Min-Max Game Theory and Optimal Control with Indefinite Cost under a Singular Estimate for e At B in the Absence of Analyticity

Abstract

In this paper, we continue the study of the abstract dynamics of [4, 5, 7, 8, 10] — characterized by a singular estimate for e At B, in the absence of analyticity for e At — in two natural directions, both over an infinite horizon. They are: (i) the min-max game theory problem, where in part we extend and in part we complement the theory of [9, Chapter 6, Part II, Sections 6.19 through 6.26, pp. 608—630], in the stable case; and (ii) the optimal control problem, with indefinite quadratic cost, where likewise in part we extend and in part we complement the theory of [9, Chapter 6, Appendix 6A, pp. 630—638], in the stable case. In turn, each topic may be seen as an extension of [4, 7, 10]. Throughout this paper, and for both topics (i) and (ii), the following strategy is applied. The first part of our analysis — Sections 1 through 3 for the min-max problem; and Proposition 6.1 through Proposition 6.5 for the indefinite cost problem — relies only on the regularity properties of the operators L, L*, W, W*, etc., given in (1.11)–(1.14) in Section 1 below, which continue to hold true under the assumed singular estimate for e At B, as in the abstract parabolic case of [9, Volume 1], in spite of the lack of analyticity for e At . Therefore, with this key observation in mind, the corresponding abstract parabolic treatment of [9, Chapter 6], which hinges only on said regularity properties of L, L*, W, W*, goes through verbatim. By contrast, the final stage of our analysis — which in [9, Chapter 6], made explicit use of the analyticity of the s.c. semigroup e At , a property presently not available — needs now appropriate modifications, in both the proofs and the final statements, of the type already performed in the final stage of the analysis of [10]. Accordingly, the treatment of this paper is brisk, as it heavily hinges either on [9, Chapter 6] or on [10]. For clarity, the corresponding building blocks of the respective theories are explicitly singled out and displayed in formal statements, for both problems. This paper aims at serving at least the following purpose: in giving an affirmative answer to open questions raised over the past few years in control-theoretic circles about a possible ‘extension’ of the min-max game theory problem and of the indefinite cost problem to systems of PDE’s consisting of a parabolic-like component strongly coupled with a hyperbolic-like component, such as they arise in the “structural acoustic problem.” The general solution provided here is based on the abstract assumption that a singular estimate for e At B holds true, as in [4, 7, 8, 10, 11, 13]. As seen in [5, 4, 7, 8, 11, 13], such an assumption is, in fact, a dynamical property naturally satisfied in acoustic problems, with either a structurally elastic or a thermoelastic or a composite (sandwich) flexible wall. Once this key feature is realized and extracted at the abstract level, then a solution of both problems treated here readily follows from the work of [9, Volume 1], modulo the additional analysis of [10]. As the “acoustic models” of the type that motivate [10, 11, 13] and the present paper are naturally stable, we limit our study to the case where the free dynamics semigroup e At is indeed uniformly (exponentially) stable. A corresponding study with no stability assumption is surely available by invoking instead [9, Chapter 6, Sections 6.1 through 6.18] for the min-max theory, rather than [9, Sections 6.19 through 6.26].