A max-plus dual space fundamental solution for a class of operator differential Riccati equations. (English) Zbl 1347.49005
Summary: A new fundamental solution semigroup for operator differential Riccati equations is developed. This fundamental solution semigroup is constructed via an auxiliary finite horizon optimal control problem whose value functional growth with respect to the time horizon is determined by a particular solution of the operator differential Riccati equation of interest. By exploiting semiconvexity of this value functional, and the attendant max-plus linearity and semigroup properties of the associated dynamic programming evolution operator, a semigroup of max-plus integral operators is constructed in a dual space defined via the Legendre-Fenchel transform. It is demonstrated that this semigroup of max-plus integral operators can be used to propagate all solutions of the operator differential Riccati equation that are initialized from a specified class of initial conditions. As this semigroup of max-plus integral operators can be identified with a semigroup of quadratic kernels, an explicit recipe for the afore-mentioned solution propagation is also rendered possible.
MSC:
| 49J27 | Existence theories for problems in abstract spaces |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49L20 | Dynamic programming in optimal control and differential games |
| 49M29 | Numerical methods involving duality |
| 93C20 | Control/observation systems governed by partial differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47F05 | General theory of partial differential operators |
| 15A80 | Max-plus and related algebras |
Keywords:
operator differential Riccati equations; optimal control; infinite-dimensional systems; fundamental solution; dynamic programming; semigroups; max-plus methods; Lengendre-Fenchel transformReferences:
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