Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case

Abstract

We consider the abstract dynamical framework of [LT3, class (H.2)] which models a variety of mixed partial differential equation (PDE) problems in a smooth bounded domain Ω ⊂ ℝⁿ, arbitraryn, with boundaryL ₂-control functions. We then set and solve a min-max game theory problem in terms of an algebraic Riccati operator, to express the optimal quantities in pointwise feedback form. The theory obtained is sharp. It requires the usual “Finite Cost Condition” and “Detectability Condition,” the first for existence of the Riccati operator, the second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the parameterγ, here calledγ _c (criticalγ),γ _c≥0, such that a complete theory is available forγ >γ _c, while the maximization problem does not have a finite solution if 0 <γ <γ _c. Mixed PDE problems, all on arbitrary dimensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable include: second-order hyperbolic equations with Dirichlet control, as well as with Neumann control, the latter in the one-dimensional case; Euler-Bernoulli and Kirchhoff equations under a variety of boundary controls involving boundary operators of order zero, one, and two; Schroedinger equations with Dirichlet control; first-order hyperbolic systems, etc., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solution of the min-max problem implies solution of theH ^∞-robust stabilization problem with partial observation.