A max-plus primal space fundamental solution for a class of differential Riccati equations

Abstract

Abstract-A class of differential Riccati equations (DREs) is considered whereby the evolution of any solution can be identified with the propagation of a value function of a corresponding optimal control problem arising in L2-gain analysis. By exploiting the semigroup properties inherited from the attendant dynamic programming principle, a max-plus primal space fundamental solution semigroup of max-plus linear max-plus integral operators is developed that encapsulates all such value function propagations. Using this semigroup, a new one-parameter fundamental solution semigroup of matrices is developed for the aforementioned class of DREs. It is demonstrated that this new semigroup can be used to compute particular solutions of these DREs, and to characterize finite escape times (should they exist) in a relatively simple way compared with that provided by the standard symplectic fundamental solution semigroup.

Representation of fundamental solution groups for wave equations via stationary action and optimal control

“…Then W t (x) ≥ J µ t (x, w) for all w ∈ W 1 [0, t]. Furthermore, if there exists a mild solution s → ξ * s as per (15) such that…”

“…and let s → ξs ∈ C([0, t]; X 1 ) denote the corresponding mild solution (15) with ξ0 = x and w = w. Define p s . = ∇ x W t−s ( ξs ) and note that…”

“…The proof of a special case [5,Theorem 11] of Theorem 3.3 employs a homotopy argument to verify the corresponding quadratic representation analogous to (27). Here, motivated by [15,Theorem 2] and [14,Theorem 5], an alternative approach to the proof of Theorem 3.3 as stated is developed by exploiting semiconvex duality. This development commences with the definition of a parameterised terminal cost ϕ :…”