contrôlabilité d’équations aux di˙érencesmazanti/...contrôlabilité d’équations aux...

Contrôlabilité d’équations aux di�érences

Guilherme Mazanti

CRAN, Nancy15 octobre 2018

Laboratoire de Mathématiques d’OrsayUniversité Paris-SudUniversité Paris-SaclayFrance

Introduction Explicit solution Stability Relative controllability Approximate and exact controllability

Outline

1 Introduction

2 Explicit solution

3 A word on stability

4 Relative controllability

5 Approximate and exact controllability

Contrôlabilité d’équations aux di�érences Guilherme Mazanti


IntroductionControlled di�erence equations

Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

• Λ = (Λ1, . . . , ΛN ) ∈ (0,+∞)N : vector of positive delays;

• x (t ) ∈ Ãd , u(t ) ∈ Ãm (or Òd , Òm );• Notation: Λmin = minj Λj , Λmax = maxj Λj .

Goal: study the controllability of Σ(A,B , Λ): given initial and �nalstates, can one �nd a control u steering the system from theinitial to the �nal state?

The state at time t can be either x (t ) ∈ Ãd or xt = x (t + ·)|[−Λmax,0)according to the type of controllability we consider.



IntroductionMotivation I: hyperbolic PDEs

Hyperbolic PDEs→ di�erence equations: [Cooke, Krumme; 1968],[Slemrod; 1971], [Greenberg, Li; 1984], [Coron, Bastin, d’Andréa Novel;2008], [Fridman, Mondié, Saldivar; 2010], [Gugat, Sigalotti; 2010]...

∂t yi (t , x ) + ∂x yi (t , x ) + αi (x )yi (t , x ) = 0,

yi (t , 0) =N∑j=1

mi j yj (t , Λj ) +m∑j=1

bi ju j (t ),

i ∈ û1,N ü,

t ∈ Ò+,

x ∈ [0, Λi ].

Method of characteristics: for t ≥ Λmax and j ∈ û1,N ü, one hasyj (t , Λj ) = e

−r Λj0 αj (x )dx yj (t − Λj , 0),

and then y (t ) = (yi (t , 0))i ∈û1,N ü satis�es a di�erence equationwith control u(t ) = (ui (t ))i ∈û1,mü:

yi (t , 0) =N∑j=1

mi j e−

r Λj0 αj (x )dx yj (t − Λj , 0) +

m∑j=1

bi ju j (t ).


http://dx.doi.org/10.1016/0022-247X(68)90038-3

http://dx.doi.org/10.1016/0022-247X(71)90016-3

http://dx.doi.org/10.1016/0022-0396(84)90135-9

http://dx.doi.org/10.1137/070706847

http://dx.doi.org/10.1137/070706847

http://dx.doi.org/10.1093/imamci/dnq024

http://dx.doi.org/10.3934/nhm.2010.5.299



Λ1

Λ2

Λ3

ΛN

Edges: EVertices: VEdges incidenton q ∈ V: Eq

∂2t t yi (t , x ) = ∂

2xx yi (t , x ), i ∈ E, t ∈ Ò+, x ∈ [0, Li ],

yi (t , q ) = yj (t , q ), q ∈ V, i , j ∈ Eq ,

+ conditions and control on vertices.




D’Alembert decomposition on travelling waves:

System of 2N transport equations.Can be reduced to a system of di�erence equations.



IntroductionMotivation II: NFDEs

Consider a neutral functional di�erential equation of the form

d

d t

x (t ) −N∑j=1

Aj x (t − Λj )

= Lxt + Bu(t ), (NFDE)

where xt = x (t + ·)|[−Λmax,0] and L : C([−Λmax, 0],Òd ) → Òd is alinear map.

• Under no control, the stability of Σ(A,B , Λ) is a necessarycondition for that of (NFDE) (see [Henry; 1974] and [Hale,Verduyn Lunel; 2002]).• The stabilizability of Σ(A,B , Λ) by linear feedback laws is anecessary condition for that of (NFDE) (see [Hale, VerduynLunel; 2002]).


http://dx.doi.org/10.1016/0022-0396(74)90089-8

http://dx.doi.org/10.1093/imamci/19.1_and_2.5





IntroductionRational dependence

Rational dependence of the delays Λ1, . . . , ΛN plays a key role inthe behavior of Σ(A,B , Λ).

De�nitionLet Λ = (Λ1, . . . , ΛN ) ∈ (0,+∞)N .• We say that Λ is rationally dependent if there existn1, . . . , nN ∈ Ú (or Ñ) not all zero such that

n1Λ1 + · · · + nNΛN = 0.Otherwise, we say that Λ is rationally independent.• We say that Λ is commensurable if there exist λ > 0 andk1, . . . , kN ∈ Î

∗ such thatΛj = λk j , [j ∈ û1,N ü.

Otherwise, we say that Λ is incommensurable.



IntroductionThe commensurable case

Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

The controllability problem for Σ(A,B , Λ) is easy if Λ1, . . . , ΛNare commensurable. Assume to simplify that Λj = j λ:x (t ) = A1x (t − λ) + A2x (t − 2λ) + · · · + AN x (t − Nλ) + Bu(t ).

Set

X (t ) =

©«x (t )

x (t − λ)...

x (t − (N − 1)λ)

ª®®®®¬.



IntroductionThe commensurable case

Then

X (t ) =

©«

A1 A2 · · · AN−1 ANId 0 · · · 0 00 Id · · · 0 0...

.... . .

......

0 0 · · · Id 0

ª®®®®®®¬X (t − λ) +

©«

B

00...

0

ª®®®®®®¬u(t ).

System in discrete time. Controllability can be studied through aKalman-like criterion.

Does not work in the incommensurable case.



IntroductionMain problems

Goal: characterize relative, approximate, and exact controllabilityof Σ(A,B , Λ).

• Relative controllability in timeT : given x0 : [−Λmax, 0) → Ãd

and xf ∈ Ãd , does there exist u : [0,T ] → Ãm such that thesolution x : [−Λmax,T ] → Ãd of Σ(A,B , Λ) with initialcondition x0 and control u satis�es x (T ) = xf ?

• Approximate/exact controllability in timeT : givenx0 : [−Λmax, 0) → Ãd and xf : [−Λmax, 0) → Ãd , does thereexist u : [0,T ] → Ãm such that the solutionx : [−Λmax,T ] → Ãd of Σ(A,B , Λ) with initial condition x0and control u is such that x (T + ·)|[−Λmax,0) is close to xf in acertain norm / satis�es x (T + ·)|[−Λmax,0) = xf ?




Goal: characterize relative, approximate, and exact controllabilityof Σ(A,B , Λ).

• Relative controllability in timeT : given x0 : [−Λmax, 0) → Ãd

and xf ∈ Ãd , does there exist u : [0,T ] → Ãm such that thesolution x : [−Λmax,T ] → Ãd of Σ(A,B , Λ) with initialcondition x0 and control u satis�es x (T ) = xf ?• Approximate/exact controllability in timeT : givenx0 : [−Λmax, 0) → Ãd and xf : [−Λmax, 0) → Ãd , does thereexist u : [0,T ] → Ãm such that the solutionx : [−Λmax,T ] → Ãd of Σ(A,B , Λ) with initial condition x0and control u is such that x (T + ·)|[−Λmax,0) is close to xf in acertain norm / satis�es x (T + ·)|[−Λmax,0) = xf ?




Previous results:• Stability analysis for the uncontrolled system: [Cruz, Hale;1970], [Henry; 1974], [Melvin; 1974], [Hale; 1975], [Silkowski; 1976],[Avellar, Hale; 1980], [Hale, Verduyn Lunel; 1993], [Michiels et al;2009], [Chitour, M., Sigalotti; 2016]...• Stabilization by linear feedbacks: [Hale, Verduyn Lunel; 2002],[Hale, Verduyn Lunel; 2003].• Relative controllability: introduced in [Chyung; 1970], [Olbrot;1972], [Klamka; 1976], originally for delays in control. Results inthe case N = 2 and integer delays in [Diblík, Khusainov,Růžičková; 2008] and [Pospíšil, Diblík, Fečkan; 2015].• Spectral and approximate controllability inLp ([−Λmax, 0],Ã

d ): [Salamon; 1984] (for more general NFDEs,using duality) and [O’Connor, Tarn; 1983].


http://dx.doi.org/10.1016/0022-0396(70)90114-2

http://dx.doi.org/10.1016/0022-0396(70)90114-2

http://dx.doi.org/10.1016/0022-0396(74)90089-8

http://dx.doi.org/10.1016/0022-247X(74)90149-8

https://mathscinet.ams.org/mathscinet-getitem?mr=397216

https://books.google.fr/books/about/Star_shaped_Regions_of_Stability_in_Here.html?id=DHWIGwAACAAJ&redir_esc=y

http://dx.doi.org/10.1016/0022-247X(80)90289-9

http://dx.doi.org/10.1007/978-1-4612-4342-7

http://dx.doi.org/10.1137/080724940

http://dx.doi.org/10.1137/080724940

http://dx.doi.org/10.3934/nhm.2016010


http://dx.doi.org/10.1080/00207720310001609039

https://doi.org/10.1109/TAC.1970.1099416

https://doi.org/10.1109/TAC.1972.1100090

https://doi.org/10.1109/TAC.1972.1100090

https://doi.org/10.1016/0005-1098(76)90046-7

http://dx.doi.org/10.1137/070689085

http://dx.doi.org/10.1137/070689085

http://dx.doi.org/10.1063/1.4912420


https://doi.org/10.1137/0321018


Explicit solutionWell-posedness

Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

• Solution with initial condition x0 : [−Λmax, 0) → Ãd andcontrol u : [0,T ] → Ãm : x satisfying Σ(A,B , Λ) for t ≥ 0 andx (t ) = x0(t ) for −Λmax ≤ t < 0.• Notation: xt = x (t + ·)|[−Λmax,0).• Existence and uniqueness follow easily; no regularityassumptions!

• If x0,u ∈ Lp for some p , then x ∈ Lp , xt ∈ Lp .• Ck solutions exist under a compatibility condition:

limt→0

x(r )0 (t ) =

N∑j=1

Aj x(r )0 (−Λj ) + Bu

(r )(0), r ∈ û0, k ü.




Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

• Solution with initial condition x0 : [−Λmax, 0) → Ãd andcontrol u : [0,T ] → Ãm : x satisfying Σ(A,B , Λ) for t ≥ 0 andx (t ) = x0(t ) for −Λmax ≤ t < 0.• Notation: xt = x (t + ·)|[−Λmax,0).• Existence and uniqueness follow easily; no regularityassumptions!• If x0,u ∈ Lp for some p , then x ∈ Lp , xt ∈ Lp .• Ck solutions exist under a compatibility condition:

limt→0

x(r )0 (t ) =

N∑j=1

Aj x(r )0 (−Λj ) + Bu

(r )(0), r ∈ û0, k ü.




Two ways to see the solution: x : [−Λmax,T ] → Ãd orxt = x (t + ·)|[−Λmax,0) : [−Λmax, 0) → Ãd .

Example

x (t ) =

(18 1

0 18

)x (t − 1) +

(17

14

− 1417

)x

(t −√22

)+

(19 0

− 1219

)x

(t − 1

π

)+

(01

)u(t ),

u(t ) = sin(4t ) +1

2cos(5t ).

t

x (t ) = (x1(t ), x2(t ))

−1 10



Explicit solutionExplicit solution

Technique to study controllability: explicit formula for thesolution.

Proposition ([Chitour, M., Sigalotti; 2016], [M.; 2017])

Let x0 : [−Λmax, 0) → Ãd and u : [0,T ] → Ãm . The solutionx : [−Λmax,T ] → Ãd of Σ(A,B , Λ) with initial condition x0 andcontrol u is given for t ∈ [0,T ] byx (t ) =

∑(n,j )∈ÎN×û1,N ü−Λj ≤t−Λ ·n<0

Ξn−e jAj x0(t − Λ · n) +∑n∈ÎNΛ ·n≤t

ΞnBu(t − Λ · n),

where Ξn is de�ned inductively for n ∈ ÎN by Ξ0 = Idd andΞn =

∑Nk=1nk ≥1

AkΞn−ek for n = (n1, . . . , nN ) ∈ ÎN \ {0}.

• Term due to the initial condition + term due to the control.



https://dx.doi.org/10.1137/16M1073157



Technique to study controllability: explicit formula for thesolution.

Proposition ([Chitour, M., Sigalotti; 2016], [M.; 2017])

Let u : [0,T ] → Ãm . The solution x : [−Λmax,T ] → Ãd ofΣ(A,B , Λ) with initial condition 0 and control u is given fort ∈ [0,T ] by

x (t ) =∑n∈ÎNΛ ·n≤t

ΞnBu(t − Λ · n),

where Ξn is de�ned inductively for n ∈ ÎN by Ξ0 = Idd andΞn =

∑Nk=1nk ≥1

AkΞn−ek for n = (n1, . . . , nN ) ∈ ÎN \ {0}.

• Term due to the initial condition + term due to the control.



https://dx.doi.org/10.1137/16M1073157



Idea for the explicit formula:

x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ),

x (t − Λj ) =N∑k=1

Ak x (t − Λj − Λk ) + Bu(t − Λj ).

Then

x (t ) =N∑j=1

N∑k=1

AjAk x (t − Λj − Λk ) +N∑j=1

AjBu(t − Λj ) + Bu(t ).

Proceed inductively until x is evaluated at times in [−Λmax, 0).




x (t ) =∑n∈ÎNΛ ·n≤t

ΞnBu(t − Λ · n).

One may have di�erent n yielding the same Λ · n. Goal: groupthem in the above formula.

• Equivalence relation on ÎN : n ∼ n′ ⇐⇒ Λ · n = Λ · n′.• Equivalence classes: [n]Λ or simply [n].Remark: [n] = {n} if Λ is rationally independent!• NΛ = Î

N /∼.

Set Ξ̂Λ[n] =

∑n′∈[n]

Ξn′ . Then x (t ) =∑[n]∈NΛΛ ·n≤t

Ξ̂Λ[n]Bu(t − Λ · n).

In particular, di�erent [n] ⇐⇒ di�erent Λ · n.



A word on stabilityHale–Silkowski criterion

Stability for u = 0:

Σ0(A, Λ) : x (t ) =N∑j=1

Aj x (t − Λj ), t ≥ 0.

Theorem ([Hale; 1975], [Silkowski; 1976])The following are equivalent:

• max(θ1,...,θN )∈[0,2π]N

ρ©«N∑j=1

Aj ei θj ª®¬ < 1;

• Σ0(A, Λ) is exponentially stable for some Λ ∈ (0,+∞)N withrationally independent components;• Σ0(A, Λ) is exponentially stable for every Λ ∈ (0,+∞)N .



https://books.google.fr/books/about/Star_shaped_Regions_of_Stability_in_Here.html?id=DHWIGwAACAAJ&redir_esc=y



• Obtained by spectral methods. Stability depends on the realparts of the roots of det

(Id−

∑Nj=1 Aj e

−sΛj)= 0 (exponential

polynomial, see [Avellar, Hale; 1980]).• For rationally dependent delays: [Michiels et al.; 2009].

• Our previous explicit formula can be adapted for the case oftime-varying matrices Aj (t ):

x (t ) =N∑j=1

Aj (t )x (t − Λj ), t ≥ 0.

One can then generalize Hale–Silkowski criterion: [Chitour, M.,Sigalotti; 2016]


http://dx.doi.org/10.1016/0022-247X(80)90289-9

http://dx.doi.org/10.1137/080724940





• Obtained by spectral methods. Stability depends on the realparts of the roots of det

(Id−

∑Nj=1 Aj e

−sΛj)= 0 (exponential

polynomial, see [Avellar, Hale; 1980]).• For rationally dependent delays: [Michiels et al.; 2009].• Our previous explicit formula can be adapted for the case oftime-varying matrices Aj (t ):

x (t ) =N∑j=1

Aj (t )x (t − Λj ), t ≥ 0.

One can then generalize Hale–Silkowski criterion: [Chitour, M.,Sigalotti; 2016]


http://dx.doi.org/10.1016/0022-247X(80)90289-9

http://dx.doi.org/10.1137/080724940




Relative controllabilityDe�nition

Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

De�nitionWe say that Σ(A,B , Λ) is relatively controllable in timeT > 0 if,for every x0 : [−Λmax, 0) → Ãd and xf ∈ Ãd , there existsu : [0,T ] → Ãm such that the unique solution x of Σ(A,B , Λ) withinitial condition x0 and control u satis�es x (T ) = xf .

It su�ces to consider zero initial conditions. Thenx (t ) =

∑[n]∈NΛΛ ·n≤t

Ξ̂Λ[n]Bu(t − Λ · n).



Relative controllabilityRelative controllability criterion

Theorem ([M.; 2017])The following statements are equivalent:

• Σ(A,B , Λ) is relatively controllable in timeT ;

• Span{Ξ̂Λ[n]Bw

�� [n] ∈ NΛ, Λ · n ≤ T , w ∈ Ãm}= Ãd ;

• \ε0 > 0 s.t., [ε ∈ (0, ε0), [x0 : [−Λmax, 0) → Ãd , and[xf : [0, ε] → Ãd , \u : [0,T + ε] → Ãm s.t. the solution x ofΣ(A,B , Λ) with initial condition x0 and control u satis�esx (T + ·)|[0,ε] = xf ;

• \ε0 > 0 s.t., [p ∈ [1,+∞], [ε ∈ (0, ε0), [x0 ∈ Lp ((−Λmax, 0),Ãd ),and [xf ∈ Lp ((0, ε),Ãd ), \u ∈ Lp ((0,T + ε),Ãm ) s.t. the solution xof Σ(A,B , Λ) with initial condition x0 and control u satis�esx ∈ Lp ((−Λmax,T + ε),Ãd ) and x (T + ·)|[0,ε] = xf .


https://dx.doi.org/10.1137/16M1073157


Relative controllabilityRelative controllability criterion

Remarks:

• Can also be generalized to other spaces (e.g., Ck ).• Generalizes Kalman criterion: for x (t ) = Ax (t − 1) + Bu(t ),one hasSpan

{Ξ̂Λ[n]Bw

�� [n] ∈ NΛ, Λ · n ≤ T , w ∈ Ãm}

= Ran(B AB A2B · · · A bT cB

).

• Relative controllability 6=⇒ stabilizability by a lineartime-invariant feedback law.



Relative controllabilityRational dependence of the delays

What happens to the relative controllability of Σ(A,B , Λ) if wechange Λ?• Z (Λ) = {n ∈ ÚN | Λ · n = 0}.• We write Λ 4 L if Z (Λ) ⊂ Z (L) (L is more rationally dependentthan Λ).• We write Λ ≈ L if Λ 4 L and L 4 Λ (L is as rationallydependent as Λ).• Remark: Z (Λ) = {0} ⇐⇒ Λ is rationally independent.

Theorem ([M.; 2017])If Λ 4 L and Σ(A,B , L) is relatively controllable in timeT , thenΣ(A,B , Λ) is relatively controllable in time κT , withκ = maxj ∈û1,N ü

ΛjLj.


https://dx.doi.org/10.1137/16M1073157


Relative controllabilityRational dependence of the delays

The converse is false, but we have the following.

Theorem ([M.; 2017])Let Λ = (Λ1, . . . , ΛN ) ∈ (0,+∞)N andT > 0. For every ε > 0, thereexists a commensurable L = (L1, . . . , LN ) ∈ (0,+∞)N satisfyingL < Λ and 1 ≤ Λj

Lj< 1 + ε for every j ∈ û1,N ü such that, if

Σ(A,B , Λ) is relatively controllable in timeT , then Σ(A,B , L) isalso relatively controllable in timeT .

Idea of the proof: Construct L as a small perturbation of Λ sothat, if Λ · n, L · n ∈ [0,T ], then

Λ · n = Λ · n′ ⇐⇒ L · n = L · n′.


https://dx.doi.org/10.1137/16M1073157


Relative controllabilityMinimal controllability time

• For x (t ) = Ax (t − 1) + Bu(t ), one has

Span{Ξ̂Λ[n]Bw

�� [n] ∈ NΛ, Λ · n ≤ T , w ∈ Ãm}

= Ran(B AB A2B · · · A bT cB

).

• In this case, by Cayley–Hamilton theorem, relativecontrollability in some timeT > 0 ⇐⇒ relativecontrollability in time d − 1.

Theorem ([M.; 2017])If Σ(A,B , Λ) is relatively controllable in some timeT > 0, then itis also relatively controllable in timeT = (d − 1)Λmax.

Idea: prove �rst for commensurable delays using stateaugmentation + Kalman criterion + Cayley–Hamilton. Conclude bythe previous results.


https://dx.doi.org/10.1137/16M1073157



Assume to simplify that Λ is rationally independent. Criterion:Span

{ΞnBw

�� n ∈ ÎN , Λ · n ≤ (d − 1)Λmax, w ∈ Ãm}= Ãd .

Finitely many vectors, but their number tends to +∞ asΛmaxΛmin→ +∞.

Theorem ([M.; 2017])If Λ is rationally independent, then Σ(A,B , Λ) is relativelycontrollable in some timeT > 0 if and only if

Span{ΞnBe j

�� n ∈ ÎN , |n|1 ≤ d − 1, j ∈ û1,mü} = Ãd .Number of vectors to compute: m

(N+d−1d−1

). Independent of Λ.

Idea of the proof: Λ ≈ Lε where Lε is a rationally independentperturbation of (1, 1, . . . , 1).


https://dx.doi.org/10.1137/16M1073157




{ΞnBe j

�� n ∈ ÎN , Λ · n ≤ (d − 1)Λmax, j ∈ û1,mü} = Ãd .

Finitely many vectors, but their number tends to +∞ asΛmaxΛmin→ +∞.


Span{ΞnBe j


(N+d−1d−1




https://dx.doi.org/10.1137/16M1073157




{ΞnBe j

�� n ∈ ÎN , Λ · n ≤ (d − 1)Λmax, j ∈ û1,mü} = Ãd .Finitely many vectors, but their number tends to +∞ asΛmaxΛmin→ +∞.


Span{ΞnBe j


(N+d−1d−1




https://dx.doi.org/10.1137/16M1073157


Approximate and exact controllabilityDe�nition

Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

Recall that xt = x (t + ·)|[−Λmax,0).

De�nition

• We say that Σ(A,B , Λ) is approximately controllable in timeT > 0 if, [x0, xf ∈ L2((−Λmax, 0),Ãd ) and [ε > 0,\u ∈ L2((0,T ),Ãm) s.t. the solution x of Σ(A,B , Λ) withinitial condition x0 and control u satis�es ‖xT − xf ‖L2 < ε.

• We say that Σ(A,B , Λ) is exactly controllable in timeT > 0 if,[x0, xf ∈ L

2((−Λmax, 0),Ãd ), \u ∈ L2((0,T ),Ãm) s.t. the

solution x of Σ(A,B , Λ) with initial condition x0 and controlu satis�es xT = xf .




Σ(A,B , Λ) : x (t ) =N∑j=1

Aj x (t − Λj ) + Bu(t ), t ≥ 0.

Recall that xt = x (t + ·)|[−Λmax,0).

De�nition

• We say that Σ(A,B , Λ) is approximately controllable in timeT > 0 if, [x0, xf ∈ L2((−Λmax, 0),Ãd ) and [ε > 0,\u ∈ L2((0,T ),Ãm) s.t. the solution x of Σ(A,B , Λ) withinitial condition x0 and control u satis�es ‖xT − xf ‖L2 < ε.• We say that Σ(A,B , Λ) is exactly controllable in timeT > 0 if,[x0, xf ∈ L

2((−Λmax, 0),Ãd ), \u ∈ L2((0,T ),Ãm) s.t. the

solution x of Σ(A,B , Λ) with initial condition x0 and controlu satis�es xT = xf .




If x0 = 0, our explicit formula for x readsx (t ) =

∑n∈ÎNΛ ·n≤t


ForT ≥ 0, let E (T ) be the operator that maps a control u toxT = x (T + ·)|[−Λmax,0), the state at timeT with initial conditionzero:

E (T ) :

L2((0,T ),Ãm) → L2((−Λmax, 0),Ã

d )

u 7→∑n∈ÎN

Λ ·n≤T +·

ΞnBu(T + · − Λ · n).

• Exactly controllable ⇐⇒ E (T ) is surjective

⇐⇒ \c > 0 s.t. ‖E (T )∗x ‖L2 ≥ c‖x ‖L2 [x .

• Approximately controllable ⇐⇒ RanE (T ) is dense

⇐⇒ E (T )∗ is injective.




If x0 = 0, our explicit formula for x readsx (t ) =

∑n∈ÎNΛ ·n≤t


ForT ≥ 0, let E (T ) be the operator that maps a control u toxT = x (T + ·)|[−Λmax,0), the state at timeT with initial conditionzero:

E (T ) :

L2((0,T ),Ãm) → L2((−Λmax, 0),Ã

d )

u 7→∑n∈ÎN

Λ ·n≤T +·

ΞnBu(T + · − Λ · n).

• Exactly controllable ⇐⇒ E (T ) is surjective⇐⇒ \c > 0 s.t. ‖E (T )∗x ‖L2 ≥ c‖x ‖L2 [x .

• Approximately controllable ⇐⇒ RanE (T ) is dense⇐⇒ E (T )∗ is injective.



Approximate and exact controllabilityCase N = d = 2, m = 1

No known criterion in the general case. We consider hereΣ(A,B , Λ) : x (t ) = A1x (t − Λ1) + A2x (t − Λ2) + Bu(t ),

with x (t ) ∈ Ã2, u(t ) ∈ Ã, and assume that Λ1 > Λ2.

(Recall that C(A,B) = (B ,AB , . . . ,Ad−1B) and that the pair (A,B) is controllable if and

only if C(A,B) has full rank.)

Theorem ([Chitour, M., Sigalotti; Preprint])

LetT ≥ 0.

• (A1,B) and (A2,B) not controllable =⇒ Σ(A,B , Λ) neitherapproximately nor exactly controllable in timeT .• RanA1 ⊂ RanB =⇒ Σ(A,B , Λ) neither approximately norexactly controllable in timeT .


http://arxiv.org/abs/1708.06175



No known criterion in the general case. We consider hereΣ(A,B , Λ) : x (t ) = A1x (t − Λ1) + A2x (t − Λ2) + Bu(t ),

with x (t ) ∈ Ã2, u(t ) ∈ Ã, and assume that Λ1 > Λ2.(Recall that C(A,B) = (B ,AB , . . . ,Ad−1B) and that the pair (A,B) is controllable if and

only if C(A,B) has full rank.)


LetT ≥ 0.

• (A1,B) and (A2,B) not controllable =⇒ Σ(A,B , Λ) neitherapproximately nor exactly controllable in timeT .• RanA1 ⊂ RanB =⇒ Σ(A,B , Λ) neither approximately norexactly controllable in timeT .






• RanA1 1 RanB and exactly one of the pairs (A1,B) and(A2,B) is controllable =⇒ the following are equivalent:• Σ(A,B , Λ) is exactly controllable in timeT .• Σ(A,B , Λ) is approximately controllable in timeT .• T ≥ 2Λ1.






• If (A1,B) and (A2,B) are controllable, let B⊥ ∈ Ã2 be theunique vector such that det(B ,B⊥) = 1 and BTB⊥ = 0. Set

β =detC(A1,B)detC(A2,B)

, α = det(B (A1 − βA2)B

⊥).

Let S ⊂ Ã be the set of all possible complex values of theexpression β + α1−

Λ2Λ1 .

• Σ(A,B , Λ) is approximately controllable in timeT if and onlyifT ≥ 2Λ1 and 0 < S.

• Σ(A,B , Λ) is exactly controllable in timeT if and only ifT ≥ 2Λ1 and 0 < S.





S ={β + α

1−Λ2Λ1

}.

β

|α |1−

Λ2Λ1

Λ2Λ1∈ Ñ

β

|α |1−

Λ2Λ1

Λ2Λ1< Ñ



Approximate and exact controllabilityIdeas of the proof

The non-trivial case is when (A1,B) and (A2,B) are controllable.One can reduce to the normal form

x (t ) =

(α β

0 0

)x (t − 1) +

(0 10 0

)x (t − L) +

(01

)u(t )

with L = Λ2Λ1∈ (0, 1). Moreover, approximately/exactly

controllable in timeT ⇐⇒ in time 2.




Approximate controllability: injectivity of E (2)∗. Reduction to�nite dimension and state augmentation when L ∈ Ñ. WhenL < Ñ, if E (2)∗x = 0, then x1 satis�es

x1(t ) = −1

β

(αχ(−1,−L)(t ) + χ(−L,0)(t )

)x1 ◦ϕ(t )

(χ : characteristic functions; ϕ: translation by L modulo 1).

Letting y (t ) = eγt x1(t ), where eγ(1−L) = −β , we gety (t ) =

(αe−γχ(−1,−L)(t ) + χ(−L,0)(t )

)y ◦ϕ(t ). (Y)

• 0 ∈ S ⇐⇒ αe−γ = 1 ⇐⇒ y = y ◦ϕ ⇐⇒ y is constant(ergodicity of ϕ). One gets non-trivial elements in KerE (2)∗.• Otherwise, we have(

1 − |αe−γ |2) w 0

L−1|y (t )|2 dt = 0.

Ok if |αe−γ | , 1. Otherwise, go back to (Y) and expandy |(L−1,0) in Fourier series to obtain that y vanishes.




Approximate controllability: injectivity of E (2)∗. Reduction to�nite dimension and state augmentation when L ∈ Ñ. WhenL < Ñ, if E (2)∗x = 0, then x1 satis�es

x1(t ) = −1

β

(αχ(−1,−L)(t ) + χ(−L,0)(t )

)x1 ◦ϕ(t )

(χ : characteristic functions; ϕ: translation by L modulo 1).Letting y (t ) = eγt x1(t ), where eγ(1−L) = −β , we get

y (t ) =(αe−γχ(−1,−L)(t ) + χ(−L,0)(t )

)y ◦ϕ(t ). (Y)

• 0 ∈ S ⇐⇒ αe−γ = 1 ⇐⇒ y = y ◦ϕ ⇐⇒ y is constant(ergodicity of ϕ). One gets non-trivial elements in KerE (2)∗.• Otherwise, we have(

1 − |αe−γ |2) w 0

L−1|y (t )|2 dt = 0.

Ok if |αe−γ | , 1. Otherwise, go back to (Y) and expandy |(L−1,0) in Fourier series to obtain that y vanishes.




Exact controllability: show that \c > 0 s.t. [x ,‖E (2)∗x ‖L2 ≥ c‖x ‖L2 . (X)

• If L ∈ Ñ, equivalent to approximate controllability andcharacterized by detM , 0 for some matrix M , withc = |M −1 |−12 .

• If L < Ñ and 0 < S, approximate L by rationals and obtain anuniform upper bound on |M −1 |2 along this approximation.• If L < Ñ and 0 ∈ S, one can construct a sequence of points inS tending to 0 and obtain a sequence (xn )n∈Î with L2 norm 1such that E (2)∗xn → 0, contradicting (X).




Exact controllability: show that \c > 0 s.t. [x ,‖E (2)∗x ‖L2 ≥ c‖x ‖L2 . (X)

• If L ∈ Ñ, equivalent to approximate controllability andcharacterized by detM , 0 for some matrix M , withc = |M −1 |−12 .• If L < Ñ and 0 < S, approximate L by rationals and obtain anuniform upper bound on |M −1 |2 along this approximation.• If L < Ñ and 0 ∈ S, one can construct a sequence of points inS tending to 0 and obtain a sequence (xn )n∈Î with L2 norm 1such that E (2)∗xn → 0, contradicting (X).



Approximate and exact controllabilityControllability to constants

What if we want to go from x0 = 0 to a constant state in timeT ?


Σ(A,B , Λ) is approximately controllable if and only if it isapproximately controllable to constants (i.e., every constant statecan be approximately reached by a suitable control from a zeroinitial condition).

• Main idea: if Σ(A,B , Λ) is approximately controllable toconstants, using a kind of commutation between E (T ) andintegration (inspired by the �nite-dimensional ideas of[Gohberg, Shalom; 1989]), one can approximate anypolynomial.• Does it also hold for exact controllability? True forN = d = 2 and m = 1, unknown in general.



http://dx.doi.org/10.1007/BF01199458






• Main idea: if Σ(A,B , Λ) is approximately controllable toconstants, using a kind of commutation between E (T ) andintegration (inspired by the �nite-dimensional ideas of[Gohberg, Shalom; 1989]), one can approximate anypolynomial.

• Does it also hold for exact controllability? True forN = d = 2 and m = 1, unknown in general.



http://dx.doi.org/10.1007/BF01199458






• Main idea: if Σ(A,B , Λ) is approximately controllable toconstants, using a kind of commutation between E (T ) andintegration (inspired by the �nite-dimensional ideas of[Gohberg, Shalom; 1989]), one can approximate anypolynomial.• Does it also hold for exact controllability? True forN = d = 2 and m = 1, unknown in general.



http://dx.doi.org/10.1007/BF01199458

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