Min–Max Game Theory and Non-standard Differential Riccati Equations Under the Singular Estimate for e At B and e At G in the Absence of Analyticity

Abstract

We consider an abstract dynamical system which is characterized by “singular estimates”, as it arises from many concrete hyperbolic/parabolic coupled PDE models, subject to boundary/point control and (deterministic) disturbance. If B is the control operator, G the disturbance operator and e At the free dynamic semigroup (which is not assumed to be analytic), then e At B and e At G are assumed to possess singular estimates. For such a system, we study a min–max game theory problem with quadratic cost functional over a finite horizon. The problem is fully solved in feedback form via a Riccati operator which satisfies a non-standard differential Riccati equation. Several PDE-illustrations are presented.