at point p ={x, y(·)} ∈ Rn × Q[−τ, 0) the invariant derivative (i–derivative) ∂yV [x, y(·)] with respect to y(·), if for anyY (·) ∈ E[x, y(·)] the corresponding function (5.2) has at

zero right-hand derivativedψ

Y(0)

dξinvariant with respect to

Y (·) ∈ E[x, y(·)]1. And in this case we set

∂yV [p] =dψ

Y(0)

dξ.

�

Remark 5.1. Existence of the invariant derivative de-pends on local properties of function (5.2) in the rightneighborhood of zero, so in Definition A.1 we can substi-tute the set E[x, y(·)] by EΔ[x, y(·)] for some Δ > 0. �

Example 5.1. Let in the functional

V [y(·)] =

0∫−τ

β[y(s)]ds (5.3)

1I.e. the valuedψY (0)

dξis the same for all Y (·) ∈ E[x, y(·)].

Appendix 113

β : Rn → R is a continuous function. We empha-size that we calculate the invariant derivative at pointh = {x, y(·)} ∈ Rn×Q[−τ, 0) (containing x) though func-tional (5.3) does not depend on x. Let Y (·) be an arbitraryfunction of E[x, y(·)], then (5.2) has the form

ψY(ξ) = V [yξ(·)] =

0∫−τ

β[Y (ξ + s)]ds =

ξ∫−τ+ξ

β[Y (s)]ds.

Calculating the derivativedψ

Y(0)

dξand taking into account

that Y (0) = x, Y (−τ) = y(−τ) we obtain

dψY(0)

dξ=

d

dξ

( ξ∫−τ+ξ

β[Y (s)]ds)

ξ=+0=

= β[Y (0)]− β[Y (−τ)] = β[x]− β[y(−τ)].

Thusdψ

Y(0)

dξ= β[x] − β[y(−τ)] is invariant with re-

spect to Y (·) ∈ E[x, y(·)] and depends only on {x, y(·)}.Hence functional (5.3) has at every point h = {x, y(·)} ∈Rn×Q[−τ, 0) the invariant derivative ∂yV [x, y(·)] = β[x]−β[y(−τ)]. �

Let us emphasize once more, though functional (5.3)depends only on y(·) nevertheless its invariant derivative∂yV [x, y(·)] is defined on pairs {x, y(·)} ∈ H . It means,for calculating invariant derivatives of regular functionalsthe very important role play “boundary values” of “testfunctions” {x, y(·)}. For this reason, for example, func-tional (5.3) does not have invariant derivatives on func-tions y(·) ∈ L2[−τ, 0) though functional (5.3) is definedon L2[−τ, 0) (if the integral in the right-hand side of(5.3) is the Lebesgue integral). The matter is, functionsy(·) ∈ L2[−τ, 0) are not defined at separate points2, so,

2These functions are not defined on sets of measure zero.


generally speaking, one value β[y(−τ)] is not also defined.However, if a function y(·) ∈ L2[−τ, 0) is continuous fromthe right at the point s = −τ , then for (5.3) we can cal-culate at point {x, y(·)} ∈ Rn × L2[−τ, 0) the invariantderivative ∂yV = β[x]− β[y(−τ)].

Singular functionals (3.7), (3.8) also have invariantderivatives. However, these derivatives are defined onlyfor sufficiently smooth functions.

Example 5.2. Let in functional (3.7) the function P [z]is continuous differentiable and a function y(·) ∈ Q[−τ, 0)has right-hand side derivative at point s = −τ . Then (3.7)has at y(·) the invariant derivative

∂yV [y(·)] =∂P [y(−τ)]

∂zy(−τ) .

Indeed, to calculate the invariant derivative we should con-struct the function (5.2)

ψY(ξ) = V [yξ(·)] = P [y(ξ − τ)] , ξ ∈ [0, Δ) .

Obviously, ψY(ξ) has right-hand side derivative at ξ = 0

only if the function y(s),−τ ≤ s < 0, has right-hand sidederivative at point s = −τ , and in this case

∂yV [y(·)] =dψ

Y(0)

dξ=

∂P [y(−τ)]

∂zy(−τ) .

�

Remark 5.2. For calculating of the invariant derivativeof singular functional (3.7) we did not use continuationsY (·) ∈ E[x, y(·)] of the function y(·). �

In Definition 5.1 we introduced the notion of the in-variant derivatives with respect to y(·). Now for func-tional (5.1) we give a general definition of its derivativeswith respect to x and y(·).

Appendix 115

Let p = {x, y(·)} ∈ Rn×Q[−τ, 0) and Y (·) ∈ E[x, y(·)],then we can construct the function

ψY(z, ξ) = V [x + z, yξ(·)] , (5.4)

z ∈ Rn, ξ ∈ [0, Δ], yξ(·) = {Y (ξ + s),−τ ≤ s < 0}.

Definition 5.2. Functional (5.1) has at point

p = {x, y(·)} ∈ Rn × Q[−τ, 0) gradient∂V [p]

∂xand par-

tial invariant derivative ∂yV [p], if for any Y (·) ∈ E[x, y(·)]the function (5.4) has at zero gradient

∂ψY(0)

∂zand right-

hand side derivativedψ

Y(0)

dζ, invariant with respect to

Y (·) ∈ E[x, y(·)]. And in this case we set

∂V [p]

∂x=

∂ψY(0)

∂z, ∂yV [p] =

dψY(0)

dξ.

�

Consider some rules and formulas which allow to cal-culate invariant derivatives of different functionals withoutusing the definition. For invariant derivatives basic rulesof differential calculus of finite-dimensional functions arevalid.

If functionals V [x, y(·)], W [x, y(·)] : Rn × Q[−τ, 0) →R have at point h = {x, y(·)} ∈ H invariant derivatives∂yV [x, y(·)] and ∂yW [x, y(·)] then the sum, difference andthe product of these functionals have invariant derivativesat point h and

∂y

(V [x, y(·)] + W [x, y(·)]

)= ∂yV [x, y(·)] + ∂yW [x, y(·)] ,

∂y

(V [x, y(·)]−W [x, y(·)]

)= ∂yV [x, y(·)]− ∂yW [x, y(·)] ,

∂y

(V [x, y(·)] ·W [x, y(·)]

)= ∂yV [x, y(·)] ·W [x, y(·)] ++ V [x, y(·)] · ∂yW [x, y(·)] .


Moreover, if W [x, y(·)] �= 0 then

∂y

(V [x, y(·)]W [x, y(·)]

)=

=∂yV [x, y(·)] ·W [x, y(·)]− V [x, y(·)] · ∂yW [x, y(·)]

W 2[x, y(·)] .

5.1.2 Examples

Two examples of calculating the invariant derivatives offunctionals defined on Q[−τ, 0) we discussed in the previ-ous section. In this subsection we calculate invariant deriv-atives of more complicated functionals.

Example 5.3. Let in the functional

V [y(·)] =

0∫−τ

α(

0∫ν

β[y(s)] ds )dν (5.5)

α : R → R is a continuous differentiable function,β : Rn → R is a continuous function. The integral inthe right-hand side of (5.5) does not depend on x, so

∂V [p]

∂x= 0 .

In order to calculate the invariant derivative with re-spect to y(·) let us fix an arbitrary Y (·) ∈ E[x, y(·)] andconsider

ψY(ξ) =

0∫−τ

α( 0∫

ν

β[Y (ξ + s)]ds)dν =

=

0∫−τ

α( ξ∫ν+ξ

β[Y (s)]ds)dν .

Appendix 117

One can calculate

dψY(0)

dξ=

d

dξ

⎛⎝ 0∫−τ

α( ξ∫ν+ξ

β[Y (s)]ds)

dν

⎞⎠

ξ=+0

=

= β[Y (0)]

0∫−τ

α( 0∫

ν

β[Y (s)]ds)

dν −

−0∫

−τ

α( 0∫

ν

β[Y (s)]ds)

β[Y (ν)]dν =

= β[Y (0)]

0∫−τ

α( 0∫

ν

β[Y (s)]ds)

dν +

+

0∫−τ

α( 0∫

ν

β[Y (s)]ds)

d[ 0∫

ν

β[Y (s)] ds]

=

= β[Y (0)]

0∫−τ

α( 0∫

ν

β[Y (s)]ds)

dν +

+

0∫−τ

d[α( 0∫

ν

β[Y (s)]ds)]

=

= β[Y (0)]

0∫−τ

α( 0∫

ν

β[Y (s)]ds)dν +

+ α(0)− α( 0∫

β[Y (s)]ds)

.


Taking into account that Y (0) = x and Y (s) = y(s),−τ ≤s < 0, we obtain

∂yV [p] = β[x]

0∫−τ

α( 0∫

ν

β[y(s)]ds)

dν +

+ α(0)− α( 0∫−τ

β[y(s)]ds)

.

�

Example 5.4. Suppose in the functional

V [x, y(·)] =

0∫−τ

ω[x, s, y(s)] ds (5.6)

ω : Rn × [−τ, 0] × Rn → R is a continuous differentiablefunction. The corresponding function (5.4) has the form

ψY(z, ξ) =

0∫−τ

ω[x + z, s, Y (ξ + s)] ds ,

where Y (·) ∈ E[x, y(·)].Obviously

∂V [p]

∂x=

0∫−τ

∂ω[x, s, y(s)]

∂xds

(it is necessary to note, we can obtain this partial derivativeby direct differentiating of (5.6) with respect to x).

One can represent the function ψY(z, ξ) as

ψY(z, ξ) =

ξ∫−τ+ξ

ω[x + z, s− ξ, Y (s)] ds ,

Appendix 119

then

dψY(0, 0)

dξ=

=d

dξ

⎛⎝ ξ∫−τ+ξ

ω[x + z, s− ξ, Y (s)] ds

⎞⎠

ξ=+0

=

= ω[x, 0, Y (0)]− ω[x,−τ, Y (−τ)]−0∫

−τ

∂ω[x, s, Y (s)]

∂sds ,

where∂ω

∂sis the derivative with respect to the second vari-

able. Taking into account that Y (0) = x and Y (s) = y(s),−τ ≤ s < 0, we obtain the following formula of the invari-ant derivative

∂yV [x, y(·)] = ω[x, 0, x]−ω[x,−τ, y(−τ)]−0∫

−τ

∂ω[x, s, y(s)]

∂sds .

�

Example 5.5. Consider a functional

V [x, y(·)] =

0∫−τ

0∫ν

ω[x, s, y(s)] ds dν , (5.7)

ω : Rn × [−τ, 0] × Rn → R is a continuous differentiablefunction. For this functional the corresponding function(5.4) has the form

ψY(z, ξ) =

0∫−τ

0∫ν

ω[x + z, s, Y (ξ + s)] ds dν ,

Y (·) ∈ E[x, y(·)].


One can easily calculate

∂V [p]

∂x=

0∫−τ

0∫ν

∂ω[x, s, y(s)]

∂xds dν .

One can represent the function ψY(z, ξ) as

ψY(z, ξ) =

0∫−τ

ξ∫ν+ξ

ω[x + z, s− ξ, Y (s)] ds dν ,

then

dψY(0, 0)

dξ=

d

dξ

⎛⎝ 0∫−τ

ξ∫ν+ξ

ω[x + z, s− ξ, Y (s)] ds dν

⎞⎠

ξ=+0

=

= τ ω[x, 0, Y (0)]−0∫

−τ

ω[x, s, Y (s)] ds−

−0∫

−τ

0∫ν

∂ω[x, s, Y (s)]

∂sds dν ,

where∂ω

∂sis the derivative with respect to the second vari-

able. Taking into account that Y (0) = x and Y (s) = y(s),−τ ≤ s < 0, we obtain the following formula of the invari-ant derivative

∂yV [x, y(·)] = τ ω[x, 0, x]−0∫

−τ

ω[x, s, y(s)] ds−

−0∫

−τ

0∫ν

∂ω[x, s, y(s)]

∂sds dν .

�

Appendix 121

Example 5.6. Consider a functional

V [x, y(·)] =

0∫−τ

⎡⎣⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠⎤⎦ dν ,

(5.8)Γ is n× n symmetric constant matrix.

The corresponding function (5.4) has the form

ψY(ξ) =

0∫−τ

⎡⎣⎛⎝ 0∫

ν

Y (s + ξ) ds

⎞⎠′

Γ

⎛⎝ 0∫

ν

Y (s + ξ) ds

⎞⎠⎤⎦ dν ,

Y (·) ∈ E[x, y(·)].One can represent the function ψ

Y(ξ) as

ψY(ξ) =

0∫−τ

⎡⎢⎣⎛⎝ ξ∫

ν+ξ

Y (s) ds

⎞⎠′

Γ

⎛⎝ ξ∫

ν+ξ

Y (s) ds

⎞⎠⎤⎥⎦ dν ,

thendψ

Y(0)

dξ=

=d

dξ

⎛⎜⎝

0∫−τ

⎡⎢⎣⎛⎝ ξ∫

ν+ξ

Y (s) ds

⎞⎠′

Γ

⎛⎝ ξ∫

ν+ξ

Y (s) ds

⎞⎠⎤⎥⎦ dν

⎞⎟⎠

ξ=+0

=

=

0∫−τ

⎡⎣(Y (0)− Y (ν)

)′Γ

⎛⎝ 0∫

ν

Y (s) ds

⎞⎠⎤⎦ dν +

+

0∫−τ

⎡⎣⎛⎝ 0∫

ν

Y (s) ds

⎞⎠′

Γ(Y (0)− Y (ν)

)⎤⎦ dν ,


hence∂yV [x, y(·)] =

=

0∫−τ

⎡⎣(x− y(ν)

)′Γ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠⎤⎦ dν +

+

0∫−τ

⎡⎣⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ(x− y(ν)

)⎤⎦ dν =

= 2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν − 2

0∫−τ

y′(ν) Γ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠ dν =

= 2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν − 2

0∫−τ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ y(ν) dν =

= 2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν +

+ 2

0∫−τ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ d

⎡⎣ 0∫

ν

y(s) ds

⎤⎦ =

= 2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν +

+

0∫−τ

d

⎡⎣⎛⎝ 0∫

ν

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫

ν

y(s) ds

⎞⎠⎤⎦ =

= 2 x′ Γ

0∫−τ

0∫ν

y(s) ds dν −

−⎛⎝ 0∫−τ

y(s) ds

⎞⎠′

Γ

⎛⎝ 0∫−τ

y(s) ds

⎞⎠ .

�

Appendix 123

Example 5.7. Let us calculate the invariant derivativeof the functional

V [x, y(·)] =

0∫−τ

0∫−τ

γ[x; s, y(s); u, y(u)] ds du .

For a function Y (·) ∈ E[x, y(·)] we construct

ψY(z, ξ) =

0∫−τ

0∫−τ

γ[x + z; s, Y (s + ξ); u, Y (u + ξ)] ds du ,

thendψ

Y(0, 0)

dξ=

=d

dξ

⎛⎜⎝

ξ∫−τ+ξ

ξ∫−τ+ξ

γ[x; s− ξ, Y (s); u − ξ, Y (u)] ds du

⎞⎟⎠

ξ=+0

=

=

0∫−τ

γ[x; 0, Y (0); u, Y (u)] du−

−0∫

−τ

γ[x;−τ, Y (−τ); u, Y (u)] du +

+

0∫−τ

γ[x; s, Y (s); 0, Y (0)] ds−

−0∫

−τ

γ[x; s, Y (s);−τ, Y (−τ)] ds−

−0∫

−τ

0∫−τ

∂γ[x; s, Y (s); u, Y (u)]

∂sds du−


−0∫

−τ

0∫−τ

∂γ[x; s, Y (s); u, Y (u)]

∂uds du ,

hence∂yV [x, y(·)] =

=

0∫−τ

γ[x; 0, x; u, y(u)] du−0∫

−τ

γ[x;−τ, y(−τ); u, y(u)] du+

+

0∫−τ

γ[x; s, y(s); 0, x] ds−0∫

−τ

γ[x; s, y(s);−τ, y(−τ)] ds−

−0∫

−τ

0∫−τ

∂γ[x; s, y(s); u, y(u)]

∂sds du−

−0∫

−τ

0∫−τ

∂γ[x; s, y(s); u, y(u)]

∂uds du .

Also one can easily check that

∂V [x, y(·)]∂x

=

0∫−τ

0∫−τ

∂γ[x; s, y(s); u, y(u)]

∂xds du .

�

5.2 Derivation of generalized Riccati equa-

tions

In this section we give deduction of GREs (3.10) – (3.16).Let us denote by W [x, y(·)] the optimal value of the

cost functional for the problem (3.1) – (3.6) at a position{x, y(·)} ∈ H . Let us assume that the functional W [x, y(·)]

Appendix 125

is invariantly differentiable at this position, then we canconstruct the function

α(u) =∂W ′[x, y(·)]

∂x

[A x+Aτy(−τ)+

0∫−τ

G(s) y(s)ds+B u]+

+ ∂yW [x, y(·)] + Z[x, y(·)] + u′N u . (5.9)

Optimal control u∗(x, y(·)) should minimize the functionα(u) and, moreover, α(u∗(x, y(·))) = 0. The function α(u)is a quadratic function with respect to u ∈ Rr, so the valueu∗ minimizing the function α(u) one can find using relation

∂α(u)

∂u= 0 , (5.10)

because∂2α(u)

∂u2= N > 0 .

From (5.10) it follows

∂α(u)

∂u= 2Nu + B′ ∂W [x, y(·)]

∂x= 0 ,

hence

u∗(x, y(·)) = −1

2N−1B′ ∂W [x, y(·)]

∂x. (5.11)

Substituting (5.11) into (5.9) we obtain

α(u∗(x, y(·)) =∂W ′[x, y(·)]

∂x

[Ax + Aτy(−τ) +

+

0∫−τ

G(s)y(s)ds− 1

2BN−1B′ ∂W [x, y(·)]

∂x

]+∂yW [x, y(·)]+

+1

4

∂W ′[x, y(·)]∂x

B N−1 B′ ∂W [x, y(·)]∂x

+Z[x, y(·)] . (5.12)


Let us suppose that the optimal value of the cost func-tional has the quadratic form

W [x, y(·)] = x′Px + 2 x′0∫

−τ

D(s)y(s)ds +

+

0∫−τ

0∫−τ

y′(s)R(s, ν)y(ν)ds dν +

+

0∫−τ

y′(s) F (s) y(s) ds +

+

0∫−τ

0∫ν

y′(s) Π(s) y(s)ds dν . (5.13)

Gradient and the invariant derivative of this functional are

∂W [x, y(·)]∂x

= 2 Px + 2

0∫−τ

D(s) y(s) ds , (5.14)

∂yW [x, y(·)] = 2 x′D(0) x− 2 x′D(−τ) y(−τ)−

− 2 x′0∫

−τ

dD(s)

dsy(s) ds +

+ x′0∫

−τ

R(0, ν) y(ν) dν − y′(−τ)

0∫−τ

R(−τ, ν) y(ν) dν −

−0∫

−τ

0∫−τ

y′(s)∂R(s, ν)

∂sy(ν) ds dν +

( 0∫−τ

y′(s)R(s, 0) ds)

x−

Appendix 127

−( 0∫−τ

y′(s) R(s,−τ) ds)

y(−τ)−

−0∫

−τ

0∫−τ

y′(s)∂R(s, ν)

∂νy(ν) ds dν +

+ x′F (0) x− y′(−τ) F (−τ) y(−τ)−

−0∫

−τ

y′(s)dF (s)

dsy(s)ds +

+ τ x Π(0) x−0∫

−τ

y′(s) Π(s) y(s)ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s)ds dν . (5.15)

Substituting (5.14) and (5.15) into (5.12) we obtain

α(u∗(x, y(·))) = 2(x′P ′+

0∫−τ

y′(s) D′(s) ds)[

A x+Aτy(−τ)+

+

0∫−τ

G(s) y(s)ds− B N−1B′(P x +

0∫−τ

D(s) y(s) ds)]

+

+ 2 x′D(0) x− 2 x′D(−τ) y(−τ)− 2 x′0∫

−τ

dD(s)

dsy(s) ds +

+ x′0∫

−τ

R(0, ν) y(ν) dν − y′(−τ)

0∫−τ

R(−τ, ν) y(ν) dν −


−0∫

−τ

0∫−τ

y′(s)∂R(s, ν)

∂sy(ν) ds dν +

( 0∫−τ

y′(s) R(s, 0) ds)

x−

−( 0∫−τ

y′(s) R(s,−τ) ds)

y(−τ)−

−0∫

−τ

0∫−τ

y′(s)∂R(s, ν)

∂νy(ν) ds dν +

+ x′F (0) x− y′(−τ) F (−τ) y(−τ)−

−0∫

−τ

y′(s)dF (s)

dsy(s)ds +

+ τ x Π(0) x−0∫

−τ

y′(s) Π(s) y(s)ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s)ds dν +

+(x′P ′ +

0∫−τ

y′(s) D′(s) ds)

B N−1 B′ ·

·(Px +

0∫−τ

D(s) y(s) ds)

+ Z[x, y(·)] =

= 2 x′[P ′A−1

2P ′B N−1B′ P+D(0)+

1

2F (0)+

1

2τ Π(0)

]x+

+ 2 x′[P ′Aτ −D(−τ)

]y(−τ) +

Appendix 129

+2 x′0∫

−τ

[P ′G(s)− 1

2P ′B N−1B′D(s)− dD(s)

ds+

12

R(0, s)]y(s) ds+

+2( 0∫−τ

y′(s)[D′(s) A−1

2D′(s) B N−1B′ P+

1

2R(s, 0)

]ds)

x+

+( 0∫−τ

y′(s)[2 D′(s) Aτ −R(s,−τ)

]ds)

y(−τ) +

+

0∫−τ

0∫−τ

y′(s)[2 D′(s) G(ν)−D′(s) B N−1B′D(ν)−

− ∂R(s, ν)

∂s− ∂R(s, ν)

∂ν

]y(ν) ds dν −

− y′(−τ)

0∫−τ

R(−τ, ν) y(ν) dν −

−0∫

−τ

y′(s)[dF (s)

ds+ Π(s)

]y(s) ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν +

− y′(−τ) F (−τ) y(−τ) + Z[x, y(·)] =

= 2 x′[P ′A−1

2P ′B N−1B′ P+D(0)+

1

2F (0)+

1

2τ Π(0)

]x+

+ 2 x′[P ′Aτ −D(−τ)

]y(−τ) +

+2 x′0∫

−τ

[P ′G(s)− 1

2P ′B N−1B′D(s)− dD(s)

ds+

12

R(0, s)]y(s) ds+


+2 x′0∫

−τ

[A′D(s)−1

2P ′B N−1B′D(s)+

1

2R′(s, 0)

]y(s) ds3+

+( 0∫−τ

y′(s)[2 D′(s) Aτ −R(s,−τ)

]ds)

y(−τ) +

+

0∫−τ

0∫−τ

y′(s)[2 D′(s) G(ν)−D′(s) B N−1B′D(ν)−

−∂R(s, ν)

∂s− ∂R(s, ν)

∂ν

]y(ν) ds dν −

−( 0∫−τ

y′(ν) R′(−τ, ν) dν)

y(−τ)4 −

−0∫

−τ

y′(s)[dF (s)

ds+ Π(s)

]y(s) ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν −

− y′(−τ) F (−τ) y(−τ) + Z[x, y(·)] =

= 2 x′[P ′A−1

2P ′B N−1B′ P+D(0)+

1

2F (0)+

1

2τ Π(0)

]x+

+ 2 x′[P ′Aτ −D(−τ)

]y(−τ) +

+ 2 x′0∫

−τ

[P ′G(s)− P ′B N−1B′D(s)− dD(s)

ds+ A′D(s) +

3Here we used the following property of matrix algebra: if z′ M y is a scalar,

then(z′ M y

)′= y′ M ′ z.

4See previous footnote.

Appendix 131

+1

2R(0, s) +

1

2R′(s, 0)

]y(s) ds +

+( 0∫−τ

y′(s)[2 D′(s) Aτ−R(s,−τ)−R′(−τ, s)

]ds)

y(−τ)+

+

0∫−τ

0∫−τ

y′(s)[2 D′(s) G(ν)−D′(s) B N−1B′D(ν)−

−∂R(s, ν)

∂s− ∂R(s, ν)

∂ν

]y(ν) ds dν −

−0∫

−τ

y′(s)[dF (s)

ds+ Π(s)

]y(s) ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν −

− y′(−τ) F (−τ) y(−τ) + Z[x, y(·)] =

= 2 x′[1

2P ′A +

1

2A′P − 1

2P ′B N−1B′ P +

+1

2D(0) +

1

2D′(0) +

1

2F (0) +

1

2τ Π(0)

]x5 +

+ 2 x′[P ′Aτ −D(−τ)

]y(−τ) +

+ 2 x′0∫

−τ

[P ′G(s)− P ′B N−1B′D(s)− dD(s)

ds+ A′D(s)+

+1

2R(0, s) +

1

2R′(s, 0)

]y(s) ds +

5Here we used the property of quadratic forms: x′ L x = x′(

12

L + 12

L′)

x for

arbitrary n× n matrix L and x ∈ Rn.


+( 0∫−τ

y′(s)[2 D′(s) Aτ−R(s,−τ)−R′(−τ, s)

]ds)

y(−τ)+

+

0∫−τ

0∫−τ

y′(s)[D′(s) G(ν)+G′(s) D(ν)−D′(s) B N−1B′D(ν)−

− ∂R(s, ν)

∂s− ∂R(s, ν)

∂ν

]y(ν) ds dν6 −

−0∫

−τ

y′(s)[dF (s)

ds+ Π(s)

]y(s) ds−

−0∫

−τ

0∫ν

y′(s)dΠ(s)

dsy(s) ds dν −

− y′(−τ) F (−τ) y(−τ) + Z[x, y(·)] .Taking into account that functional Z[x, y(·)] has the

form (3.7) we obtain

α(u∗(x, y(·))) =

= x′[P ′A + A′P − P ′BN−1B′ P +

+ F (0) + D(0) + D′(0) + Φ0 + τ Π(0)]x +

+2 x′[P ′Aτ −D(−τ)

]y(−τ)+

+2 x′0∫

−τ

[P ′G(s)− P ′B N−1B′D(s)− dD(s)

ds+

6Here we used that

2

0∫−τ

0∫−τ

y′(s) D′(s) G(ν) y(ν) ds dν =

0∫−τ

0∫−τ

y′(s)[D′(s) G(ν)+G′(s) D(ν)

]y(ν) ds dν .

Appendix 133

+A′D(s) + R(0, s) + Φ1(s)]y(s) ds+

+( 0∫−τ

y′(s)[2 D′(s) Aτ −R(s,−τ)−R′(−τ, s)

]ds)

y(−τ)+

+

0∫−τ

0∫−τ

y′(s)[D′(s) G(ν)+G′(s) D(ν)−D′(s) B N−1B′D(ν)−

− ∂R(s, ν)

∂s− ∂R(s, ν)

∂ν+ Φ2(s, ν)

]y(ν) ds dν +

+

0∫−τ

y′(s)[Φ3(s)− dF (s)

ds− Π(s)

]y(s) ds +

+

0∫−τ

0∫ν

y′(s)[Φ4(s)− dΠ(s)

ds

]y(s) ds dν +

+ y′(−τ)[Φ5 − F (−τ)

]y(−τ) .

Because {x, y(·)} is an arbitrary element of H so thequadratic functional α(u∗(x, y(·))) equal to zero if its coef-ficients will be equal to zero.

Thus we obtain the system of generalized Riccati equa-tions (3.10) – (3.16).


5.3 Explicit solutions of GREs (proofs of

theorems)

5.3.1 Proof of Theorem 3.2

Lemma 5.1. Let n × n matrix P be the solution of thematrix equation

P A + A′P + M = P K P (5.1)

where M is a symmetric n×n matrix. Then n×n matrices

D(s) = e−[P K−A′](s+τ)PAτ , (5.2)

R(s, ν) =

{Q(s) D(ν) for (s, ν) ∈ Ω1 ,D′(s) Q′(ν) for (s, ν) ∈ Ω2 ,

(5.3)

where

Ω1 ={

(s, ν) ∈ [−τ, 0]× [−τ, 0] : s− ν < 0}

,

Ω2 ={

(s, ν) ∈ [−τ, 0]× [−τ, 0] : s− ν > 0}

,

andQ(s) = A′τe

[P K−A′](s+τ) . (5.4)

are solutions of system

dD(s)

ds+[P K − A′

]D(s) = 0 , (5.5)

∂R(s, ν)

∂s+

∂R(s, ν)

∂ν= 0 , (5.6)

with boundary conditions

D(−τ) = P Aτ , (5.7)

R(−τ, s) = A′τD(s) . (5.8)

Proof.1) Matrix D(s).

Appendix 135( )

First let us calculate matrix D(s). The solution of sys-tem (5.5) on the interval [−τ, 0] has the form

D(s) = e−[P K−A′](s+τ)CD ,

and the constant CD can be found from the boundary con-dition (5.7)

D(−τ) = CD = P Aτ .

ThusD(s) = e−[P K−A′](s+τ)P Aτ .

2) Matrix R(s, ν).Now let us check that matrix (5.3) is the solution of sys-

tem (5.6).

Region Ω1. In this region

R(s, ν) = Q(s) D(ν) . (5.9)

Substituting (5.9) into (5.6) we have

dQ(s)

dsD(ν) + Q(s)

dD(ν)

dν= 0 .

Taking into account (5.5) we can replacedD(s)

dsby

−[PK − A′

]D(s), then we obtain

dQ(s)

dsD(ν)−Q(s)

[PK − A′

]D(ν) = 0 ,

or (dQ(s)

ds−Q(s)

[PK − A′

])D(ν) = 0 .

Because D(ν) �= 0, hence Q(s) should be the solution ofthe following equation

dQ(s)

ds−Q(s)

[PK − A′

]= 0 . (5.10)


The solution of this equation on the interval [−τ, 0] has theform

Q(s) = CQe[PK−A′](s+τ) ,

and the constant CQ can be found from the boundary con-dition (5.8)

Q(−τ) = CQ = A′τ ,

Thus in the region Ω1 the matrix R(s, ν) has the form (5.4).

Region Ω2. In this region

R(s, ν) = D′(s) Q′(ν) . (5.11)

Substituting (5.11) into (5.6) we have

dD′(s)ds

Q′(ν) + D′(s)dQ′(ν)

dν= 0 .

Taking into account (5.5) we can replacedD′(s)

dsby

−D′(s)[PK −A′

]′, then we obtain

−D′(s)[PK − A′

]′Q′(ν) + D′(s)

dQ′(ν)

dν= 0 .

or

D′(s)(

dQ′(ν)

dν−[PK − A′

]′Q′(ν)

)= 0 ,

Because D′(s) �= 0 hence Q(s) is the solution of the follow-ing equation

dQ′(ν)

dν−[PK − A′

]′Q′(ν) = 0 . (5.12)

One can see equation (5.12) is the same as (5.10) if we it.So in the region Ω2 matrix R(s, ν) has the form (5.4).

Appendix 137( )

Now let us check the property R(s, ν) = R′(ν, s). Onecan see that

R′(s, ν) =

{D′(ν) Q′(s) for (s, ν) ∈ Ω1 ,Q(ν) D(s) for (s, ν) ∈ Ω2 ,

hence, after interchanging s and ν, we obtain (it is nec-essary to note, if we interchange s and ν, then it is alsonecessary to interchange Ω1 and Ω2)

R′(ν, s) =

{D′(s) Q′(ν) for (s, ν) ∈ Ω2 ,Q(s) D(ν) for (s, ν) ∈ Ω1 ,

thus the condition R(s, ν) = R′(ν, s) is satisfied. �

Proof of Theorem 3.2. Let matrix P be the solu-tion of the matrix equation (3.22) and matrices D(s) andR(s, ν) have the form (3.23) – (3.25).

If we choose weight matrices Φ0, . . ., Φ5 as (3.26) then,substituting these matrices and matrices P , D(s), R(s, ν)into GREs (3.10) – (3.17) we obtain, using Lemma 5.1,identity.

Hence matrices P , D(s), R(s, ν) are solutions of GREs(3.10) – (3.17) corresponding to the weight matrices (3.26).�


Lemma 5.2. Let P be the solution of the exponentialmatrix equation

P A+A′P+e−[P K−A′] τP Aτ+A′τP e−[P K−A′]′ τ+M = PKP ,(5.13)

where M is a symmetric n × n matrix. Then the matrixP and matrices D(s), R(s, ν), defined by (5.2) – (5.4), aresolutions of the following system

PA + A′P + D(0) + D′(0) + M = PKP , (5.14)


dD(s)

ds+[PK − A′

]D(s) = 0 , (5.15)

∂R(s, ν)

∂s+

∂R(s, ν)

∂ν= 0 , (5.16)

with boundary conditions (5.7) – (5.8).

Proof. The difference between this system and system(5.1) – (5.6) consists only in the presence of the term D(0)in (5.14), hence matrices D(s), Q(s) and R(s, ν) have thesame forms as in the Lemma 5.1 (see (5.2) – (5.4)).

Substituting

D(0) = e−[P K−A′] τP Aτ

into (5.14) we obtain the exponential matrix equation(5.13).

Thus, solving equation (5.13) we find matrix P andthen, substituting this matrix into (5.2) – (5.4) we obtainD(s), Q(s) and R(s, ν). �

Using direct substitution one can check that if the ma-trix P is the solution of EME (5.13) then the triple P , D(s)and R(s, ν) satisfy system (5.14) – (5.16), (5.7), (5.8). �

Proof of Theorem 3.3. Let matrix P be the solu-tion of the matrix equation (3.28) and matrices D(s) andR(s, ν) have the form (3.23) – (3.25).

If we choose weight matrices Φ0, . . ., Φ5 as (3.29) then,substituting these matrices and matrices P , D(s), R(s, ν)into GREs (3.10) – (3.17) we obtain, using Lemma 5.2,identity.

Hence matrices P , D(s), R(s, ν) are solutions of GREs(3.10) – (3.17) corresponding to the weight matrices (3.29).�

Appendix 139( )


Theorem can be proved by the direct substitution of thecorresponding matrices to the GREs. �

5.4 Proof of Theorem 1.1. (Solution rep-resentation)

Proof. Note, to prove formula (1.24) it is sufficient toshow that the derivative of (1.24) is equal to the right-sideof equation (1.19).

Differentiating (1.24) and substituting∂F [t, ξ]

∂tby the

right-side of (1.21) we obtain

x(t) =∂F [t, t0]

∂tx0 +

+

0∫−τ

∂F [t, t0 + τ + s]

∂tAτ (t0 + τ + s) y0(s) ds +

+

0∫−τ

⎡⎣ s∫−τ

∂F [t, t0 + s− ν]

∂tG(t0 + s− ν, ν) dν

⎤⎦ y0(s) ds +

+

t∫t0

∂F [t, ρ]

∂tu(ρ) dρ + u(t) =

=

⎡⎣A(t)F [t, t0] + Aτ (t)F [t− τ, t0] +

0∫−τ

G(t, s)F [t + s, t0] ds

⎤⎦ x0 +

+

0∫−τ

{A(t) F [t, t0 + τ + s] + Aτ (t) F [t− τ, t0 + τ + s] +


+

0∫−τ

G(t, η) F [t + η, t0 + τ + s] dη

}×

×Aτ (t0 + τ + s) y0(s) ds +

+

0∫−τ

[ s∫−τ

{A(t) F [t, t0 + s−ν]+Aτ (t) F [t− τ, t0 + s−ν]+

+

0∫−τ

G(t, η) F [t+η, t0+s−ν] dη

}G(t0+s−ν, ν) dν

]y0(s) ds+

+

t∫t0

[A(t) F [t, ρ] + Aτ (t) F [t− τ, ρ] +

+

0∫−τ

G(t, s) F [t + s, ρ] ds

]u(ρ) dρ + u(t) =

= A(t)

[F [t, t0] x

0+

0∫−τ

F [t, t0+τ+s] Aτ (t0+τ+s) y0(s) ds+

+

0∫−τ

( s∫−τ

F [t, t0 + s− ν] G(t0 + s− ν, ν) dν

)y0(s) ds +

+

t∫t0

F [t, ρ] u(ρ) dρ

]+ Aτ (t)

[F [t− τ, t0] x

0 +

+

0∫−τ

F [t− τ, t0 + τ + s] Aτ(t0 + τ + s) y0(s) ds +

Appendix 141( )

+

0∫−τ

( s∫−τ

F [t−τ, t0 +s−ν] G(t0 +s−ν, ν) dν

)y0(s) ds+

+

t∫t0

F [t−τ, ρ] u(ρ) dρ

]+

⎡⎣ 0∫−τ

G(t, η) F [t + η, t0] dη

⎤⎦ x0 +

+

0∫−τ

⎛⎝ 0∫−τ

G(t, η) F [t + η, t0 + τ + s] dη

⎞⎠ ·

·Aτ (t0 + τ + s) y0(s) ds +

+

0∫−τ

[ s∫−τ

{ 0∫−τ

G(t, η) F [t + η, t0 + s− ν] dη

}·

·G(t0 + s− ν, ν) dν

]y0(s) ds +

+

t∫t0

⎡⎣ 0∫−τ

G(t, η) F [t + η, ρ] dη

⎤⎦ u(ρ) dρ + u(t) =

= A(t) x(t) + Aτ (t) x(t− τ) +

+

0∫−τ

G(t, η)(F [t + η, t0] x

0)

dη +

+

0∫−τ

G(t, η)

( 0∫−τ

F [t + η, t0 + τ + s] ·

·Aτ (t0 + τ + s) y0(s) ds

)dη +


+

0∫−τ

G(t, η)

{ 0∫−τ

[ s∫−τ

F [t + η, t0 + s− ν] ·

·G(t0 + s− ν, ν) dν

]y0(s) ds

}dη +

+

0∫−τ

G(t, η)

⎡⎣ t∫

t0

F [t + η, ρ] u(ρ) dρ

⎤⎦ dη + u(t) =

= A(t) x(t) + Aτ (t) x(t− τ) +

+

0∫−τ

G(t, η)

{F [t + η, t0] x

0 +

+

0∫−τ

F [t + η, t0 + τ + s] Aτ(t0 + τ + s) y0(s) ds +

+

0∫−τ

⎡⎣ s∫−τ

F [t + η, t0 + s− ν] G(t0 + s− ν, ν) dν

⎤⎦ y0(s) ds +

+

t∫t0

F [t + η, ρ] u(ρ) dρ

}dη + u(t) =

= A(t) x(t) + Aτ (t) x(t− τ) +

0∫−τ

G(t, η) x(t + η) dη + u(t) .

Initial Conditions

The theorem is proved. �

143

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164

Index

automatic step size con-trol, 118

conditional representation,16

consistent, 112converse theorem, 62

discrete model, 101

exponential matrix equa-tion, 78

– solution, 83extrapolation by contin-

uation, 111extrapolational operator,

111

generalized Riccati equa-tions, 73

– explicit solutions,77

gradient methods, 84

improved Euler method,119

initial value problem, 19interpolation-extrapolation

operator, 112interpolational operator,

109

linear system with de-lays, 11

– time-invariant , 11– with discrete delay,

64– with distributed de-

lay, 66Lipschitz condition, 112LQR problem, 70Lyapunov-Krasovskii quadratic

functionals, 46

phase space, 25piece-wise constant inter-

polation, 103

residual, 114Runge-Kutta-Fehlberg method,

120Runge-Kutta-like method,

113

solution– asymptotically sta-

ble, 40– exponentially sta-

ble, 40– stable, 40

stationary solution method,83

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