MIN-MAX GAME-THEORY AND ALGEBRAIC RICCATI-EQUATIONS FOR BOUNDARY CONTROL-PROBLEMS WITH CONTINUOUS INPUT-SOLUTION MAP .2. THE GENERAL-CASE

We consider the abstract dynamical framework of [LT3, class (H.2)] whi ch models a variety of mixed partial differential equation (PDE) probl ems in a smooth bounded domain OMEGA subset-of R(n), arbitrary n, with boundary L2-control functions. We then set and solve a min-max game t heory problem in terms of an algebraic Riccati operator, to express th e optimal quantities in pointwise feedback form. The theory obtained i s sharp. It requires the usual ''Finite Cost Condition'' and ''Detecta bility Condition,'' the first for existence of the Riccati operator, t he second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the pa rameter gamma, here called gamma(c) (critical gamma) gamma(c) greater- than-or-equal-to 0, such that a complete theory is available for gamma > gamma(c), while the maximization problem does not have a finite sol ution if 0 < gamma < gamma(c). Mixed PDE problems, all on arbitrary di mensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable inclu de: second-order hyperbolic equations with Dirichlet control as well a s 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; Schroedinge r equations with Dirichlet control; first-order hyperbolic systems, et c., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solut ion of the min-max problem implies solution of the H(infinity)-robust stabilization problem with partial observation.