Kalman–Popov–Yakubovich Lemma and the S-procedure: A ...shendongacademy.com/2019AdapContr/gusev2006.pdfKALMAN–POPOV–YAKUBOVICH LEMMA AND THE S-PROCEDURE 1769 respect are not

ISSN 0005-1179, Automation and Remote Control, 2006, Vol. 67, No. 11, pp. 1768–1810. c© Pleiades Publishing, Inc., 2006.Original Russian Text c© S.V. Gusev, A.L. Likhtarnikov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 11, pp. 77–121.

TOPICAL ISSUE

Kalman–Popov–Yakubovich Lemma

and the S-procedure: A Historical Essay

S. V. Gusev and A. L. Likhtarnikov

St. Petersburg State University, St. Petersburg, RussiaReceived January 26, 2006

Abstract—A history of two fundamental results of the mathematical system theory—the Kal-man–Popov–Yakubovich lemma and the theorem of losslessness of the S-procedure—was pre-sented. The studies directly concerned with these statements were reviewed. The recent pub-lications using the theorem of losslessness of the S-procedure to derive the Kalman–Popov–Yakubovich lemma and its generalizations were analyzed.

PACS number: 02.30.Yy

DOI: 10.1134/S000511790611004X

1. INTRODUCTION

1.1. On the Subject of this Essay

This essay is devoted to the studies concerned with the two subjects mentioned in the title intowhich V.A. Yakubovich made a vital contribution. The first result is called the Kalman–Popov–Yakubovich lemma, the second, the theorem of losslessness of the S-procedure. Both won generalrecognition as the fundamental statements reflecting the basic principles of the mathematical systemtheory.

It is generally agreed by the publishers that the mathematical books (and the main part oftheir content) become a thing of the past in twenty-five years on the average, and the scientificpapers, approximately in five years. Stated differently, the resolved problems become forgotten or,on the contrary, developed over these periods. By paraphrasing a little the figurative expression ofP. Halmos, we emphasize that the fifty-year-old papers can be full of life, whereas some books diein childbed.

The two results under consideration soon will reach their semi-centennial age. As will be shownbelow, today they are full of life as never before. Time passes, notations of the theorem formulationsare modified, new fields of application and problems are considered, but the results themselves cometo be in even greater demand as the tools for tackling the control theory problems. It comes as nosurprise that these two theorems live so long. As can be seen even from the simplistic understandingof their formulations, both the theorem of losslessness of the S-procedure and the Kalman–Popov–Yakubovich lemma have a clear algebraic nature and pertain to the fundamental mathematicalfacts as the statements about the linear inequalities for quadratic forms. A recent derivation ofthe Kalman–Popov–Yakubovich lemma from the theorem of losslessness of the S-procedure seemssurprising. For a long time these two theorems lived, figuratively speaking, as good neighbors, andafter so many years it became known that they are also kinsfolk.

The present authors encountered two great obstacles prior to writing this paper. The firstobstacles lies in the fact that they are not science historians whose position implies a much widerunderstanding of the scientific context of the field of studies under discussion. Our resources in this

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KALMAN–POPOV–YAKUBOVICH LEMMA AND THE S-PROCEDURE 1769

respect are not too large. That is why this paper is rather an essay of the development of the well-known achievements of our teacher whose works served as textbooks for us. The present writerscould learn when they read the papers of Yakubovich that are cited in what follows, and whenattended his lectures, and when they had the honor and privilege to be his coauthors. Preparationof the present text served as a continuation of this process when the related works of differentauthors written at different times were read anew.

Afterwards, the second obstacle manifested itself as a contradiction when the draft of the paperwas prepared and the publications of other authors were listed. The point is that the two resultsmentioned in the title have an incredible number of applications in diverse fields of the mathematicalsystem theory. In all their attempts to cover as much as possible of the history, facts, and results,the present authors kept in mind the origin of their coordinate system and did not digress very muchfrom the domain of activity of their protagonist, Yakubovich. At the same time, it was plannedto compile a sufficiently full review of the works where these theorems are used as the tools forsystem studies. If we could succeed in this problem, the list of references would occupy the entireissue leaving no space to this and other papers. Therefore, we have to admit our boundedness andmake our apologies to those authors whose publications got only a brush treatment and were notmentioned at all because we had no time to consider them with due diligence.

1.2. How to Discriminate the Lemma from Its Corollaries and Distant Counterparts

The Kalman–Popov–Yakubovich lemma is so popular now that its corollaries and even distantcounterparts often get its name. This hinders understanding by the readers, especially by thosewho familiarize themselves with a wide range of works where the lemma is applied to the problemsfrom various areas of the mathematical system theory. To avoid alternative versions, we propose asimple criterion for “identification” of the lemma. The lemma makes use of the

(a) the so-called “frequency” inequality involving the parameter which in applications usuallyhas the sense of frequency and varies on some set of the complex plane,

(b) the matrix inequality, and(c) the matrix equation called the Lur’e equation.In what follows, by the Kalman–Popov–Yakubovich lemma is meant the assertion of equivalence

of the statements: the frequency inequality is satisfied for all permissible values of the parameter, thematrix inequality is solvable, and finally, the matrix Lur’e equation is solvable. Precise formulationsare given in what follows.

1.3. Circumstances of Lemma’s Origin, First Proofs, Names

The Kalman–Popov–Yakubovich lemma which was first formulated and proved in [1] where itwas stated that if the strict frequency inequality are satisfied, then (a) and (b) are equivalent. Thecase of nonstrict frequency inequality was considered in [2] where its relation to solvability of theLur’e equations was established. Both papers considered scalar-input systems. The constraint onthe control dimensionality was removed in [3, 4].

This result occurs in the literature under the names of the Yakubovich lemma [2], Kalman–Popov–Yakubovich lemma [5, 6, and others], or Kalman–Yakubovich–Popov (KYP) lemma. Spe-cial cases of this assertion are known as the positive real lemma and the bounded real lemma.Yakubovich calls this result the “frequency theorem.” In this paper we make use of the denomina-tion “Kalman–Popov–Yakubovich lemma” where the names follow in alphabetical order.

AUTOMATION AND REMOTE CONTROL Vol. 67 No. 11 2006

1770 GUSEV, LIKHTARNIKOV

1.4. Areas of Lemma Application

The Kalman–Popov–Yakubovich lemma and S-procedure stem from the studies of stability ofthe nonlinear control systems. The series of publications of Yakubovich [7–24] that followed theproof of the lemma and were devoted to stability, instability, and oscillations of the nonlinearsystems defined to a large extent the path of future studies in this domain. The results obtainedwere reviewed in [25–31].

The close relation between the Kalman–Popov–Yakubovich lemma and the problems of linearquadratic optimization was first noted by Popov [4, 6]. A significant contribution to these studieswas made by J.C. Willems [32]. The relation between the Kalman–Popov–Yakubovich lemma andthe linear-quadratic optimization underlies the infinite-dimensional counterparts of the lemma [33].The works of Yakubovich [34, 35] based on the application of the Kalman–Popov–Yakubovichlemma and the S-procedure laid the groundwork for the studies of linear-quadratic optimizationunder quadratic constraints. These studies were developed in [36–39].

The Kalman–Popov–Yakubovich lemma was first used to design adaptive control in [40]. Moregeneral results were obtained in [41–43]. The development of this line of studies is reflected in[44–46]. The results are reviewed in [29, 47].

The infinite-dimensional variants of the Kalman–Popov–Yakubovich lemma [33, 48–58] are usedto study the control systems obeying different classes of the partial differential equations: par-abolic [59, 60] and hyperbolic systems of the first and second orders, equations of Euler–Bernoulli,Kirchoff, and Schrödinger, equations of oscillations of string and membrane [61–65], nonstandardRiccati equations arising in the problems with boundary control of plate oscillation damping, Ham-merstein integral with weakly singular kernel [66, 67], theory of passive scattering systems [68],bounded control operators, semigroups of the class C0 and the Pritchard–Salamon systems [69–72],distributed-parameter sampled-data systems, and approximation theory for systems with boundarycontrol [73–75]. Now, there exist reviews of this domain of lemma development [76–78].

The Kalman–Popov–Yakubovich lemma finds numerous applications in the stochastic systemtheory. It is used by the theory of representation of the stochastic processes by linear models [79–81]and the estimation theory [82]. In [83, 84] it is used to design the optimal universal controllersunder singular stochastic perturbations. It finds application in the problems of absolute stability ofstochastic systems (see the review [85] of P.V. Pakshin and V.A. Ugrinovskii under this cover). TheKalman–Popov–Yakubovich lemma is used in the H∞-optimization [86–89] and the multi-objectiveoptimization [90] where it is employed to reduce the original problem to linear matrix inequalities.

Among the applications of the S-procedure that are not concerned with the control theory, wemention its use in the problem of minimization of the sign-indefinite quadratic form on a quadraticsurface—ellipsoid, in particular [91]. This minimization is the main operation in the so-called trust-region method used in the problems of global optimization (see the review [92]). The S-procedureallows one to demonstrate that in fact the traditional methods of local optimization determine theglobal minimum. This result established in [91] was rediscovered more than once [93–95]. A widerange of issues concerned with the S-procedure is discussed in the reviews [96, 97].

1.5. Relation between the Lemma and the Theorem of S-procedure

The Kalman–Popov–Yakubovich lemma and the S-procedure appeared as two mutually comple-menting methods for studies of the absolute stability problems [3]. And today the S-procedure andthe Kalman–Popov–Yakubovich lemma often adjoin in applications as two most important tools ofproblem solution.

It deserves noting that these results initially appeared as statements of absolutely differentnature. The Kalman–Popov–Yakubovich lemma traditionally is considered as an algebraic re-



sult proved on the basis of the factorization theorems. Since the time of the first publicationsof Yakubovich, the S-procedure was closely related with convex analysis and the duality theory.Owing to this difference, the long-continued active studies on the development and generalizationof these results were disconnected.

The first proof of the Kalman–Popov–Yakubovich lemma indicating to the intrinsic relation ofthis result with the S-procedure was established by A. Rantzer [98]. As follows from the abstractof this interesting work, its aim was to obtain a simple proof of the Kalman–Popov–Yakubovichlemma. This aim deserves attention in the light of numerous declarations about extreme difficultyof proving the lemma. It deserves noting that despite a substantial simplification of the proof, itwas not possible to avoid complaints about the difficulty of understanding it.

An important result of this work is represented by a new approach enabling one to considerthe statement of the Kalman–Popov–Yakubovich lemma as a special example of the losslessness ofthe S-procedure with many relations. This idea was used in [99] to obtain an unexpected curiousgeneralization of the Kalman–Popov–Yakubovich lemma to the case where the frequency conditionis defined over a finite frequency interval, rather than over the entire imaginary axis. This approachwas used in [100–102] to generalize the Kalman–Popov–Yakubovich lemma to the systems definedin a form implicit in the derivative.

This, to our opinion, interesting history of the studies on the fundamental and, at the same time,topical area of the mathematical system theory defines the subject matter of the present paperwhich focuses on the mathematical results directly concerned with the Kalman–Popov–Yakubovichlemma and S-procedure. The numerous important applications of these results and their relationswith other lines of research in the system theory were left out of the scope of this survey. Some ofthese matters are covered in other publications in this issue.

1.6. How This Paper is Organized

The next section presents in short the history of origination of the Kalman–Popov–Yakubovichlemma and S-procedure which is closely related with the studies of stability of the nonlinear systems.The third section presents the modern formulations of the Kalman–Popov–Yakubovich lemma forthe finite-dimensional systems. Presented is A.N. Churilov’s generalization of the lemma to the caseof the frequency condition defined over an arbitrary straight line or circumference. Additionally,the results concerning the properties of the solutions of the Lur’e equation and the linear matrixinequality were described. These properties are of great importance in the stability theory, optimalcontrol, H∞-optimization, and other applications of the lemma. Further, this section indicatesto the relation between the Kalman–Popov–Yakubovich lemma and the solution of the algebraicRiccati equation and the generalized Lur’e equations.

The fourth section is devoted to the infinite-dimensional generalizations of the Kalman–Popov–Yakubovich lemma. The historical context of lemma generalization to the distributed systems andthe main publications on the lemma since the early 1970’s till now are discussed.

The fifth section describes in brief the first results on losslessness of the S-procedure. The sixthsection is devoted to the relation between the theorems of losslessness of the S-procedure withconvexity of the image associated with the S-procedure of mapping. The results on the imageconvexity in the finite-dimensional and infinite-dimensional cases are presented. Consideration isgiven to the relation between the losslessness of the S-procedure and the Lagrange duality or theFenchel duality in some problems of mathematical programming. A case of losslessness of theS-procedure for the Hermit forms which is important for applications is considered in Section 7.A.L. Fradkov’s criterion for losslessness of the S-procedure for an arbitrary form is formulated andexamples where this criterion is met are presented. Relation is shown between the S-procedure for



the quadratic forms and the linear extremal problem on the cone of positive semi-definite matrices.The formulation of the generalized S-procedure by T. Iwasaki, G. Meinsma, and M. Fu is presented.

Section eight presents the results relating the S-procedure and the Kalman–Popov–Yakubovichlemma. The statements of the lemma are formulated in the general form corresponding to consider-ation of the systems defined in a form implicit in the derivative. Given is an extended formulation ofthe Kalman–Popov–Yakubovich lemma complemented by the assertions about the relation betweenthe lemma statements and the losslessness of the S-procedure as well as the presence of Fenchelduality in some extremal problem. The lemma is generalized to the case where the frequencycondition is satisfied over a bounded frequency interval.

2. HISTORY OF ORIGINATION OF THE KALMAN–POPOV–YAKUBOVICH LEMMAAND THE S-PROCEDURE

The history of origination of the lemma and S-procedure is concerned with the studies of stabilityof the nonlinear automatic control systems. The first results in this domain were based on the directLyapunov method [103]. In a brief article [104] published in 1944, A.I. Lur’e and V.N. Postnikovproposed a new form of the Lyapunov function for studying the nonlinear systems. Considerationwas given there to a system of indirect control whose equation in the general case is as follows:

d

dtx = Ax + Bφ(σ),

d

dtξ = φ(σ), σ = Cx + ρξ, (1)

where A ∈ Mn, B ∈ Mn,1, C ∈ M1,n. Here and below, Mm,n is the set of, generally speaking,complex m × n matrices, Mn = Mn,n. In [104], A,B, and C are some particular real matrices,n = 3. The graph of the nonlinear function φ which in [104] has the sense of the friction forces liesin the first and third quadrants. This condition can be set down as the inequality

σφ(σ) � 0.

To study system stability, consideration is given to the Lyapunov function given by

V = x∗Hx +σ∫

0

φ(ς) dς, (2)

where H ∈ HMn. Here and below ∗ stands for transposition in the real case and the Hermitconjugation in the complex case. In (2), H is the real matrix. By determining the matrix H fromthe conditions

V (x(t)) � 0 and ddt

V (x(t)) � 0 on the solutions of the system (1), (3)

and using some additional assumptions about the system, one can prove stability of the set ofsolutions of the equation V (x) = 0. The conditions for the parameters of the special third-ordersystem which provide stability of its stationary set were determined in [104]. This approach wasextended in [105] to the arbitrary-order systems admitting reduction of the system matrix to thediagonal form. Let

Λ′(H) =

(A∗H + HA HB

B∗H 0

)and G =

(0 12A

∗C∗

12CA CB + ρ

). (4)

Then, the derivative of the function V along the trajectories of the system (1) is as follows:

V ′(x) =

(x

φ(σ)

)∗(Λ′(H) − G)

(x

φ(σ)

). (5)



The inequalityd

dtV (x(t)) � 0 will be satisfied on the solutions of the system (1) if V ′(x) � 0 for

all x �= 0. When determining the matrix H that meets this condition, Lur’e replaces the inequality

V ′(x) � 0 by the equation V ′(x) = −|h(

xφ(σ)

)|2 where h = (h1, . . . , hn+1) is an unknown

vector. The equation must be satisfied for any function φ satisfying (13). Therefore, by equatingthe coefficients at the identical powers of φ, it can be represented as

Λ′(H) − G = −h∗h. (6)

If A is a Hurwitz matrix, then it follows from the solution of the Lur’e equations that there will beH � 0 such that (11) is satisfied.

Generally speaking, this condition is insufficient for judging about stability of system (1). Finalconclusion about system stability is made on the basis of additional study using specificity of thesystem at hand. This method was used in [104] to determine the conditions for the parameters ofa special third-order system which provide stability of its stationary set representing a segment ofthe straight line in the phase space of the system.

Another approach was suggested by I.G. Malkin [106]. Instead of function (5) depending on thenonlinearity of φ, he considered the quadratic form

(xu

)∗(Λ′(H) − G)

(xu

)

obtained from (5) by replacing the function φ by a free variable u and proposed to seek a matrix Hsatisfying the linear matrix inequality relative to H

Λ′(H) − G < 0. (7)

Yakubovich noted [107] that for all x �= 0 and all φ satisfying (13) the condition V ′(x) < 0 isequivalent to the matrix inequality

Gxx − HA − A∗H − (Gxu − HB)G−1uu (Gxu − HB)∗ > 0, (8)

where the matrices Gxx, Gxu, Gux, and Guu are defined by the block representation

G =

(Gxx GxuGux Guu

). (9)

The inequalities (7) and (8) are known to be equivalent. provided that Guu > 0. Hence, inequality(7) and the above condition for V ′(x) are equivalent.

As was shown in [108], satisfaction of the inequality(

A−1B1∗

)G

(A−1B

1

)> 0 (10)

is necessary for existence of H = H∗ satisfying inequality (8). We note that in [108] considerationwas given to the cases of both the real and complex matrices A,B,G,H.

If there exists a matrix H > 0 satisfying (7), then, obviously, on the nonzero solutions ofsystem (1) the corresponding function V satisfies the inequalities

V (x(t)) > 0 andd

dtV (x(t)) < 0 (11)

and guarantees asymptotic stability of system (1).



Popov proposed [109] a new method of studying system (1) for stability doing without theLyapunov function. In this method, the stability criteria are expressed in terms of frequency, thatis, in terms of the transfer function W (λ) = (λI −A)−1B of the linear part of system (1) from theinput φ to the output x. Let the matrix G(ϑ) have the aforementioned block structure, the blocksobeying the equations

Gxx = 0, Gux = ρ−1C(ϑA + I) Gxu = G∗xu, Guu = 2ϑ(1 + ρ−1CB).

Then, the Popov frequency condition is representable as follows: there exists ϑ > 0 such that forall real ω

(W (iω)

1

)∗G(ϑ)

(W (iω)

1

)� 0. (12)

As was shown in [109], if (12) is satisfied and the function φ meets the conditions

φ(0) = 0, σφ(σ) > 0 for σ �= 0, (13)

then the system (1) is global asymptotically stable.The same paper demonstrated that if the function

V (x, ξ) = x∗Hx + ξ2 + βσ∫

0

φ(ς) dς (14)

satisfies the conditions (11) for any function φ satisfying (13), then (12) is satisfied as well. Popovformulated the following problem: “If condition (12) is satisfied, is it possible to construct a Lya-punov function of the form (14)?”

Solution of Popov’s problem for the case where the nonstrict inequality (12) is replaced by thestrict inequality

(W (iω)

1

)∗G(ϑ)

(W (iω)

1

)> 0, (15)

was announced by Yakubovich [1]. A detailed proof made by Yakubovich was published in theappendix to [110].

Yet the main matter of [1] lies in the first formulation of the Kalman–Popov–Yakubovich lemmafrom which follows the solution of this problem. We cite the original formulation of this statementfrom [1] using the above notation and assuming as in [1] that G(ϑ) = G = const, Gxx = 0, Guu = 1.

For the inequality (8) to have solution H = H∗, it is necessary and sufficient that (15) is satisfiedfor −∞ < ω < +∞.

Necessity is proved by direct calculation of the left side of the inequality (15). Sufficiencyis proved by recurrent depression of dimensionality. It was also noted in [1] that necessity ofcondition (15) follows directly from necessity of condition (10) because the left part of (8) is notchanged by the replacement of A by A − iωI.

R. Kalman [2] established a relation between the Popov condition (12) and solvability of theLur’e equation (19). Let G(ϑ) = G = const, Gxx = 0, Guu = γ. Then, with our notation the resultof Kalman may be formulated as follows:Let the pair A,B be controllable and γ � 0. Then, the vector h satisfying the Lur’e equation (19)for some matrix H = H∗ exists if and only if the inequality (12) is satisfied for all real ω.



Now we describe the scheme of the proof given in [110] that if the Popov frequency conditionis satisfied with the replacement of (12) by (15), then system (1) has a Lyapunov function of the

form (14). The inequalityd

dtV (x(t), ξ(t)) � 0 will be satisfied on other-than-stationary solutions of

system (1) if V ′(x, ξ) � 0 for all x, ξ meeting (13). We follow the Lur’e approach [105] and replacethe last condition by a simpler sufficient condition

S(x, ξ) � 0,

where S(x, ξ) = V ′(x, ξ) + τσf(σ), τ � 0 is a parameter (in the work of Lur’e, τ = 1). Theintroduced function is denoted by S in [110] where Lur’e’s technique was called the S-procedure.Let us assume that τ = −2ρ−1. Then,

S(x, ξ) =

(x

φ(σ)

)∗(Λ′(H) − G(ϑ))

(x

φ(σ)

),

that is, coincides with the above function V ′(x). Therefore, if (8) with G = G(ϑ) is satisfied, then

by virtue of (13)d

dtV (x(t), ξ(t)) < 0 on other-than-stationary solutions of system (1).

Inequality (8) is satisfied by virtue of the Popov condition and the result of Yakubovich. Itfollows from the Hurwitz stability of the matrix A and inequality (8) that H > 0, which impliesthat condition (11) is satisfied and system (1) is asymptotically stable. Similar reasoning based onthe result of Kalman allows one to prove that (14) is the Lyapunov function of system (1) if thePopov condition with nonstrict inequality is satisfied.

3. KALMAN–POPOV–YAKUBOVICH LEMMA

In 1964 Yakubovich [3] and Popov [4] established generalizations of the Kalman–Popov–Yakubo-vich lemma to the case of multivariable systems. We begin with a generalization of Kalman’s resultconcerning satisfaction of the nonstrict frequency inequality and formulate it using a notationdiffering from the original notations of these publications.

We introduce the Popov matrix

Π(λ) =

(W (λ)Im

)∗G

(W (λ)Im

). (16)

Let Γ = iR be a set of purely imaginary numbers.

Theorem 1 (Kalman–Popov–Yakubovich lemma, singular case). Let the pair A,B be control-lable. Then, the following statements are equivalent for any matrix G ∈ HMn+m:

(1) The Popov frequency condition is satisfied, that is, for all λ ∈ Γ \ SpA

Π(λ) � 0. (17)

(2) There exists a matrix H ∈ HMn such that

Λ′(H) − G � 0. (18)

(3) There exists a solution of the Lur’e equation, that is, there exist matrices H ∈HMn andh ∈ Mn+m,m such that

Λ′(H) − G = −h∗h. (19)

If A,B, and G are real matrices, then the matrices H,h in (18) and (19) can be taken to be real.



We note that the additional assumption of [4] can be eliminated [6]. Moreover, [4] consideredonly the real case. In [3], consideration was given only to Assertions (1) and (2) under an irrelevantadditional assumption of Hurwitz stability of the matrix A, both the real and complex cases werediscussed, and the result of Yakubovich generalized to the case of multivariable systems.

Theorem 2 (Kalman–Popov–Yakubovich lemma, regular case).Let A be a Hurwitz matrix. Then, the following statements are equivalent for any matrix G ∈HMn+m:

(1+) The strict Popov frequency condition is satisfied, that is, there exists δ > 0 such that forall λ ∈ Γ

Π(λ) � δI. (20)

(2+) There exits a matrix H ∈ HMn such that

Λ′(H) − G < 0. (21)

If A,B,G are real matrices, then the matrix H in (21) can be taken to be real.

The following studies concerned with the Kalman–Popov–Yakubovich lemma were continuedalong several lines.

3.1. Relaxation of the Lemma Conditions

The first such attempt was made in [111] which asserted that in the singular case the Kalman–Popov–Yakubovich lemma is satisfied for m = 1 without the assumption of controllability of thepair A,B, provided that A is a Hurwitz matrix. This assertion is incorrect. It suffices to consider

the counterexample of n = 1, A = −1, B = 0, G =(

0 11 0

). It is possible to demonstrate,

however, that the assertion becomes correct if one assumes additionally that Π(iω) �≡ 0, whichfollows for example from the results of [112]. For the singular case, the conditions were relaxed in[113, 114]. We consider the following frequency conditions:

(1a) For all λ ∈ Γ, x ∈ Cn, u ∈ Cm, meeting the equation λx = Ax + bu, the inequality(

xu

)∗G

(xu

)� 0

is satisfied;(1b) For all λ ∈ Γ, (17) is satisfied and there exists λ0 ∈ Γ such that Π(λ0) > 0.We call λ ∈ C the uncontrollable eigenvalue of the pair A,B if λ is the eigenvalue of the matrix

A + Bk for all k ∈ Mm,n. The uncontrollable eigenvalue λ of the pair A,B is nondefective ifthere exists k ∈ Mm,n such that the algebraic multiplicity λ of the matrix A + Bk is equal to itsgeometrical multiplicity.

Assertions (1a) and (2) were proved to be equivalent [114] in the case where all uncontrollableeigenvalues of the pair A,B lie on the imaginary axis and are nondefective. An infinite-dimensionalcounterpart of this assertion was obtained in [115]. Condition (1b) was proved to entail Condition (2)[113] if the pair A,B has no uncontrollable purely imaginary eigenvalues.

It was proved in [27, Theorem 1.2.7.] that in the regular case the controllability condition canbe replaced by that of stabilizability of the pair A,B if the frequency condition is replaced by thefollowing condition:



(1+a ) There exists δ > 0 such that the inequality(

xu

)∗G

(xu

)� δ

(|x|2 + |u|2

)

is satisfied for all λ ∈ Γ, x ∈ Cn, u ∈ Cm satisfying the equation λx = Ax + bu. As follows fromthe results of [100] which will be discussed in detail in Section 8, Assertions (1+a ) and (2+) areequivalent without any assumptions about the matrices A and B.

In [14] Yakubovich obtained a result in a sense lying in between the statements of the Kalman–Popov–Yakubovich lemma for the regular and singular cases. We present a special case of thisassertion in order to avoid the rather bulky formulations of the original work.

Theorem 3. Let the pair A,B be stabilizable, the matrix A have no purely imaginary eigenvalues,rankB = m, Guu = 0. Then, the following assertions are equivalent:

(1◦) For all ω ∈ R Π(iω) > 0, and limω→∞ ω2Π(iω) > 0.(2◦) There exists a matrix H ∈ HMn satisfying the inequalities (18) and A∗H +HA∗−Gxx < 0.

We note that for m = 1 this result was proved in the first publication of the Kalman–Popov–Yakubovich lemma [1]. A similar assertion was obtained in [14] for Guu � 0. A detailed proof canbe found in [27].

3.2. Kalman–Szegő Lemma

When examining the nonlinear discrete-time systems for stability, construction of the Lyapunovfunction gives rise to the need for a corresponding analog of the Kalman–Popov–Yakubovich lemma.This result which was christened the Kalman–Szegő lemma was first formulated in [116] for simplyconnected systems. Its generalization to the multivariable systems is usually called the generalizedKalman–Szegő lemma or Kalman–Popov–Yakubovich lemma for discrete-time systems.

We assume that Γ = {λ ∈ C | |λ| = 1},

Λ′(H) =

(A∗HA − H A∗HB

B∗HA B∗HB

). (22)

A general formulation of the Kalman–Popov–Yakubovich lemma for discrete-time systems wasgiven in [6] in the singular case and [117] in the regular case. With our notation, these assertionscoincide with the formulations of Theorems 1 and 2. The proof is based on the linear fractionaltransformation [117] known as the Cayley transformation which enables one to reduce the general-ized Kalman–Szegő lemma to that of Kalman–Popov–Yakubovich. This method of proof was firstused in [118] for the case of scalar control.

3.3. Generalization of Churilov

By using the linear fractional transformation, Churilov [112] extended the Kalman–Popov–Yakubovich lemma to the case where Γ is an arbitrary line or circumference on the complex plane.

Let the matrix Θ =

(ϑ11 ϑ12ϑ21 ϑ22

)be Hermit and satisfy the condition detΘ < 0. We assume that

Γ = {λ ∈ C | (λ, 1)Θ(λ, 1)∗ = 0}, Λ′(H) =(

A BIx 0

)∗(Θ ⊗ H)

(A BIx 0

), (23)



where Θ ⊗ H is the Kronecker product of Θ and H. For ϑ11 = 0, Γ is a straight line, and forϑ11 �= 0, a circumference on the complex plane.

It follows from [112, 113] that Assertion (1a) implies Assertion (2) if Γ has no uncontrollableeigenvalues of the pair A,B. We note that Assertion (2) obviously implies Assertion (1) for any Aand B. It follows from the results of [102] that if the pair A,B is controllable, then Assertions (1)–(3) are equivalent, and Assertions (1+) and (2+) are equivalent to the following assertion:

(3+) There exist matrices H, h = (hx, hu), hx ∈ Mm,n, hu ∈ Mm, satisfying Eq. (19) and suchthat det hu �= 0 and Sp(A − Bh−1u hx) ∩ Γ = ∅.

Assertions (1+a ) and (2+) were shown in [100] to be equivalent for any A and B. In the case

of Θ =

(0 11 0

), Γ is the imaginary axis, and Λ′ assumes the form (4). In this case, the these

assertions coincide with the above formulations of the Kalman–Popov–Yakubovich lemma for the

continuous-time systems. In the case of Θ =

(1 00 −1

), Γ is a unit circle, and Λ′ assumes the

form (22). In this case, these assertions reinforce the aforementioned formulations of the Kalman–Popov–Yakubovich lemma for the discrete-time systems.

3.4. Existence of Solutions of the Matrix Inequalitywith the Given Spectral Characteristics

Existence of a positive definite or semi-definite matrix H satisfying (18) or (21) is of interestwhen using the Kalman–Popov–Yakubovich lemma in the problems of stability. The followingsimple assertion based on the properties of the solution of the Lyapunov equation is valid.

Assertion 1. Let A be a Hurwitz matrix. Then, H � 0 if H satisfies (18) and Gxx � 0, or H > 0if H satisfies (21) and Gxx � 0 or (18) and Gxx < 0.

We present examples of using assertion (1) and consider the system

d

dtx = Ax + Bu, (24)

y = Cx + Du,

where A ∈ Mn, B ∈ Mn,m, C ∈ Mm,n, D ∈ Mm, x is the state, and the input u and output yhave the same dimensionality. Let T (λ) = D + C(λI −A)−1B be the system transfer matrix fromthe input u to the output y. The matrix T is called positive real [119] if SpA ⊂ clC− and

T (λ) + T (λ)∗ � 0 for Reλ � 0. (25)

The matrix T is called strictly positive real if SpA ⊂ C− and there exists δ > 0 such that

T (λ) + T (λ)∗ > δI for λ ∈ iR.

The strictly positive real matrix is positive real [120].Let W (λ) = (λI − A)−1B. We determine the matrix G by assuming that Gxx = 0, Gxu = C,

Guu = D + D∗, and the Popov matrix Π(λ) by virtue of (16). Then, T (λ) + T (λ)∗ = Π(λ).Theorem 4 follows from the Kalman–Popov–Yakubovich lemma (regular case) and Assertion 1,

Theorem 4 (lemma of strictly positive real matrices). T is a strictly positive real matrix if andonly if there exists a positive definite matrix H satisfying (21). In the case under consideration,any solution of (21) is positive definite.



Other definitions of the strictly positive real matrices and the corresponding variants of thelemma differing in some conditions from the above variant can be met in the literature. A similarcriterion for real positiveness of T was obtained in [119].

Theorem 5 (lemma of positive real matrices). Let SpA ⊂ clC−, A have no multiple purely imag-inary roots, and the matrices A, B, C, and D define the minimal realization of T . Then, for ex-istence of a positive definite matrix H satisfying Eq. (19) together with some matrix h ∈ Mn+m,m,it is necessary and sufficient that T be a positive real matrix.

Another corollary of the Kalman–Popov–Yakubovich lemma related with positive definitenessof H is known as the lemma of boundedly real matrices [121]. We formulate the most popularvariant of this assertion pertaining to the case of strict matrix inequality (21) and continuous time.Let γ > 0. We define the matrix G by taking

Gxx = −C∗C,Gxu = −C∗D, Guu = γ2I − D∗D. (26)

Then, Π(λ) = T ∗(λ)T (λ).

Theorem 6. To satisfy the inequality

T ∗(λ)T (λ) < γ2I for all λ ∈ C+, (27)

it is necessary and sufficient that a matrix H > 0 satisfying (21) exist.

This assertion finds numerous applications in the problems of H∞-optimization because condi-tion (27) is equivalent to the condition T ∈ H∞(C+), ‖T‖H∞ < γ. A similar assertion was shown[121] to be true in the case of nonstrict inequalities if the matrices A,B,C define the minimalrealization of the transfer function W .

Restricted spectrum of the matrix A is a disadvantage of the above theorems. In his attempts todetermine a more convenient criterion for positive semi-definiteness of H, Willems [32] formulatedthe following assertion: if the pair A,B is controllable and for all λ ∈ C, Reλ � 0, the inequalityΠ(λ) > 0 is satisfied, then there exists H � 0 satisfying (18). Later he published a counterexample[122] demonstrating that, generally speaking, this assertion is not true. The necessary and sufficientcondition for existence of the positive semi-definite solution of inequality (18) in the frequency termswas obtained in [123].

There exists the following generalization of assertion (1). Let the curve Γ and the operator Λ′

be defined by (23). The curve Γ divides the complex plane in the open areas

Ω± = {λ ∈ C | ± (λ, 1)Θ(λ, 1)∗ > 0}. (28)

Assertion 2. Let Gxx � 0, SpA ∩ Γ = ∅ and Condition (1a) be satisfied. Then, there existsa nonsingular H satisfying (18) and having the same number of eigenvalues in the left half-plane(with regard for multiplicity) as the number of eigenvalues of the matrix A lying in the domain Ω+.

This assertion can be readily obtained from the results of [113, 124, 125]. Its variant for thesign-indefinite matrix H finds use in the studies of self-oscillations of the nonlinear systems.

Let T (λ) be the transfer function of system (25) and γ > 0, Γ,Ω±,Λ′ obey (23), (28). Wedefine the matrix G by the equalities (26). It follows from the Kalman–Popov–Yakubovich lemma(regular case) and Assertion 2 that T satisfies the conditions T ∈ H∞(Ω+), ‖T‖H∞ < γ if and onlyif there exists a matrix H > 0 satisfying (21). This result generalizes Theorem 6 and allows oneto reduce the estimation of the H∞-norm of the transfer function, which is analytical within somecircle or half-plane, to the solution of the linear matrix inequality (21).



3.5. Extremal Solutions of the Matrix Inequality

The study of the set of solutions of inequality (18) for continuous-time systems was startedin [32]. It was proved that in the case where the pair A, B is controllable and this set is nonemptythere are matrices H± satisfying (18) and such that any solution H of inequality (18) satisfies theinequality

H− � H � H+. (29)

This assertion was shown in [126] to be true for any Γ,Λ′ satisfying (23) under some additionalassumption about the matrix Π(λ). It was shown in [102] that this additional condition maybe rejected. The conditions for nondegeneracy of the matrices H satisfying (18) for which theinequalities (H+)−1 � H−1 � (H−)−1 are valid were also obtained in [127, 128].

3.6. Properties of the Solutions of the Lur’e Equation

We first consider the case of continuous time and assume that the pair A,B is controllable.It was proved in [129, 130] that if the condition (20) is satisfied, then there will be matrices H,h = (hx, hu), hx ∈ Mm,n, hu ∈ Mm, satisfying the Lur’e equation (19) such that A − Bh−1u hx isthe Hurwitz matrix.

A refined formulation of this result was presented in [32]. It was proved that for Guu > 0 theextremal solutions of inequality (18) H± together with some matrices h± = (h±x , h

±u ), h

±x ∈ Mm,n,

h±u ∈ Mm, satisfy the Lur’e equation (19). At that, if the strict inequality (20) is satisfied, thenSp(A − B(h±u )−1h±x ) ⊂ C±, where C± = {λ ∈ C | ± Reλ > 0} are the right and left open half-planes. If the nonstrict inequality (18) is satisfied, then Sp(A − B(h±u )−1h±x ) ⊂ clC±.

The result [129] was extended in [131, 132] to the discrete-time systems. The result of [32]was generalized in [126] to the case where Γ and Λ′ obey relations (23). Let the pair A,B becontrollable and (17) be satisfied. Then, there exist h± ∈ Mm,k such that the pairs H+, h+ andH−, h− satisfy (19). It was shown in [102] that if the pair A,B is controllable, then Assertions (1+),(2+), and (3+) are equivalent to the following assertion:

(3+a ) For that sign of “±” which makes the inequality ±Θ11 � 0 satisfied, there exist matricesH±, h± = (h±x , h±u ), h±x ∈ Mm,n, h±u ∈ Mm, satisfying Eq. (19) and such that

det h±u �= 0, Sp(A − B(h±u )−1h±x ) ⊂ Ω±. (30)

We explain relations (30). If Θ11 = 0, then Γ is the straight line. In this case, (30) is satisfied forboth matrices h+ and h−. If Θ11 �= 0, then Γ is a circumference. It may happen in this case that(30) is satisfied only for that matrix of h+ or h− which corresponds to the bounded domain Ω+

or Ω−.

3.7. Algebraic Riccati Equation

Let us consider the case of continuous time and assume that h = (hx, hu), hx ∈ Mm,n, hu ∈ Mm.Then, Eq. (19) is equivalent to the system

A∗H + HA + h∗xhx = Gxx, HB + h∗xhu = Gxu, h

∗uhu = Guu. (31)

Let Guu > 0. We use the second and third equations of (31) to determine

hu = G1/2uu , hx = G−1/2uu (Gux − B∗H). (32)



By eliminating hx from the first equation of (31), we obtain the algebraic Riccati equation

HPH + HQ + Q∗H + R = 0, (33)

P = BG−1uuB∗, Q = A − BG−1uuGux, R = GxuG−1uuGux − Gxx. (34)

One can easily see that with the assumption of Guu > 0 solvability of the Lur’e equation (19) isequivalent to solvability of the algebraic Riccati equation (33), (34).

In the system theory, this equation first appeared in [133] in connection with the problemsof optimal control. A vast literature is devoted to this equation (see the bibliography in themonographs [134–136]). The relation between this equation and the Kalman–Popov–Yakubovichlemma was first noted in [32] where the properties of the solutions of this equation were used toexamine the properties of the matrix inequality (18). Some of these results were presented in thelast section.

The solution H− of Eq. (33) is called stabilizing if Q + PH− is a Hurwitz matrix. Let H−

be the stabilizing solution. We determine the matrix h− = (h−x , h−u ) by means of Eqs. (31), (32).Then, Q + PH− = A − B(G−1uu (Gux − B∗H)) = A − B(h−u )−1h−x . Consequently, determination ofthe stabilizing solution is coordinated with inclusion (30).

Let us consider the matrix H =(

Q −PR −Q∗

)which is called the Hamiltonian matrix corre-

sponding to Eq. (33). One can readily see that (33) is equivalent to equation

H(

I−H

)=

(I

−H

)(Q + PH). (35)

As was shown in [137], the function ϕ(λ) defined on the imaginary axis by the equality ϕ(iω) =(det Guu)−1|det(iωI − A)|2 detΠ(iω) is a polynomial. It was shown in [138] that ϕ(λ) =(−1)n det(λI − H). It follows from the result of [137] that the polynomial ϕ is not affected bytransformations of the feedback, that is, by replacement of A by A + BK and G by T ∗GT under

an arbitrary matrix K ∈ Mm,n, where T =(

I 0K I

).

Summation provides that if the pair A,B has no uncontrollable eigenvalues on the imaginaryaxis, then the equality ϕ(iω) = 0 implies the equality detΠ(iω) = 0 and, therefore, the frequencycondition (1+) is equivalent to the following conditions:

(1+b ) Guu > 0, ϕ(iω) �= 0 ∀ω ∈ R;(1+c ) Guu > 0, SpH ∩ iR = ∅.It follows that if the pair A,B is stabilizable, then Eqs. (33) and (35) have a stabilizing solution

if and only if H has no purely imaginary eigenvalues.The algebraic Riccati equation (33) is encountered in many sections of mathematics without

relation to the Kalman–Popov–Yakubovich lemma . In the general case, Eq. (33) is considered forarbitrary P,R ∈ HMn, Q ∈ Mn. The case where the inequality P � 0 is satisfied is the beststudied. It is the case of the equation obtained from the Lur’e equation. In this case, the generalresult on solvability of the Riccati equation was obtained in [139].

Theorem 7. Let P � 0 and the pair of matrices Q,P have no uncontrollable eigenvalues thatare symmetrical relative to the imaginary axis. Then, the following assertions are equivalent:

(R1) For all ω ∈ R such that det(iωI − Q) �= 0, the inequality

P + P (iωI + Q∗)R(iωI − Q)P � 0

is satisfied.



(R2) There exists H ∈ HMn satisfying the nonstrict Yakubovich inequality

HPH + HQ + Q∗H + R � 0.

(R3) There exists H ∈ HMn satisfying the algebraic Riccati equation (33).(R4) The Jordan blocks of the matrix H corresponding to the purely imaginary eigenvalues, if

any, have even dimensionality.

This result was presented also in [140]. This assertion was obtained in [141, 142] under a morerestrictive assumption of controllability of the pair Q,P .

Let Guu > 0. We assume that k = −h−1u hx ∈ Mm,n. Then, the Lur’e equations (31) can be setdown as

A∗H + HA + k∗Guuk = Gxx, HB − k∗Guu = Gxu. (36)

The question of solvability of the Lur’e equation can be formulated [137] as the problem of repre-

sentability of the quadratic form Ψ(x, u) =

(xu

)G

(xu

)as

Ψ(x, u) =

(xu

)∗Λ′(H)

(xu

)+ (u − kx)∗Guu(u − kx), (37)

where H ∈ HMn, k ∈ Mm,n are the matrices to be determined. It was noted in the worksof Yakubovich on construction of the optimal controls in differential games [143, 144] that insuch problems one has to consider representation (37) under the sign-indefinite nondegenerate ma-trix Guu, which is equivalent to the solution of the algebraic Riccati equation (33), (34) underthe sign-indefinite nondegenerate matrix P . This problem was considered in [135, 145–149] andother works. We present the result of [150] where the solvability conditions are formulated in thefrequency terms and complement the conditions of Theorem 7.

Theorem 8. Let Gxu = 0, det P �= 0. Then, for Eq. (33), (34) to have solutions, it sufficesthat Condition (R4) of Theorem 7 be satisfied and x∗W (λ)Π(λ)−1W (−λ)∗x �≡ 0 for each nonzerox ∈ Cn.

The case of Gxu �= 0 can be reduced to that considered by means of the changes A = Ã−BG−1uu Gux,Gxx = G̃xx − GxuG−1uu Gux.

3.8. Generalized Lur’e Equation

In connection with the questions of H∞ optimization, the following problem was posed in [151].

Let u =

(vw

), v ∈ Rmv , w ∈ Rmw . We consider the corresponding block representations

B = (Bv, Bw), Guu =

(Gvv GwvGvw Gww

), Gxu = (Gxv , Gxw). Let Gww = 0, Gwv = 0, Gvw = 0. It is

desired to determine a representation of the form Ψ(x, u) as

Ψ(x, u) =

(xu

)∗Λ′(H)

(xu

)+ (v − kx)∗Gvv(v − kx) + x∗Kx, (38)



where H, K ∈ HMn, k ∈ Mmv,n, are the matrices to be determined. Representation (38) can beput down as the following generalized system of the Lur’e equations:

A∗H + HA + k∗Gvvk + K = Gxx, HBv − k∗Gvv = Gxv, HBw = Gxw. (39)

For an arbitrary Hermit matrix M , we denote by n±(M) the number of positive (negative) eigen-values M . We assume that

Δn± = limω→∞

n±(Π(iω)) − n±(Gvv).

Then, Δn± � 0. The condition n±(K) � Δn± is necessary for solvability of Eqs. (39). Thenecessary and sufficient conditions n±(K) = Δn± for solvability of Eq. (39) with the minimum-rank matrix K as well as the exact description of all solutions of this equation were given in [151].

4. INFINITE-DIMENSIONAL KALMAN–POPOV–YAKUBOVICH LEMMA

4.1. First Generalizations of the Lemma: Historical Context

Until the 1960’s inclusive, the application of the control theory methods was mostly oriented tothe lumped-parameter control systems which have finite-dimensional phase space and usually aredescribed in mathematical terms by systems of ordinary differential or integro-differential equations.Equations of mathematical physics including the partial differential operators with respect to thespatial variables are used in the majority of the mathematical models of the distributed-parametertechnical systems such as the gas- and hydro-dynamic plants, nuclear reactors, and so on.

Importance of the problems of control of the distributed-parameter systems increased dramati-cally by the 1970’s. We describe some attributes of this growth. The P.K.C. Wang bibliography ofthe works on the stability theory and control of the distributed-parameter systems published until1967 [152] got in nothing but fifteen pages. Since four years, in 1971 the First International IFACSymposium on the distributed-parameter systems was held in Canada. After two more years, in1973 a book Recent Mathematical Development in Control edited by D.J. Bell came off the pressin London [153]. It included papers and materials of the discussions on new applications of themathematical methods to the control theory. This book identified five research mainstreams suchas stability of the nonlinear systems, optimal control, filtering theory, control of systems obeyingthe partial differential equations, and finally, the algebraic system theory.

It was namely at that time—the late 1960’s and early 1970’s—that publications started toappear on the concepts of designing the control systems for thermonuclear fusion [154] and soon. This path of research was often called in the publications “The Problem of the TwentiethCentury.” Its topicality (and estimated cost) still continue to grow, but now it is a matter of “TheProblem of the Twenty-first Century.” The processes of experiment control played an important,although ancillary relative to the phenomenon under study, part in the physical experiments alreadysince the Nineteenth century. In the next century, the problems where the role of process controlceased to play an ancillary part came to the foreground. The problem of creating a thermonuclearplasma beam was discussed in some publications as that of optimal control where an externalelectromagnetic field played the role of the distributed control in the system. Suggested weredifferent mathematical formulations of the problem of control of thermonuclear fusion, includingthe variational and optimization problems to which the principle of maximum was applied, whichhad physical sense.

Therefore, the new areas of application of the methods developed by the theory of control ofthe lumped-parameter system seemed attractive, but some difficulties existed there. One of theauthors of the first publications on the methods of control of the distributed-parameter systems



proper, A.G. Butkovskii wrote about these difficulties in the preface to one of his books that “Wewill not keep back the fact that the gist of these problems is such that it requires a relativelycomplicated, nontraditional for the engineer. . .mathematical apparatus. Taking into account thisspecificity, the present writer aims at the utmost clarity and obviousness in the formulations ofthe discussed problems, somewhere renouncing for that the formal generality of the mathematicaldescription. . . ” [155, pp. 8, 9].

It is common knowledge that in the 1970’s the majority of experts on the control theory whohad the engineering background were unfamiliar with the language of functional analysis used bythe contemporary mathematical physics where the boundary problems can be represented as theunbounded operators in the Hilbert or, in the general case, Banach spaces. Yet, here the researchersencountered conceptual difficulties. In the theory of control of the distributed-parameter systems,the boundary problems often are nonuniform, the boundary conditions themselves usually being adynamic system. Variants of such systems with boundary control and observation are diverse andoccur in thermal, biological, chemical, nuclear, and other controlled plants.

On the other hand, the monographs of J.L. Lions and E. Madgenes [156] and Lions [157] werewritten namely in this language. They presented the most advanced modern mathematical ap-proach to the theory of optimal control of the distributed-parameter systems and in 1971, 1972were translated into Russian. There was need for comprehensible introductory books on functionalanalysis for the experts in the control theory, and such books did appear ([158] and others).

Familiarization with the notions and problems of optimal control of the distributed-parametersystems was facilitated by the fact that [156, 157] leaned heavily on the achievements of the strongnational schools of optimal control and mathematical physics. The Pontryagin principle of maxi-mum found application to the problems of optimal control of the distributed systems as early as inthe 1960’s (see [159] and others). The Sobolev spaces, variational formulations of the nonuniformprimordial-boundary problems, and other notions related with the theory of generalized functionswere taught at the university courses on the partial differential equations and widely used in theworks on mathematical physics and mechanics of continua. The problems of optimal control of thedistributed-parameter systems which were discussed in the later monographs [160–162] were of thesame variational nature and allured first of all by their desire to reformulate and present in otherterms the new physical and engineering problems that were of practical interest. The bibliographiesof [156–162] enabled one to master the formulations of a wide range of the problems of optimizationof the distributed-parameter systems and the methods of their solution.

4.2. First Publications and Differences in the Approaches to Generalizationof the Lemma to the Infinite-dimensional Case

The above situation in the development of the theory of control of the distributed systems wasreadily comparable with the problems where the Kalman–Popov–Yakubovich lemma was used. Thework of Yakubovich [25] that was published in 1973 was of a generalizing nature and summarized,within certain limits, the development of the matrix Kalman–Popov–Yakubovich lemma duringthe first decade of its existence. The ideas of applying the Kalman–Popov–Yakubovich lemmato the problems of absolute stability and instability, adaptation, determination of the criteria fordissipativity and convergence, as well as to the proofs of existence of the periodic and almost-periodic forced modes and autooscillations were becoming more understandable. This comparisonnaturally suggested the idea of extending the lemma to the case of operators in the Hilbert space.

When formulating in the fall of 1971 the problem of generalization of the lemma to the case ofpartial differential equations to A.L. Likhtarnikov, one of the present writers and then a fourth-year under-graduate, Yakubovich advanced (possibly, not for the first time) the idea of provingthe frequency theorem for the operators in the Hilbert space. The result obtained after some time



by the student did not satisfy the advisor. The variational formulations of the theorems wereincomplete, and the condition for solvability of the operator inequality was not represented in the“frequency form.” That is why Vladimir Andreevich got down to business himself. After additionalwork, these results were published in the joint paper [163].

The paper of Yakubovich [33] published in two parts in 1974 and 1975 was a pioneering progressin the generalization of the lemma of the matrix inequalities to the case of the operators in theHilbert space. It seems that the mentioned above difficulties of the potential readers influencedthe choice by Yakubovich of his wording because the formulations contained some excessive as-sumptions. One of them—and as will be shown below not the most essential—was the requirementon boundedness of the operators involved in the problem conditions which later was eliminated.The author also made the following remark in the preface to [33] (Part I): “We make use of someconsiderations and techniques that are well known in the theory of optimal control. . . . We presenttheir proofs, all the more so since they are very simple, in order to spare the readers the difficultiesof finding the sources where these proposals are presented in a different form which is not what isprecisely necessary for us (in particular, consideration is given to the finite-dimensional spaces).”

This work was conceived both to generalize the matrix lemma and to reach some methodologicalaims. In particular, the text contained the definitions of the basic notions of the control theoryand included, for example, simple derivations of the operator variant of the Lur’e equations andassertions about factorization of the Hermit operator called the “Popov function” Π(iω) (thisfunction sometimes is called the spectral density) which is involved in the “frequency conditions”of the theorems proved in the paper. We note that in the infinite-dimensional case the problemof factorization of the Popov function varies depending on the formulation of the linear-quadraticproblem of optimal control and must be solved anew with the advent of new formulations (see, forexample, [164]).

In the title of [33] the author emphasized the expected application domain of its results, theproblems of design of optimal control in the linear system with quadratic performance index.We note that the linear-quadratic problem of optimal control plays here two roles. First, theproblem of optimal control over an infinite interval is considered using the Bellman’s concept ofdynamic programming, and then its result is used to prove the Kalman–Popov–Yakubovich lemma.Second, when the lemma is already proved, it is used to solve other problems of linear-quadraticoptimization, for example, in the case of the finite time interval. As the result, the reader had anunderstandable introduction to the theory of optimal control where different variants of the notionsin common use such as controllability and stabilizability, relations between different properties ofthe control systems, operator form of the Lur’e equations, and so on were explained. Finally, thelemma formulations themselves were presented in two—nondegenerate and degenerate—forms.

In 1975, after a year since the publication of [33, Pt. I], five papers on the frequency theorem forthe infinite-dimensional case appeared simultaneously [33, Pt. II, 48–51]. The sufficient conditionsfor solvability (in the form of the nonstrict frequency condition) of the Lur’e equations were obtainedin [48] for the case of bounded operators. Two assumptions about the infinite-dimensional case weresimultaneously made there: full controllability of the pair A,B for the variant where reachabilityof any point of the state space is required (this requirement reduces this result in applications tothe finite-dimensional case) and condition for stabilizability. The scheme of proof follows the linesof the Lions book [157].

Although in [49] it was assumed that the control space is one-dimensional, the work itself isof interest by the ideas of proofs that differ basically from the other proofs and demonstrate theprofound relations of the lemma with the theory of functions. The paper [50], if judged by itsname, is devoted not to the frequency theorem as such but to its most popular application, theproblem of absolute stability of the nonlinear systems. Yet along the way used in [50] it is namely



the operator inequalities of the lemma that make the essence of the result and basis of the methodof constructing the Lyapunov functional. The work [51] is concerned with the infinite-dimensionalKalman–Szegő lemma. This subject is discussed in what follows.

Five more papers on the infinite-dimensional variant of the lemma and its applications werepublished in 1975, 1976 [52–55, 163]. The subject of the infinite-dimensional Kalman–Szegő lemmaand its applications was developed in [52] in continuation of [51]. The following two publications[163, 53] discussed generalizations of the lemma to the case of the Lions variational approach toformulating the problems of control systems in the Hilbert spaces. We recall that on the whole thisapproach to the formulation of problems for the operator equations on the scale of Hilbert spaces (orin other terms, in the framed Hilbert spaces) was developed for the nonuniform boundary problems.That is why it is suitable for the problems of optimization with the boundary control and/orobservation. In this approach, essential are the conditions for the system input—the operatoracting from the space of controls to the space of system states. Let, for example, X0 (B : U → X0)be instead of X−1 the range space of the operator B. This condition eliminates the generalizedfunctions from the range space of B and creates difficulties in applying the general operator schemeto the most interesting cases of systems with control on the domain boundary—for example, atconstructing the linear or nonlinear pulse controllers. The papers [54, 55] are devoted to theapproach of V.A. Brusin [50] to the problems of absolute stability of the nonlinear distributed-parameter systems.

In 1977, only two publications on the infinite-dimensional case of the lemma [56, 57] existed.In [56], the results of the first paper [33] were extended to the case of unbounded operators or,more precisely, the generators for the semigroups of the class C0. This work not only expandedthe lemma to one more, new area of problem formulations, but also demonstrated that with minormodifications the proofs used in [33] are suitable for the case of unbounded operators. The paper [57]was devoted to the applications of the lemma to the problems of absolute stability. It generalizedand reinforced both the finite-dimensional criteria for absolute stability and the criteria obtainedearlier for the distributed-parameter systems.

4.3. Lemma for the “Nondegenerate Case”

Here we present the results of [56] whose formulations are most obvious. Let X and U beHilbert spaces whose elements are called, respectively, states and controls, A : D(A) ⊂ X → X isthe generating operator of the semigroup of class C0, B : U → X is the linear bounded operator, and

G : X×U → X×U is the bounded self-conjugate operator. We represent G as G =(

Gxx GxuGux Guu

)

and determine on X × U the Hermit form Ψ(x, u) = (Gxxx, x) + 2Re(Gxuu, x) + (Guuu, u).

Theorem 9 (nondegenerate case). Let the pair A,B be stabilizable, that is, there exists an oper-ator C : X → U such that the spectrum Sp(A + BC) lies in the left half-plane. Then, the followingassertions are equivalent:

(1+∞) For some δ > 0, the “strict frequency condition”

Ψ(x, u) � δ(|x|2 + |u|2

)(40)

is satisfied for all x ∈ D(A), u and ω such that iωx = Ax + Bu.(2+∞) There exist δ > 0 and bounded self-conjugate operator H : X → X such that

2Re(Ax + Bu,Hx) − Ψ(x, u)+ � −δ(|x|2 + |u|2

).

Additionally, (1+∞) implies the following:



(3+∞) There exist the linear bounded operators H = H∗ : X → X, k : U →X, r : U → U suchthat the identity

Ψ(x, u) = 2Re(Ax + Bu,Hx) + |r(u − k∗x)|2 (41)

is valid for all x and u.At that, the operator r is determined from the relation Guu = r∗r. After the choice of r, the

operators k and H with the aforementioned properties are determined uniquely.

These assertions proved in [56] are valid under some other a priori assumptions about theoperators A and B, except for stabilizability; for example, if the pair A,B is L2-controllable orL2-stabilizable.

4.4. Lemma for the “Degenerate Case”

Theorem 10 (degenerate case). Let us assume that (a) the pair A, B is L2-controllable and(b) the pair −A,−B is L2-controllable. Then, the following assertions are equivalent:

(1∞) The “frequency condition”

Ψ(x, u) � 0 (42)

is satisfied for all ω ∈ R, x ∈ D(A), u ∈ U such that iωx = Ax + Bu.(2∞) There exists a linear operator H = H∗ : D(A) ⊂ X → X satisfying

2Rex∗H(Ax + bu) − Ψ(x, u) � 0

(for all x ∈ D(A) and u ∈ U).(3∞) There exist linear operators H = H∗ : D(A) ⊂ X → X, k : U → X, r : U → U such that

for all x ∈ D(A) and u the identity

Ψ(x, u) = 2Re(Ax + Bu,Hx) + |ru − k∗x|2 (43)

is valid.

Note. If the pair (A,B) satisfies condition (a) and does not satisfy condition (b), then thetheorem assertion is retained at the replacement of the “frequency condition” 1∞ by the followingcondition:

for any a ∈ D(A), the functional

J(x(.), u(.)) =∞∫

0

Ψ(x(t), u(t))dt (44)

is bounded from below on the set of pairs x(.), u(.) belonging to the space L2[R+,X × U ] andsatisfying the equation dx/dt = Ax + Bu and the initial condition x(0) = a.

We make some comments on Theorems 9 and 10. First, relations (41) and (43) are different;at that, (43) is representable as (41) only if the operator Guu is invertible (positive definite, inparticular) on the space U . Second, the identities (43) and (41) are equivalent to the operatorLur’e equation (

A∗H + HA HBB∗H 0

)− G = −h∗h,

where h = (−rk∗, r) for (41) and h = (−k∗, r) for (43).



4.5. Kalman–Szegő Lemma for the Infinite-dimensional Case

The first (finite-dimensional) result for the discrete-time systems was obtained in [116]. Bymeans of the well-known Cayley transformation, the Kalman–Szegő lemma was generalized in[33] to the case of bounded operators in the Hilbert space. This work was followed by [51, 52]where the previous results were dispensed with the excessive assumptions, the proofs were carriedout independently of the case of continuous-time systems, and the relations between the lemmaand the problems of optimal control of the discrete-time systems were shown in more detail. LetA : X → X, B : U → X be bounded linear operators and X and U be the Hilbert spaces. Weconsider a control system whose state obeys the discrete-time equation

x(t + 1) = Ax(t) + Bu(t), t = 0, 1, 2, . . . , (45)

where x(t) describes the system state and u(t) is the control at time t. We denote by l2(X) andl2(U) the Hilbert spaces of the quadratic summable sequences with values in X and U , respectively.

The notions required for formulation of the lemma are defined by analogy with the continuouscase. For example, system (45) is l2-controllable if for any a ∈ X there exists a control ua ∈ l2(U)such that x(.) ∈ l2(X) is satisfied for the solution of the system with the initial condition x(0) = a.We present the formulation of the most difficult generalization.

Theorem 11 (degenerate case of the Kalman–Szegő lemma). We assume that (c) the pair A,Bis l2-controllable and (d) the pair −A,−B is l2-controllable. Then, the following assertions areequivalent:

(1◦∞) The “frequency condition”Ψ(x, u) � 0,

is satisfied for all x ∈ X,u ∈ U and λ such that |λ| = 1, λx = Ax + Bu.(2◦∞) There exists a linear operator H = H

∗ : X → X satisfying

(Ax + Bu)∗H(Ax + Bu) − x∗Hx − Ψ(x, u) � 0

(for all x ∈ X, u ∈ U).(3◦∞) There exist linear operators H = H∗ : X → X, k : U → X, r : U → U such that for all

x ∈ X and u ∈ U there exists the identity

Ψ(x, u) = (Ax + Bu)∗H(Ax + Bu) − x∗Hx + |ru − k∗x)|2.

Note. If the pair A,B satisfies condition (c) and does not satisfy (d), then the theorem assertionis retained with the replacement of the condition (1◦∞) by the following condition:

for any a ∈ X, the functional

J [x(.), u(.)] =∞∑

t=0

Ψ[x(t), u(t)]

is bounded from below on the set of pairs x(.), u(.) belonging to the space l2(X) × l2(U) andsatisfying Eq. (45) with the initial condition x(0) = a.

4.6. Lemma, Linear-quadratic Problem of Optimal Control,Dissipative Systems, and Diffusion Theory

As was already noted, in [25, 33] and subsequent publications with his coauthors [51, 52, 163,56], Yakubovich used the linear-quadratic problem of optimal control as the method of proving the



lemma. To our opinion, the regularities arising in various problems for the dynamic systems stand“behind the back” of this elegant proof. The progress in studies takes place within the frameworkof certain problem formulations dividing the paths of research into “flows” of a kind moving inone direction, but remaining each within its own river-bed. Different authors rediscover the sameregularities and often give different names and applications to the relations used. And the lemmais not an exception. We turn for definiteness to the formulation of the linear-quadratic problemin [56]. Let the linear control system be defined in the standard form as

x′(t) = Ax(t) + Bu(t), x(0) = a, (46)

where x(t) is the state and u(t), control. We also assume that the quadratic performance functional

J [x(.), u(.)] =∞∫

0

Ψ[x(t), u(t)]dt (47)

is the criterion for quality of control. The problem lies in determining a process for which functional(47) assumes the least value denoted by E(a). One can readily see that the frequency condition(40) formulated above for the nondegenerate case is sufficient for existence and uniqueness of theoptimal process [xa(t), ua(t)] in the optimal control system (46), (47). At that, the value of theminimum E(a) = −(Ha, a) is a continuous quadratic form in the space X.

It turns out that ua(t) = k∗xa(t), and if a ∈ D(A), then the process is strong continuouslydifferentiable with respect to t. Moreover, it is easy to prove that for the arbitrary elementsa1, a2 ∈ D(A) and u ∈ U valid is the inequality

(Ha1, a1) − (Ha2, a2) +t2∫

t1

Ψ[x(t), u]dt � 0, (48)

where x(t) is the solution of (46) with u(t) = u = const over the integration interval [t1, t2] and a1and a2 are the initial values of the respective optimal processes defined at the points t1, t2 (see also[165]).

In the dynamic programming theory, the form E(x) = (Hx, x) is called the Bellman function, inthe theory of dissipative scatter systems, the storage function, and in the stability theory (Hx, x)is the Lyapunov function.

Relation (48) was introduced in the system theory in 1972 by Willems [166] as a definition ofdissipativity of the nonlinear systems of ordinary differential equations (Willems dissipativity orW -dissipativity). It was treated as a balance of an abstract energy in the system, a nonnegativefunction (Hx, x) to within the sign was called the storage function, and the form Ψ[x, u], the supplyrate. In 1976 this inequality was studied by D.J. Hill and P.J. Moylan for the ordinary differentialequations [167]. Their result was named the nonlinear KYP-lemma. Inequality (48) was used in[19] as a “nonfrequency” equivalent of the KYP-lemma. Different variants of this assertion andits applications to the problems of analysis and design of the nonlinear adaptive systems wereconsidered in [166–171]. With its help, some results on the dichotomy and absolute stability ofuncertain nonlinear systems were obtained in [58].

Notions allied to the Willems-dissipativity theory underlie the circle of theories which usuallyare referred to as the scatter theories. We mean the variant suggested by P. Lax and R. Philips[172] for the particular problems of forecasting random processes, interpolation of functions, designof dynamic systems, and so on. On the basis of this theory and the works of B. Nagy, Ch. Foiyash,Kalman, and other authors, D.Z. Arov developed his variant of the theory of passive systems[173–175]. His first publications on this subject date back to the 1960’s. In the early 1970’s this



path of research was already shaped [34], and a certain stage of its development was completed. Weindicate only some traits of the theory under consideration. Its physical source is represented bythe theory of electrical circuits (Darlington), and in the control theory this path includes problemswhich, in a certain context, can be attributed to the lemma which is the subject matter of thepresent paper. In the continuous-time version, the passive system is defined by the “input–output”equations:

x′(t) = Ax(t) + Bu(t), x(0) = a,y(t) = Cx(t) + Du(t), t � 0,

and the quadratic relation in the two most popular variants is defined by the forms

Ψ1(u, y) = ‖u‖2 − ‖y‖2 or Ψ2(u, y) = Reu∗y. (49)

The scatter system is H-passive if for any a and u(.) its solution meets the condition

EH(x(t)) − EH(x(0)) �t∫

0

Ψ(u(s), y(s))ds, (50)

where EH(x) = (Hx, x) is the storage function. It is common knowledge that for the forms (49)inequality (50) is equivalent to the standard Kalman–Popov–Yakubovich inequality. The readerscan acquaint themselves with the state-of-the-art of the Arov passive system theory and its resultsconcerning the lemma from the recent publications [174, 175].

5. FIRST RESULTS ON LOSSLESSNESS OF THE S-PROCEDURE

As was already noted, the notion of S-procedure was introduced in [110] in order to describe amethod of constructing the Lyapunov function for the nonlinear control systems. This notion in[3] was separated from the problem of constructing the Lyapunov function and formulated in anabstract form. Let the functions Φi, i = 0, 1, . . . , k, be defined in the vector space Z. We considerthe following assertions:

(I+) Φ0(z) > 0 is satisfied for z �= 0 satisfying the inequalities Φi(z) � 0, i = 1, 2, . . . , k;(II+) There exist τi � 0, i = 1, 2, . . . , k, such that S(x) = Φ0(x) −

∑ki=1 τiΦi(x) > 0.

Obviously, (II+) implies (I+). By the S-procedure is meant the replacement of condition (I+)by condition (II+). The problem of finding the classes of functions Φi, i = 0, 1, . . . , k for whichAssertions (I+) and (II+) are equivalent was formulated in [3].

The first assertion belongs to Yakubovich and is pertinent to the case of two quadratic forms ofthe real argument. This assertion follows from the assertion proved in [176] and stating that thereexist no absolutely stable linear systems with one nonlinearity satisfying the sector condition forwhich the fact of absolute stability can be established by means of the Lyapunov function of theLur’e-Postnikov form, but not by means of the S-procedure.

The well-known paper of Yakubovich [177] was the first publication devoted especially to theS-procedure. Along with conditions (I+) and (II+), this paper considered similar conditions wherestrict inequalities are replaced by nonstrict. We refer to the so-obtained assertions as (I) and (II).In compliance with the definition from the paper under consideration, we regard the S-procedureas lossless if (I+) is equivalent to (II+) or (I) is equivalent to (II). Otherwise, the S-procedure isnonlossless. The following popular theorem of losslessness of the S-procedure for two quadratic orHermit forms was proved in [177].



Theorem 12. Let Z be a real (complex) vector space, and Φi (i = 0, 1) be the quadratic (Hermit)forms on Z. If there exists z ∈ Z such that Φ1(z) > 0, then (I+) is equivalent to (II+) and (I) isequivalent to (II).

In the theorem condition formulated in [177] an inaccuracy was made which was corrected in[178]. The proof of the theorem is based on the fact that in virtue of the F. Hausdorff [179] andL. Dines [180] theorems the image of the map compiled of two Hermit (quadratic) forms is a convexset.

Let us consider the composite map Φ̂ = (Φ0,Φ1, . . . ,Φk). A general method of proving convexityof the image of the map Φ̂ was proposed in [177] where new proofs of the results of Hausdorff [179]and Dines were obtained. Additionally, it was already proved there that in the case of three realquadratic forms the S-procedure is, generally speaking, nonlossless.

The case of three Hermit forms was considered in [178] where the equality constraints wereconsidered along with the inequality constraints. For the equality constraints, the assertions of theS-procedure are as follows:

(I0) For z �= 0 satisfying the inequalities Φi(z) = 0, i = 1, 2, Φ0(z) � 0 is satisfied.(II0) There exist τi, i = 1, 2 such that Φ0(z) − τ1Φ1(z) − τ2Φ2(z) � 0.The case of one equality constraint and one inequality constraint was considered as well. It

was assumed that the constraints satisfy some regularity conditions which were christened thegeneralized Slater conditions [178]. They are formulated in the following section in the generalform for an arbitrary set of constraints. As was shown in [178], the S-procedure for three Hermitforms is lossless for all the aforementioned types of constraints, provided that the Slater conditionsare satisfied. Moreover, this result retains its validity if the quadratic functionals of the formΦi(z) = z∗Fiz + Re(f∗i z) + ϕi, fi ∈ Cκ, ϕi ∈ R, i = 0, 1, 2, are considered instead of the quadraticforms. If an equality constraint is imposed on the functional Φi, then it is assumed that Fi �= 0.

We note that the case of two quadratic forms and equality constraint was discussed in [181].The result obtained in this paper is known as the Finsler lemma.

Theorem 13. Let Φ0,Φ1 be quadratic forms. Then, the following assertions are equivalent:(I+0 ) For all z �= 0 satisfying the equality Φ1(z) = 0, the inequality Φ0(z) > 0 is satisfied.(II+0 ) There exists τ ∈ R such that Φ0(z) − τΦ1(z) > 0 for all z �= 0.

A similar assertion for the Hermit forms with the replacement of the strict inequalities by thenonstrict ones was obtained in [182] under an additional assumption of alternating-sign Φ1.

6. PROBLEM OF LOSSLESSNESS OF THE S-PROCEDURE:GENERAL FORMULATION

Extension of the aforementioned results to the case of more than one constraints defined eitherby inequalities or equalities is of practical interest. The first results along this path were establishedin [177, 178, 183]. Yet it is more convenient to begin not with these results, but with the generalformulation of the problem of losslessness of the S-procedure as formulated in [34] and [36].

Let Z be a set and Y, a topological vector space where the convex cone Y+ ⊂ Y is defined. Thecone Y+ defines on Y the preorder relation � defined for y1, y2 ∈ Y by the condition y1�y2, providedthat y2 − y1 ∈ Y+. If riY �= ∅, then for y1, y2 ∈ Y the relation y1 � y2 implies that y2 − y1 ∈ riY+.We denote by Y ′ the space conjugate to Y, and let Y ′+ = {τ ∈ Y ′|∀y ∈ Y ′+ 〈τ, y〉 � 0} be the dualcone to Y+. Here and below, 〈y, τ〉 = τ(y), where y ∈ Y, τ ∈ Y ′.

Let the maps Φ0 : Z → R and Φ : Z → Y be given. We consider the following assertions:



(I) The inequality Φ0(z) � 0 is satisfied for all z ∈ Z such that Φ(z) � 0.(II) There exists τ ∈ Y ′+ such that the inequality Φ0(z)− 〈τ,Φ(z)〉 � 0 is satisfied for all z ∈ Z.If Assertions (I) and (II) are equivalent, then the S-procedure is said to be lossless.A general assertion relating losslessness of the S-procedure with convexity of the set R = Φ̂(Z)

was proved in [183]. Let Y = Y ′ = Rk,

Y ′+ = {τ = (τ1, τ2, . . . , τk)|τi � 0 for i ∈ P+, τi = 0 for i ∈ P0}, (51)

where the sets of indices P0, P+ meet the conditions P0 ∩ P+ = ∅, P0 ∪ P+ = {1, 2, . . . , k}. Let usconsider the following condition imposed on the constraints:

(i) For each τ = (τ1, τ2, . . . , τk) satisfying the condition τi = ±1, i ∈ P0, τi = 1, i ∈ P+, thereexists z(τ) ∈ Z such that τiΦi(x(τ)) > 0, i = 1, 2, . . . , k.

Condition (i) is a variant of the well-known Slater conditions from mathematical programming.The following theory was proved in [183].

Theorem 14. Let the set R be convex and (i). Then, (I) is equivalent to (II).

Theorem (14) was generalized in [34, 36, 37] where it was assumed that Y is a normalized spaceand Y+ is an arbitrary convex cone.

We go after [36] in considering the following condition imposed on the constraints:(ii) There exists z∗ ∈ Z such that Φ(z∗)�0. The vector τ = 0 is a unique solution of the system

of inequalities τ � 0, τ � 0, 〈τ,Φ(z)〉 � 0 ∀z ∈ Z.Condition (ii) generalizes (i). Let R = Φ̂(Z), R+ = {(y0, y) ∈ R × Y | ∃z ∈ Z : y0 � Φ0(z),

y � Φ(z)}.

Theorem 15. Let condition (ii) be satisfied and the set R+ meet one of the following conditions:(a) R+ is convex;(b) cl R+ is convex and intY+ �= ∅;(c) R+ is almost convex, that is, there exists a convex Q such that Q ⊂ R+ ⊂ cl Q.

Then, Assertions (I) and (II) are equivalent.

Conditions (a) and (b) were introduced in [34, 36], (c), in [37]. We note that if the set R featuresone of the properties mentioned in (a)–(c), then R+ features the same properties. As was notedin [184], it suffices to require in Theorem (15) that conditions (a)–(c) be satisfied for the conicalshell R.

6.1. Results on Image Convexity. Finite-dimensional Case

Let Z ⊂�Z ,

�Z = Rn or Z̃ = Cn, Y = Rk, Φ = (Φ1, . . . ,Φk), and the maps Φi, i = 0, 1, . . . , n,

be quadratic or Hermit forms

Φi(x) = z∗Fiz, i = 0, 1, . . . , k, (52)

where Fi ∈ HMn are symmetrical real or Hermit matrices and S(Rn)(S(Cn)) is a unit sphere inRn(Cn).

The cases where the set R = Φ̂(Z) is convex are listed below:6.1.1. Z = S(Cn), n is arbitrary, k = 1, F0, F1 are arbitrary Hermit matrices [179].6.1.2. Z = Rn, n is arbitrary, k = 1, F0, F1 are arbitrary real symmetrical matrices [180].



6.1.3. Z = S(Rn), n � 3, k = 1, F0, F1 are arbitrary real symmetrical matrices [185].6.1.4. Z = Cn, n is arbitrary, k = 2, F0, F1, F2 are arbitrary Hermit matrices [185].6.1.5. Z = Cn, n is arbitrary, k is arbitrary, there are at most three linear independent matrices

among the Hermit matrices Fi, i = 0, . . . , k, [183].6.1.6. Z = C2, k is arbitrary, Fi, i = 0, . . . , k, are arbitrary real symmetrical matrices [177].6.1.7. Z = Rn, n � 3, k = 2, F0, F1, F2 are real symmetrical matrices such that α0F0 + α1F1 +

α2F2 > 0 for some α0, α1, α2 [186].6.1.8. Z = Rn, n is arbitrary, k is arbitrary, the matrices Fi, i = 0, . . . , k, are congruent to the

diagonal ones under the general transformation [183].6.1.9. Z = S(Cn), n is arbitrary, k is arbitrary, the matrices Fi, i = 0, . . . , k, are Hermit-

congruent to the Hermit–Toeplitz matrices under the general transformation [187].6.1.10. Z = Rn, n is arbitrary, k is arbitrary, the matrices Fi, i = 0, . . . , k, are congruent to

the real symmetrical Toeplitz matrices under the general transformation (is proved like the laststatement).

6.1.11. Z = Cn, n is arbitrary, k is arbitrary, the matrices Fi, i = 0, . . . , k, are Hermit-congruentto the real symmetrical three-diagonal matrices under the general transformation [187].

6.1.12. Z = Cn, n is arbitrary, k is arbitrary, the matrices Fi, i = 0, . . . , k, are Hermit-congruentto the real Hankel matrices under the general transformation [187].

6.1.13. Z = Mn,m(R), m,n are arbitrary, Φi(x) = tr(x∗Fix), Fi, i = 0, . . . , k, are arbitraryreal symmetrical matrices, k satisfies the inequality entier(

√8k+9−1

2 ) � m for m < n [188], and k isarbitrary for m � n.

6.2. Results on Image Convexity. Infinite-dimensional Case

The paper [189] which evoked great response proved that for arbitrary dimensionality of thespace Y, the S-procedure can be lossless for wider classes of the maps Φ0,Φ in the case of infinite-dimensional Z than in the finite-dimensional case. This paper and other works on this subject relyon the assertion of convexity of the set clR or almost convexity of R. These results are formulatedbriefly in this section.

6.2.1. Let�Z = L2((0,+∞),Rκ), Y = Rk, κ, k � 1. A family of the shift operators Tϑ,

ϑ ∈ (0,+∞) mapping the function z ∈�Z into the function Tϑz(t) = 0 for t � ϑ and Tϑz(t) =

z(t − ϑ) for t > ϑ is defined in the space�Z. Let Z be a linear subspace

�Z, Φ = (Φ1, . . . ,Φk),

where Φi, i = 0, 1, . . . , k, be continuous quadratic forms defined on Z. We assume that Z and themaps Φi are invariant in the operators Tϑ, that is, Tϑz ∈ Z, Φi(Tϑz) = Φi(z), i = 0, 1, . . . , k issatisfied for all ϑ ∈ (0,+∞), z ∈ Z. Then, clR is convex [189, 190].

The following example illustrates application of the above result in the problems of control. Weconsider the control system

d

dtx = Ax + Bu, (53)

where x ∈ Rn, u ∈ Rm, the matrices A,B are of corresponding sizes, t � 0. Let z(t) = (x(t), u(t)) ∈Rκ, κ = m + n. The space Z is defined as the set of trajectories z ∈ L2((0,+∞),Rκ) meeting theinitial condition x(0) = 0. We define the quadratic forms

Φi(z) =+∞∫

0

z(t)∗Giz(t)dt,



where Gi ∈ SMκ, i = 0, 1, . . . , k. One can readily see that the quadratic forms and the trajectoryspa

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