causality, strict causality and invertibility for systems in hilbert resolution spaces

SIAM J. CONTROLVol. 12, No. 3, August 1974

CAUSALITY, STRICT CAUSALITY AND INVERTIBILITYFOR SYSTEMS IN HILBERT RESOLUTION SPACES*

ROMANO M. DE SANTIS’

Abstract. The Hilbert resolution space setting is used to obtain some new sufficient conditions forthe existence and causality ofan inverse system. These conditions are applicable when the system underconsideration is characterized by a behavior which is special, in some sense, with respect to the notionof causality. Relevant conceptual connections with recent problems in system sensitivity and stabilitytheory are pointed out.

1. Introduction. Given an operator T defined on the Hilbert space H, anddenoting by I the identity operator on H, the main objective of this article is toestablish some sufficient conditions on T which insure the invertibility of (I + T)and the causality of (I + T)- 1.

The question of invertibility for (I + T) has already received considerableattention in the technical literature. In particular, Browder [2], Dolph [10] andMinty 17], among others, have given sufficient invertibility conditions for the casein which Tis a monotone operator. Similar results have been obtained by Petryshyn[19] and Shinbrot [28], who considered operators T with special compactnessproperties. These and other developments are summarized by Damborg [5] andwill not be detailed here.

In regard to the causality of (I + T)- 1, the most familiar results are perhapsthose offered by Foures and Segal [12] and Youla, Castriota and Carlin [29].These results are essentially based on the Paley and Wiener theorem [18] andtheir application is confined to the case of linear and time-invariant systems. Morerecent developments concerning systems of a more general type are also available.Sandberg [27], for example, has considered nonlinear time-variant systems and hasgiven an interesting connection between causality and energy related concepts.Damborg [3], [4] has established a sufficient condition for the causality of(/+ T)-in terms of an expression involving "incremental truncated" gain and phaseshift concepts. Saeks [25] has considered linear systems in Hilbert space and hasestablished the causality of (I + T)-1 when T is causal and satisfies an innerproduct type condition. In a similar context Porter [22] has shown that (I + T)-is causal whenever T is causal and dissipative.

A distinctive feature of the present development is that the invertibility of(I + T) and the causality of (I + T)-1 are investigated by focusing attention onthose causal systems for which the future of the output is determined by the strictpast of the input; such systems are said to be strictly causal. This approach leadsin a natural way to the utilization of the Hilbert resolution space framework pro-posed in [22] and [26], and to the exploitation of the strict causality treatmentdeveloped in [25].

Received by the editors February 15, 1972, and in revised form November 13, 1972.

" Ecole Polytechnique, Montr6al, Qu6bec, Canada..This work was done in part while the authorwas associated with the Electrical Engineering Department of the University of Michigan. This workwas supported in part by the United States Air Force Office of Scientific Research under Grant 732427and by the Canadian National Research Council under Grant A8244.

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CAUSALITY, STRICT CAUSALITY AND INVERTIBILITY 537

The interrelation of strict causality with causal invertibility has already beenemphasized in the technical literature. Relevant efforts in this direction areillustrated by the work of Zames [34] and Willems [31] (see, in particular, [31,5.2, pp. 93-101]). More recently it has been shown that some results of Gohberg

and Krein 14], 15] on the abstract theory of Volterra operators can be interpreted[7] as showing that if T is linear, completely continuous and strictly causal, then(I + T) is invertible and (I + T)- is causal. The scope and applicability of theseresults are hampered, however, on the one hand by the requirements of linearityand complete continuity and on the other hand by the adoption of a strict causalityconcept which is more restrictive than that usually considered in the technicalliterature.

The present development will show that linearity and complete continuityrequirements are not at all essential to establish results of the Gohberg and Kreintype. In particular, we will find, for example, that a sufficient condition for (1 + T)to be invertible and for its inverse to be causal is that Tbe given by the compositionof a weakly additive causal and Lipschitz continuous system with a linear boundedand strictly causal system. This type of result will be shown to be applicable to alarge class of systems which are not necessarily.strictly causal.

The paper is organized as follows. Section 2 establishes the mathematicalframework in which the study is embedded. Section 3 summarizes the definitionsand properties associated with the notions of causality and strict causality. Theaforementioned invertibility and causality results are to be found in 4 and 5.In particular, 4 gives a number of sufficient conditions which simultaneouslyinsure the invertibility of (I + T) and the causality of (I + T)- 1. In 5, (I + T)-is assumed to exist and we offer sufficient conditions for its causality. Section 6illustrates some connections between present results and problems of sensitivityand stability theory, and 7 contains some concluding remarks about the overalldevelopment.

2. Mathematical preliminaries. The reader is assumed to be familiar withthe notions of metric, linear, normed, Banach, inner product, and Hilbert spaces.The notions of linear and nonlinear mappings between such spaces are also as-sumed to be familiar. A unified treatment of such concepts is available, for example,in [20].

If x is an element of a Banach space B, the norm ofx is indicated by the symbolIxl. If T is an operator on B, then T is said to be bounded if

TI sup Txl/Ixl < o.0 xeB

The number IT[ is called the norm of T. T is continuous if for any x B and any reale > 0, there exists a real 6 > 0 such that [Tx Tyl <- when Ix Yl -< 6. T isLipschitz continuous if

I1Tll- sup Tx- Tyl/lx- Yl <= oe.0 x-yB

IIT is called the Lipschitz norm of T. Observe that when T is linear, then theconcepts of boundedness, continuity and Lipschitz continuity are equivalent. T iscalled compact if T(S), the closure under T of a bounded set S, is a compact set.When T is compact and continuous, then it is said to be completely continuous.D

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538 ROMANO M. D SANTIS

Next the structure of Hilbert resolution spaces will be presented [223, [24],[25]. Suppose that H is a Hilbert space, and v a linearly ordered set with o and t,respectively, minimum and maximum elements. A family [ {Pt},t v, oforthogonal projections on H is a resolution of the identity if it enjoys the followingtwo properties"

(Ri) PtH O, PtH H, and pkH_PH whenever k > 1"

(Rii) if {U} is a sequence of orthogonal projections in and there existsan orthogonal projection P such that {Px} -. Px, for every x e H, then P

A Hilbert space, H, equipped with a resolution of the identity, [R {P’}, iscalled a Hilbert resolution space (in short" HRS) and is denoted by the symbol[H,U.

Example 2.1. Suppose that H is given by L2[0, ), the Hilbert space ofLebesgue square integrable real functions. In L2[0, ) a family of orthogonalprojections {Pt}, [0, cx3], can be defined as follows" if x, y L2[0 o) andy ptx, then y(s) x(s) on [0, t] and y(s) 0 in It, ). When then Px

x. The family {pt}, as it enjoys properties (Ri) and (Rii), is a resolution ofthe identity. It is then possible to view [L2[0 (3), Pt] as a HRS in the followingsections.

The notion of integral on HRS will play a major role in this development.Suppose that T(s), s v, is a family of operators on a HRS indexed by s e v, andconsider the following operations"

(i) Choose a partition f of v, f {o, , "’", }, where o to,and j < j+l, J 1,2,..., N 1.

(ii) Consider the partial sumN

(1) In AP(k)T(Sk),k=l

where AP(k) P P-’ and k- <= Sk <---- k"(iii) On the set of all partitions f of v, define a partial order as follows"

’1 -" "2 if every element of ’2 is contained in(iv) Suppose that there exists an operator T such that for any e > 0 there is

a partition fl of v such that the operator norm IT Inl is less than e if f2 _< f.The operator T obtained through operations (i)-(iv) is called the integral of thefamily T(s) with respect to and is denoted by T f dPT(s).

It is useful to consider slightly different variations of the above concept ofintegral. To this purpose, the notations r dPT(s) and dPT(s) will be used toindicate the integrals which are obtained by choosing SR in operation (ii) respectivelyas follows" s k- or sk k" Similarly, the operator f dPT(s) dP will denote theintegral which is obtained by replacing equation (1) in operation (ii) by the follow-

NingI= AP(k)T(sk)AP(k)

k=l

In the course of the development it will be natural to associate with an operatorT on [H, U], the family TW. This family will lead to integrals such as f dPTW,dPTW, dPTP, and f dPT dP.

The Hilbert space whose elements are square summable sequences,/2, can be viewed as a HRSin a similar way, and this is true also for the cross-product space Lz x l’. For other important examplesof HRS, the reader is refered to [11] where reproducing kernel Hilbert spaces are considered.D

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Example 2.2. In [L/[O, oe), pt], the HRS described in Example 2.1, considerthe operator T defined according to the following rule" if y Tx, then y(t) g(t)x(t)+ k(t, s)x(s)ds, where g e L2[0, oe) and k(t, s) is a Lebesgue square integrablekernel. In this case, it is not difficult to recognize that the operators dPTdP,ldPTPs, and dPTW are well-defined. In particular, they are described asfollows"

f dPT dPx (t)= g(t)x(t),

dPrpsx (t) k(t, s)x(s) ds,

derWx (t) g(t)x(t) + k(t, s)x(s) ds.

3. Some causality properties. In the sequel it will be supposed that T is abounded operator defined on [H, pt]. As proposed by Porter [22] and Saeks 25],T will be called causal (anticausal) if PtTy PtTy2, whenever Uyl pry2((I Pt)Ty (I U)Ty2, whenever (I Pt)y (I e’)y) for all Yl, Y2c [H, U] and c v. T is memoryless if it is simultaneously causal and anticausal.T is strictly causal if T ffa dPTW.2

For later use and to gain some familiarity with these concepts, some causalityproperties are presented.

LEMMA 3.1 [25]. The following statements are equivalent" T is causal; T

?I dPTW; PT ptTpt.LEMMa 3.2 [9]. A necessary and sufficient condition for a linear and causal T

to be strictly causal is that dPTdP O.The proofs of Lemmas 3.1 and 3.2 are a direct consequence of the definitions

of causality and integral, and will be omitted for brevity. A partial illustration ofthe techniques involved in these proofs is given by the proof of the following result.

THEOREM 3.1. If T is linear and strictly causal and T2 is causal, then T2T andT1T2 are also strictly causal.

Proof By the definition of strict causality and Lemma 3.1, the followingrelations hold"

TI adPT1P and T2 fi dPT2P.From Lemma 3.2 and the definition of integral, these relations imply that, givenany e > 0, it is possible to find two partitions, f’ and f;’ of v, with the property thatfor all partitions f such that f {0, 1, "’", N} ->_ f f’ U f;’, the follow-ing holds"

N

k=l

This definition is conceptually identical to that proposed in [25] and is more restrictive than thatconsidered in [9] or the definition of strong causality used in [31]. In our terminology, a delay time in

Lz[a, b] is strictly causal only if a and b are finite.Dow

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540 ROMANO M. D SANTIS

where

N

AP(,)T1P’-k=l

andN

2 AP(,)T2P’.k=l

It follows that

Hence

Zx Z= Zl Z=l IZlZ2 Zl Zl + Z Z= TIT21 (1%.1 + IZll +where (by virtue of the linearity of T1)

N

1 AP()T1T2P’-k=l

< lZ21,

These last two equations imply that T1T2 r dPT1T2Ps. Hence T1T2 is strictlycausal. A similar argument applies for T2 T

COROLLARY 3.1. If T1, T2,..., T, are linear causal operators, and To isstrictly causal for some o {1, 2,..., n}, then the operator T1T2... T, is alsostrictly causal.

Example 3.1. Consider in [L2[0 ), pt] the operators T1, T2, T3, T4 definedas follows: if Yi Tx, 1, 2, 3, 4, then:

Yl(t)- Nl(X(t))

where Nl(. is a bounded Lipschitz continuous real function;

Y2(t) 2 g.x(t- At.),n=0

where Ato > 0, At. > At._ 1, and = g. < oo

Y3(t) k(t r)x(t)dr,

where

Y4(t) h(t)x(t),

where h e L[0, ) and lim,oo ess sup [h(t)l 0.It is easy to verify that for each x e L20 oo) and e 0, oo) the following

relations hold"

P’TlX P’TIP’XP’T2x P’T2P’xP’T4x P’T4P’x4

and (I P’)Tlx (I P’)TI(I P’)x;

P’Tax p’TaP’x

and (I- P’)T4x (I- P’)T(I- P’)x.It follows then that T and T are simultaneously causal and anticausal, hence theyare memoryless T2 and Ta are simply causal.D

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Observe that while none of the operators T T2, T3 and T4 is strictly causal,the composition T4T2 gives an example of a strictly causal operator. To prove thisfact, it is sufficient to show (see Lemma 3.2) that given any e > 0, a partition f ofv [0, oe can be found with the property that, for any other partition f’ ;,’1, ’n’} "e, the following holds’

AP(’i)TTAP(I) <= .i=1

This is easily done by choosing an element [0, ) such that

I(I v)r,,(Z V)l <= e/(2lr2l)

and by verifying that every partition

a {0,,,,..., ,}such that ]i i-] < Ato for all 1, 2, ,.., n, enjoys the desired property.

Arguments similar to the above would also show that T2 T, TT and T4 T3 areall strictly causal operators. Moreover, it is interesting to note that, for everypermutation (il, i2, i, i) of (1,2, 3, 4), the operator T/, T/T/T/ is also strictlycausal.

4. Existence and causality of (I + T)-1. The concepts of causality, strictcausality and memorylessness will now be supplemented with the concept of weakadditivity. 3 This is done via the following definition and the two subsequentlemmas.

DEFINITION. T is called weakly additive if Tx Tptx -Jr- T(I Pt)x holds forevery x e H, and all e v.

LFMMA 4.1 [6, p. 114]. The following operators are weakly additive:-everylinear operator; every memoryless operator; the linear combination of weaklyadditive operators;the composition TaT1, where T is weakly additive and T2 is

linear; the composition T T1, where T is memoryless and Ta is weakly additive.LFMMA 4.2. If T is weakly additive and causal, then for every pair p1, p).

e {pt} one has the following operator identity:

(P P)T (P p1)Tp1 + (P P1)T(P p1).

We are now in a position to state and prove the first fundamental result of thepaper.

THEOREM 4.1. Suppose that T T1NT, where:(i) N, is a causal, weakly additive, bounded and Lipschitz continuous operator;(ii) T and T are linear, bounded and causal operators;(iii) either T1, or T2, or T2T is strictly causal. Then (I + T) is invertible and

its inverse is causal, bounded and continuous.

Proof. First, it will be shown that it is sufficient to prove the theorem in thespecial case where T coincides with the identity and T is strictly causal. Indeed,if the theorem is valid in this special case, then from Theorem 3.1, one would

In a system theory context, the concept of weak additivity has been explicitly exploited by earlierauthors. In particular, it has been adopted by Zadeh [32], and, more recently, by Gersho [13].D

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542 ROMANO M. DE SANTIS

obtain the following :if T1 and T2 satisfy (ii) and (iii), then the operator (I + T2 TINI)is invertible and (I + T2T1N,)-1 is bounded, causal and continuous. This latterresult implies the validity of the theorem in the general case. To see this, note thatthe invertibility of (I + T) is equivalent to the property that for every elementy H there exists one and only one x H such that

(2) y X + Z1NlZ2X.

This, in turn, is equivalent to the existence of a unique element Tzx H such that

(3) T2y Z2x -b Z2 ZlN Z2XBut, from the invertibility of (I + T2T1NI) one has that the desired T2x exists andis given by

(4) T2x (I + T2 T1NI) -1T2y.

It follows that the solution of equation (2) also exists, is unique and has the pro-perty that

(5) x y T1NI(I -4- T2 T1NI) -1T2y.

The operator (I + TI NIT2) is then invertible and its inverse is given by

(6) (I + TNIT2)- I T1NI(I + T2T1NI)-1T2.This last equation shows also that (! + T1NtT2)-1 can be expressed by the sumand composition of causal, bounded, and continuous operators, and therefore(I + T1NIT2)-1 is itself causal, bounded, and continuous.

It remains then to show that the theorem is valid in the case where T isstrictly causal and T2 !. To verify the invertibility of (I + T) in this specialcase, one has again to show that for every y H there exists a unique x H suchthat

(7) y x + Tx.

From Lemma 3.2 one has dPT dP 0. This implies that there exists a partition{to 0, 1’ 2, N t} v such that

AP(i)TAP(i)II < 1/IIN, II,

that is,

AP(,)T1AP(,)II < 1 ! U

for each 1, 2, ..., N. Using Lemma 4.2, and the fact that N is memoryless, itfollows that

(8) IIAP(i)T1NIAP(i)[I [IAP(i)TAP(g)II <

for each 1, 2,..., N.Observe now that solving equation (7) is equivalent to finding an x H

such that

(9) AP(i)y AP(i)x + AP(i)TP’x,Dow

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where 1, 2, ..., N. Indeed, if x 6 H satisfies (7), then, from the causality of T,AP(i)x must clearly satisfy (9). Conversely, suppose that the element x H issuch that AP(i)x does satisfy (9). Then it would follow that

N N N

AP()y AP()x + AP(i)TP’x,i=1 i=1 i=1

and this equation coincides with (7).For 1, equation (9) becomes

(10) AP(I)y AP(I)X -- AP(I TAP(I

But from (8) one has that AP(I)TAP(I) the restriction of T to the Hilbert spaceAP(1)H, is Lipschitz continuous and has a Lipschitz norm smaller than 1. Apply-ing Lemma 4.3 (given below), there exists then a causal operator K such that theelement

(11) AP(I)x K 1AP(I)Y

is the unique solution of equation (10).For 2, equation (9) becomes

(12) AP(2)y AP(2)x + AP(2)TP2x.Noting that T is weakly additive (see Lemma 4.1), and using Lemma 4.2, this lastequation can be rewritten as follows:

aP(2)y- AP(2)TAP(I)X AP(2)x + AP(2)TAP(2)x,

where, once again, AP(2)TAP(e) is Lipschitz continuous and has a Lipschitznorm smaller than 1. Lemma 4.3 can then be applied again and there exists acausal operator K: which provides the following unique solution to (12):

AP(2)x K2[AP(2)(y- TAP(I)X)]

where AP()x is defined by (11). By induction, having computed AP()x, AP(2)x,.., AP(_ 1)x, the unique solution to (9), AP()x, can be computed as follows:

(13) AP(,)x=K, AP(,) y- T AP(j)xj=l

where K is a causal operator.The above recursive relations define the element x 7= iP(i)x, and this

element is the unique solution to (7). It can then be concluded that (I + T) isindeed invertible.

With regard to the causality of (I + T)-1, it is sufficient to show that forevery v and all y H, the following holds (see Lemma 3.1):

(14) U(I + T)- y pt(l + T)- 1pry.

Consider the partition f’ of v given by

’-’ {to 0,1,2, "’’, i-l,t,i, "’’, N- t},Dow

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where {0, 1, "’", ,N} is the partition f considered in the first part of the proof,and it has been assumed, without any loss of generality, that i-1 < < i. Usethe following notations"

x (I + T)-yl x2 (I + T)-y2

where yl y and y2 pry. From equation (13) the following relation holds"N

xq (I + T)-yq= K2[Ap(2)(yq- Tp-xq)-Ij=l

+ Kt[(Pt- P’-’)(yq-

+ K[(p, nt)(yq_ Tnxq)],

where q 1, 2. By inspection, from this equation it follows that

p,x px2 p2x1 p2x2 p,-,x p-,x2 ptx ptx2

This implies the validity of equation (14). The proof of the boundedness and con-tinuity of (I + T)- can be obtained in a similar way (the missing details can befound in [6]).

LEMMA 4.3 [6, p. 131]. If T is a causal, bounded and Lipschitz continuous

operator with Lipschitz norm less than 1, then (I + T) is invertible and its inverse is

causal, bounded, and Lipschitz continuous.Example 4.1. Consider on [L2[0, ), P’] the operator T given by the composi-

tion TT(T3 + Te), where T, T2, T3 and T4 are the operators defined in Example3.1. Note that T3T and T2T are linear and strictly causal, hence T3T + T2T isalso linear and strictly causal. Moreover, T is memoryless and Lipschitz contin-uous. It follows then that T satisfies the hypotheses of Theorem 4.1, and con-

sequently (I + T) is invertible and its inverse (I + T)- is causal, bounded andcontinuous.

It is of interest to observe that the statement of Theorem 4.1 is automaticallyvalid when T is linear and strictly causal. In this special case, however, that resultcan be further strengthened as follows.

THEOREM 4.2 [6, p. 141]. If T is a linear bounded and strictly causal operator,then (I + T) is invertible and its inverse is causal and bounded. Moreover, (I + T)-can be computed by the following Neumann series"

+ W)-’= + En=l

where T T and T"+ is given by the composition of T" with T.This theorem can be proved by applying techniques similar to those already

used in the case of Theorem 4.1. The proof is based on the following well-knownspecialization of Lemma 4.3 and is omitted for brevity.

LEMMA 4.4 [6, p. 140]. If T is a linear causal bounded operator with norm lessthan 1, then (I + T) is invertible and its inverse is causal and continuous. Moreover,

4 In the case where T is also completely continuous, this theorem provides the Gohberg and Krein

result remarked on in the Introduction.Dow

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(I + T)-1 can be computed by the following Neumann series"

(I+ T)-’=I+ (-1)"T".n=l

When T is not strictly causal, Theorems 4.1 and 4.2 cannot be directly applied.A number of variations on these theorems are, however, available. This is illus-trated, for example, by the following two theorems.

THEOREM 4.3. Suppose that the operator K is given by the sum K + T,where satisfies the hypotheses of Theorem 4.1, and T is a linear bounded causaloperator with the property that (I +. T)- exists and is causal. Then (I + K) isinvertible and its inverse is causal, bounded and continuous.

THEOREM 4.4 [6, p. 141]. Suppose that the operator K is given by the sumK + T, where satisfies the hypotheses of Theorem 4.2 and T is a linearbounded causal operator with norm less than 1. Then (I + K) is invertible and itsinverse is causal, bounded and continuous. Moreover, (I + K)- can be computed bythe following Neumann series"

(I + K)-I= I +n=l

5. Additional results on the causality of (I + T)-1 When an operator T ona Hilbert resolution space [H, pt] satisfies the hypotheses stated in Theorems 4.1or 4.3, then those hypotheses are also satisfied for every operator of the familyPsTPS, where too -s e v. Unfortunately, the converse of this statement is notnecessarily true. Indeed, in many system problems of interest it occurs that theoperators psTps, too s v, satisfy the hypotheses of Theorems 4.1, 4.3, while theoperator T does not. Under this assumption, nothing can in general be said aboutthe invertibility of (I + T). The next result establishes, however, that if (I + T) isinvertible, then its inverse is causal. 6

THEOREM 5.1. Suppose that T is a causal operator on [H, U] and that for eachs v, s too, PTP satisfies the hypotheses of Theorem 4.1 (or Theorem 4.3). If(I + T) is invertible, then its inverse is causal.

Proof. Given any y eH, consider the element x (I + T)-y. Clearlyy x + Tx and for each s v, one has that

Py Px + PSTx.

Using the causality of T, one finds that

Py Px + PTPx.

The proof of this result is contained in the proof of Theorem 6.2.It is noted that the causality of T is not sufficient to establish the causality of (I + T)-2. In this

regard, some illustrative counterexamples can be found in [5] and [30].Dow

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But by hypothesis, PsTPS satisfies the conditions of Theorem 4.1 (or Theorem 4.3).This implies that (I + PsTP) is invertible in PH, and therefore

Px [P + PTps] Py.

From this last equation one obtains that

Ps(I + T)-I= [e + psTps]-1psy ps(i + T)-lesy,

and, by Lemma 3.1, it follows that (I + T)- is causal.Example 5.1. Consider in [L2[0 o(), pt] the operator T given by T T3

or T T1T3, where T and T3 are the operators defined in Example 3.1. Theoperator Tis not necessarily strictly causal and in general neither Theorem 4.1 nor4.3 can be applied. Indeed, many instances can be mentioned where the operator(I + T) is not invertible. Theorem 5.1 says, however, that if (I + T) is invertible,then (I + T)- is causal. This result can be applied because, as it is easy to verify,the restriction of the operator T3 to L2[0 s),Pt] is strictly causal for everys e [0, oe), and as a consequence the restriction of T to pS[L2[O, ), pt] satisfiesthe conditions of Theorem 4.1.

In a number of other interesting situations, it may happen that while T isstrictly causal, neither T nor PsTps satisfies the hypotheses of Theorem 4.1 or4.3. As in the previous case, the invertibility of (I + T) cannot in general beascertained. However, if (I + T)-1 is known to exist, then its causality can beestab.lished.

THFOrtFM 5.2. Suppose that T is a strictly causal bounded operator and that(I + T)-1 exists and is Lipschitz continuous. Then (I + T)-1 is causal.

Instead of proving this theorem, a slight generalization of it will be proved.This also will provide an opportunity to illustrate the techniques necessary toex.end Theo,’ems 4.1 and 4.2 respectively into Theorems 4.3 and 4.4. The generaliza-tion ii-; qaestion is the following.

THEOREM 5.3. Suppose that T satisfies the following conditions:(a) T Tc + Tc, where Tc and Tc are respectively strictly causal and causal

operators.(b) Tc is bounded and Tc is Lipschitz continuous.(c) Tc is weakly additive and has Lipschitz norm less than 1.(d) (I + T) is invertible and (I + T)-1 is Lipschitz continuous.

Then (I + T)-1 is causal.Proof. Note first that, since Tc is strictly causal, there exists a sequence of

operators

N

In‘= AP(j)Tc_P-’j=l

such that {In’} Tc, and {In’+ Tc} T. Moreover, for each operator(In’+ Tc)and any element y e [H, pt], the equation

y= x + In’x + TcxDow

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has the unique solution x defined by the following recursive relations .v

AP(,)x AP( 1)(1 + Tc)- 1AP( 1)Y,

An(2)x AP(2)(I + Tc)-lAn(2)y- AP(2)TcAP(I)x- An(2)In,An(l)x,

An(j)x AP(j)(I + Tc)-1An(j)y- An(j)TcnCJ-’x

AP(N)x AP(N)(I + Tc)-lAn(u)y- An(u)Tcn’-’x An(u)In’p’-’x.

These equations clearly imply that (I + In’ + Tc) is invertible and its inverse(I + In’ + Tc)-1 is causal. In view of Lemma 3.1, to prove that (I + T)-1 iscausal, it is then sufficient to show that if y is. any element of [H, P’], then thefollowing relation holds"

(15) l(I +in,+ Tc)-ly_(I+ T)-ly[O.

Indeed, if this relation holds, then it would follow that

{U(I + In’+ Tc)-ly} P’(I + T)-

{U(I + In’+ Tc)-1p,y} pt(i + T)-

and, since from the causality of (I + In’ + Tc)- one has that

pt(i + in,+ Tc)- y pt(i + in,+ Tc)- 1pry,

one would also have that

P’(I + T)- y U(I + T)- 1p,y.

Suppose that equation (15) is not true. Then there would exist a positive realand {T,}, a subsequence of {In‘ + Tc}, such that

(16) I(I + T,)-ly_ (I + T)-ly > e.

To see that this is impossible, denote {(I + T,)-ly} and (I + T)-ly respectivelyby {x,} and x, and observe that

(17) y=x,,- T,,x,,=x- Tx.

Note also that from the boundedness of (I + T)-1, it follows that (I + T,)-1 isuniformly bounded, and consequently the sequence {x,} is also uniformly bounded,

These relations follow from the hypothesis that the Lipschitz norm of Tc is less than and Tc isweakly additive, plus the application of Lemma 4.3.D

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that is,

(18) Ixnl--I(I + Tn)-lyl -<- My,

where M is a positive number conveniently chosen. 8

Consider now the sequence {Yn} given by y. (I + T)xn and observe thatfrom (17) and (18) one has that

(19)--I(T- T,)x,I <= MIT- T,Iy.

Moreover, clearly one has also that

(I + T)-ly- (I + T)-ly, (I + T)-y- (I + T,)-ly.

From this equation and (16) it follows that

I(I / T)-

Applying (19), one obtains

I(I / T)- y (I + T)- ynl

lYBut, as the sequence T,} converges to T, IT T,I can be as small as desired.

As a consequence, the last equation implies that (I + T)-1 is not Lipschitz con-tinuous, and one obtains a contradiction to hypothesis (d).

6. Applications. Potential applications for the ideas and results developed inthe previous sections can be envisioned in a number of engineering areas, suchas system sensitivity, stability, game theory, optimal control, communicationtheory and others. In this section we shall briefly illustrate how some of theseapplications can be realized in a system sensitivity and stability theory context.

Consider the systems represented in Figs. and 2, where the blocks G, P, Mare described by linear bounded and causal operators on the Hilbert resolution

FIG. 1. The system considered in the sensitivity problem

By the boundedness of (I + T)- 1, there exists a positive real m such that for every x [H, pt]the following relation holds"

I(l +2m.

Ixl

On the other hand, one can always choose an integer N such that n >= N implies IT- TI -< m. Forn > N one has then that

I(I + r)xl I(I + T)xl_ I(T- r)xlI11 Ixl Ixl

It follows that I(I + T,)xl Ixl and therefore I(I + T,)-ll 1/m M.Dow

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FIG. 2. A feedback compensation scheme for system sensitivity improvement

space [L[0, oe), pt] and r/is an element of [L[0, c), U]. In a typical sensitivityproblem, P represents a given physical plant, and G and M are two compensatorswith the property that the compensated system (Fig. 2) is input-output equivalentto the original system (Fig. 1). This input-output equivalence can be obtained, forexample, by choosing G I + MP. The question of interest is then to determineM in such a way that the compensated system has a sensitivity with respect to the"perturbation" r/which is better than that of the original system.

While a proper review of philosophy, motivations and results related to thissensitivity problem is well beyond the scope of the present discussion, here itwill be sufficient to consider the following widely adopted sensitivity criteria (see,for example, Anderson and Newcomb [1], Porter [21], [23], and Zahm [33]).

Sensitivity reduction criteria (Zahm [33, p. 51 ]). The sensitivity of the compen-sated system is better than the sensitivity of the original system if the followingconditions are satisfied"

(i) 5 (I + PM)-1 is a well-defined operator in Lz[0,(ii) 5e is a causal operator;

(iii) (x, x *x) >__ 0 for every x e L[0,The above system sensitivity criteria reduce a good portion of the sensitivity

problem to an invertibility and causality problem. The development ofthe previoussections can then be used to gain insight into the structure of sensitivity reduction.The following theorem illustrates the types of results which are obtainable in thisregard.

THEORE 6.1. If either the plant P, or the compensator M, or PM, is strictlycausal, then the first two conditions for sensitivity reduction are satisfied and thethird condition becomes"

Re (x, (I + PM)-PMx) >[(I + PM)-IPMxl 2 2’

where x is any element in Lz[0,Proof Applying Theorem 3.1, the operator PM is strictly causal, and there-

fore, from Theorem 4.2, (I + PM) is invertible and (I + PM)- is causal. More-over, (I + PM)- is given by the following expression:

(20) (I + PM)- I + (-1)"(PM)".n=l

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The third condition for sensitivity improvement then becomes

(x,x 6’6x) (x, x) (I + PM)-ix, (.I + PM)-ix) >= 0,

and from equation (20),

(x,x)- + (-1)"(PM)"x,x+ (-1)"(PM)"x __>0.n-1 n=l

This leads to

(21) -2 Re (- 1)"(PM)"x, (-1)"(PM)"x2 >__ 0.n=l

Observe now that

(- 1)’(PM) (I + PM)-PM,

and therefore equation (21) becomes

Re ((I + PM)- PMx, x) >__ 1/21(I + PM)- PMxl.If, with Damborg [5, p. 32], we interpret Re (x, Tx)/lTxllxl and ITxl/Ixl as

the phase and gain respectively of T, then the above result can be paraphrased asfollows.

THEOREM 6.1. If either P, or PM is strictly causal, then a necessary and sufficientcondition for sensitivity improvement is that the ratio between the phase and thegain of (I + PM)-lpM be bigger than or equal to 1/2.

Let us now turn our attention to the feedback system of Fig. 3 where Kis a bounded, continuous and causal operator on [H, pt]. We will say that thisfeedback system is (bounded input-bounded output) stable if it has the property

FIG. 3. The feedback system considered in the stability problem

that in correspondence to any input y e H, the output x is a well-defined elementof H and the input-output mapping is causal bounded and continuous. Thisdefinition of stability is consistent with that used in the normed space stabilityapproach developed by Damborg [5] and Willems [30]; a discussion of its con-nections with the more classical definition used in the extended space stabilityapproach (see, for example, Zames [35]) goes beyond the scope of the presentdevelopment and can be found in [3] and [31]. Note that these latter referencesalso consider the case where K is unbounded and not necessarily defined in allof H. For our purpose it is sufficient to recall the following result.D

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LEMMA 6.1. Necessary and sufficient conditions for the basic feedback systemto be stable are that (I + K) be invertible and (I + K)-1 be causal, bounded andcontinuous.9

This lemma allows us to view most of the development of the previous sectionsin a stability context. The full exploration of this connection is beyond the intendedscope of this paper. However, we shall state a single theorem and corollary whichfoster conceptual insight.

THEOREM 6.2. Suppose that T is a linear operator and that the basic feedbacksystem is stable for K T. Then the basic feedback system is also stable for K

(T + ), where denotes an operator satisfying the hypotheses of Theorem 3.1.Proof. By Lemma 6.1, it will be sufficient to show that (I + T + ) is invertible

and that (I + T + )-1 is causal, bounded and continuous. To this purposeobserve first that if the operator [I + (I + T)- 1] is invertible, then (I + T +is also invertible and

(22) (I + T + )-1 ii + (I + T)-1-1(I + T)-I.

Indeed, suppose that [I + (I + T)-I is invertible. Then for every ythere would exist an x such that

x- [I / (I / T)-l]-l(I / T)-ly.

It would then follow that

(I + T)-ly x + (I + T)-lyand therefore

y=x + Tx + Tx.

This last equation would imply the invertibility of (I + T + ) and the validityof equation (22).

At this point the proof can be completed by showing that [I + (I + T)-is in fact invertible and that[/+ (I + T)- 1]- iscausal, bounded and continuous.This is readily done by observing that (I + T)- is (by hypothesis and Lemma 6.1)causal, bounded and continuous and by verifying that (I + T)-I satisfies theconditions, of Theorem 4.1.

COROLLARY 6.1. IlK is given by the composition ofa weakly additive, boundedand Lipschitz continuous operator with a linear, bounded and strictly causal operator,then the basic feedback system is stable.

7. Conclusions. The primary results of this paper are embedded in Theorems4.1 and 5.2, and emphasize the importance of the concept of strict causality inconnection with questions of existence and causality of an inverse system. Inparticular, Theorem 4.1 states that the strict causality of a system Tplus some otherreasonable conditions are sufficient to insure existence and causality of (I + T)- 1.Theorem 5.2 considers more relaxed conditions and establishes the causality of(I + T)- when this system exists and is Lipschitz continuous. These results canbe extended in various directions, and some examples of these extensions are

In a framework slightly different from that adopted here, this result can be found either in [5]or in [31]. In [8] it is shown that the proof used by these two authors is also applicable to the presentHRS context.D

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provided by Theorems 4.3, 5.1 and 5.3. Useful specializations are also possible.For instance, in the case of linear systems, Theorem 4.1 leads to Theorems 4.2and 4.4 which offer substantial generalizations of a well-known Gohberg andKrein result.

Finally, some connections of the theory to system sensitivity and stabilityproblems have been pointed out. In this regard, Theorem 6.1 illustrates the rele-vance of the concepts of gain and phase shift in connection with sensitivity theory.Theorem 6.2 provides formalization to the conceptual connections betweencausality, strict causality and weak additivity on the one hand, and the stabilityof a basic feedback system on the other.

Acknowledgment. This paper has benefited from a number of suggestionsand comments by Professor William A. Porter of the University of Michigan.

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[2] F. E. BROWDER, The solvability of nonlinear functional equations, Duke Math. J., 30 (1962),pp. 557-566.

[3] M. DAMBORG AND A. W. NAYLOR, Stability structure for feedback systems having unstable openloops, IEEE Trans. Automatic Control, AC-18 (1973), pp. 318-319.

[4] ---, The fundamental structure of input-output stability for feedback systems, IEEE Trans.Systems Science and Cybernetics, 1970, pp. 92-96.

[5] M. DaMBORG, Stability of the basic nonlinear operator feedback system, Tech. Rep. 37, SystemsEngineering Laboratory, University of Michigan, Ann Arbor, 1969.

[6] R. M. DE SaNTIS, Causality structure ofengineering systems, Ph.D. thesis, University of Michigan,Ann Arbor, 1971.

[7] , On some connections between causality and stability, Preprints 14th Midwest Symposiumon Circuit Theory, Denver, 1971.

[8] --., Espaces de resolution Hilbertienne et theorie de la stabilitY, Tech. Rep. EP73-R.-1, EcolePolytechnique de Montr6al, 1973.

[9] --, Causality for nonlinear systems in Hilbert resolution spaces, Math. Systems Theory,vol. 17, no. 4, to appear.

10] C. L. DOLPH aND G. J. MINTY, On nonlinear integral equations ofthe Hammerstein type, Nonlinear

Integral Equations, P. M. Anselone, ed., University of Wisconsin Press, Madison, 1964,pp. 99-154.

[11] D. L. DUTTWEILER, Reproducing kernel Hilbert space techniques for detection and estimationproblems, Tech. Rep. 7050-18, Information Systems Laboratory, Stanford University,Stanford, Calif., 1970.

[12] Y. FOURES aND I. SEGaL, Causality and analicity, Amer. Math. Soc. 78 (1955), pp. 385-405.13] A. GERSHO, Nonlinear systems with a restricted additivity property, IEEE Trans. Circuit Theory,

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Mathematical Society, Providence, R.I., 1970.[16] T. KAILATH AND D. L. DUTTWEILER, Generalized innovation processes and some applications,

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on Circuit and Systems Theory, 1971.[27] I. W. SANDnERG, Conditions for the causality of nonlinear operators defined on a function space,

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[28] M. SnINRo, A fixed point theorem and some applications, Arch. Rational Mech. Anal., 17 (1964),pp. 255-271.

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[30] J. C. WILLEMS, Stability, instability, invertibility and causality, this Journal, 7 (1969), pp. 645-671.[31] --, The Analysis ofFeedback Systems, MIT Press, Cambridge, 1971.[32] L. A. ZADEH, Optimum nonlinear filters, J. Appl. Phys., 24 (1953), pp. 396-404.[33] C. L. ZAHM, Structure of sensitivity reduction, Tech. Rep. 33, Systems Engineering Laboratory,

University of Michigan, Ann Arbor, 1968.[34] G. ZAMES, Realizability conditions for nonlinear feedback systems, IEEE Trans. Circuit Theory

CT-11 (1964), pp. 186-194.[35] ., On the input-output stability of time-varying nonlinear feedback systems, IEEE Trans.

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causality, strict causality and invertibility for systems in hilbert resolution spaces

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