ABSTRACT
We consider a min-max game theory problem for an abstract model [L-T.1, class (H.l)], which covers in particular parabolic and parabolic-like partial differential equations in a general bounded domain with both control function (“good” player) and deterministic disturbance (“bad” player) acting on the boundary of the spatial domain. The case of point control and point disturbance is also included. Specific examples encompass not only mixed problems for heat/diffusion equations, but also for wave/plate equations with a sufficiently high degree of internal damping [L-T.1, section 6].
The present paper treats the case where the original free dynamics is stable. Here, a direct treatment may be given which provides an explicit solution of all the relevant quantities directly in terms of the data of the problem. The optimal control and the worst disturbance are both synthesized in a feedback form, pointwise in time, in terms of the unique solution of an algebraic Riccati operator equation. The overall proof is a fusion of the strategy devised in [M-T.1] for hyperbolic/plate-like or Schroedinger equations with the technicalities of the corresponding linear quadratic regulator problem for boundary control parabolic equations, where the disturbance w ≡ 0 [L-T.1] – [L-T.3], [D-I.1], [F.1].
