A Max-plus Dual Space Fundamental Solution for a Class of Operator Differential Riccati Equations

A new fundamental solution semigroup for operator differential Riccati equationsis developed. This fundamental solution semigroup is constructed via an auxiliary finite horizonoptimal control problem whose value functional growth with respect to time horizon is determinedby a particular solution of the operator differential Riccati equation of interest. By exploiting semi-convexity of this value functional, and the attendant max-plus linearity and semigroup properties ofthe associated dynamic programming evolution operator, a semigroup of max-plus integral operatorsis constructed in a dual space defined via the Legendre-Fenchel transform. It is demonstrated thatthis semigroup of max-plus integral operators can be used to propagate all solutions of the operatordifferential Riccati equation that are initialized from a specified class of initial conditions. As thissemigroup of max-plus integral operators can be identified with a semigroup of quadratic kernels, anexplicit recipe for the aforementioned solution propagation is also rendered possible.