Min–max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

Weconsider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω ⊂ Rn, n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min–max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbancemay be applied on the boundary, or as a Diracmeasure at a point. The observation operator has some smoothing properties. © 2011 Elsevier Ltd. All rights reserved. 1. Mathematical setting and formulation of the min–max problem. Statement of main results In this paper we return to the abstract dynamical setting of [1, Chapter 9] (‘‘abstract hyperbolic-like equations’’), which model a large variety of mixed PDE problems of hyperbolic-like type with either boundary or point control/disturbance, all defined on a smooth bounded domainΩ ⊂ Rn. They include: second-order hyperbolic equations and Schrödinger equations with control/disturbance in the Dirichlet B.C. [1, Sections 10.5 and 10.9]; non-symmetric, non-dissipative, first-order hyperbolic systemswith boundary control/disturbance [1, Section 10.6]; Euler–Bernoulli and Kirchhoff plate equationswith different types of boundary controls/disturbances [1, Sections 10.7; 10.8], even wave equations with Neumann boundary control/disturbance, however, only in one dimension [1, Section 9.8.4, p. 857]; as well as wave and Kirchhoff equations with point control/disturbance [1, Section 9.8], as well as systems of coupled PDe equations (wave and Kirchhoff equations; wave and structurally damped Euler–Bernoulli equations [1, Sections 9.10 and 9.11] arising in noise reduction models, with point control/disturbance). One may add elastic and thermoelastic dynamics with point control disturbance [2–5]. For such abstract dynamics, we then study a min–max game theory problem over a finite time horizon and with a smoothing observation operator. This is the perfect counterpart of the optimal control problem studied in [1, Chapter 9]. In the solution of this min–max problem that we provide here, all the optimal quantities – control, disturbance, state, observed state – are expressed explicitly in terms of the data of the problem via a time-dependent, positive definite Riccati operator, solution of a non-standard differential Riccati equation. For such hyperbolic-like dynamics, the corresponding min–max problem over infinite time interval was studied in [6,7] for the stable, respectively, unstable cases leading to an algebraic non-standard Riccati equation. ✩ Research partially supported by the National Science Foundation under grant DMS-0104305 and the Air Force Office of Scientific Research under grant FA9550-09-1-0459. ∗ Correspondence to: Department of Mathematics, University of Virginia, Charlottesville, VA 22903, United States. Tel.: +1 804 324 8946. E-mail address: [email protected]. 0362-546X/$ – see front matter© 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.08.025 R. Triggiani / Nonlinear Analysis 75 (2012) 1572–1591 1573 For min–max problems for parabolic-like (analytic semigroup), we refer to [8, Section 9.8], [9,10]. For min–max problems in systems of coupled PDEs of different type (parabolic versus hyperbolic), we quote structural acoustic chambers [11]; a fluid–structure interaction model [12], as well as the abstract treatment in [13], see also [14]. Dynamical model (Setting in [8, Chapter 9], [15,16]). Let U (control), V (disturbance), Y (state) be separable Hilbert spaces. In this paper, we consider the following abstract dynamical system  ẏ(t) = Ay(t) + Bu(t) + Gw(t) in [D(A)] y(0) = y0 ∈ Y . (1.1) Here the function u ∈ L2(0, T ;U) is the control and w ∈ L2(0, T ; V ) is a deterministic disturbance. The dynamics (1.1) is subject to the following assumptions, to be maintained throughout this paper: (A.1) A : Y ⊃ D(A) → Y is the infinitesimal generator of a strongly continuous (s.c.) semigroup eAt on Y . Without loss of generality, as the dynamics (1.1) is studied here over a finite time interval [0, T ], T < ∞, at the price of replacing A with a suitable translation of A, we may assume that A−1 ∈ L(Y ). (A.2) B and G are linear operators satisfying: B ∈ L(U; [D(A)]) and G ∈ L(V ; [D(A)]), respectively. Equivalently, A−1B ∈ L(U; Y ), and A−1G ∈ L(V ; Y ). Here, [D(A)] is the dual space of the domainD(A), with respect to the pivot space Y . Thus, eAt can be extended as a s.c. semigroup on [D(A)] as well. (A.3) The observation operator R is bounded: R ∈ L(Y ; Z), (1.2) where Z (output space) is another Hilbert space. (A.4) The (closable) operators B∗eA t and G∗eA t can be extended (from D(A)) to satisfy the ‘abstract trace regularity’ [1, p. 766], [17–21]: B∗eA t : continuous Y → L2(0, T ;U); G∗eA t : continuous Y → L2(0, T ; V ), (1.3a) that ∫ T 0 ‖B∗eA ∗y‖Udt ≤ CT‖y‖ 2 Y ; ∫ T 0 ‖G∗eA ∗y‖Vdt ≤ CT‖y‖ 2 Y , y ∈ Y (1.3b) (estimate (1.3b) is first checked for all y ∈ D(A) and then extended to all of Y ). (A.5) The maps R∗ReAtB and R∗ReAtG can be extended as follows [1, p. 767], [21]: R∗ReAtB : continuous U → L1(0, T ; Y ); R∗ReAtG : continuous V → L1(0, T ; Y ), (1.4a) that ∫ T 0 ‖RReBu‖Ydt ≤ CT‖u‖U , u ∈ U; ∫ T 0 ‖RReGw‖Ydt ≤ CT‖w‖V , w ∈ V . (1.4b) Min–max game theory problem on [0, T ]. For a fixed 0 < T < ∞ and a fixed γ > 0, we associate with (1.1) the cost functional J(u, w; y0) = J(u, w, y(u, w); y0) = ∫ T 0 [ ‖Ry(t)‖Z + ‖u(t)‖ 2 U − γ 2 ‖w(t)‖V ] dt, (1.5) where y(t) = y(t; y0) is the solution of (1.1) due to u(t) and w(t). See below in (1.15a). The aim of this paper is to study the following min–max game theory problem: sup wεL2(0,T ;V ) inf uεL2(0,T ;U) J(u, w; y0), (1.6) where the infimum is taken over all u ∈ L2(0, T ;U), for w ∈ L2(0, T ; V ) fixed, and the supremum is taken over all w ∈ L2(0, T ; V ). This problem, in addition to being of interest by itself, is known to be the state space formulation of the so called H robust stabilization problem; see [22,23]. In these references, this problem was introduced and stated in terms of the transfer function in the context of finite-dimensional theory [23]. The min–max game theory problem on the infinite time horizon, [0, ∞] and thus leading to a non-standard algebraic Riccati equation, has been studied in [6,7,24,1]. In this paper, we are interested in a finite time setting, [0, T ], and the corresponding non-standard differential Riccati equation. Dual versions of (A.4), (A.5). By duality in (A.4), we obtain the following relations [8, p. 767]. First, on B∗eA t : (A.4) The map L0 satisfies [18], [1,21, Chapter 7, Theorem 7.2.1] (L0u)(t) = ∫ t 0 eA(t−τ)Bu(τ )dτ : continuous L2(0, T ;U) → C([0, T ]; Y ), (1.7)