Continuous Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on $$L^2(\Omega )$$ Under Control Constraints

Abstract

An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton–Jacobi–Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.

Differentiability of the value function on $ H^1(\Omega) $ of semilinear parabolic infinite time horizon optimal control problems under control constraints

“…With Corollary 4.5 available (91) can be verified with the same techniques as the analogous one with V replaced by L 2 (Ω) given in [19,Theorem 4.10]. The claim concerning the Riesz representor follows from (91) and Corollary 4.5.…”

“…Concerning some basic properties of problem (75) we can proceed as in Lemma 4.7, Remark 4.2, and Step (ii) of Theorem 2.1 of [19]. First, note that the second order sufficient optimality condition in the sense of (57) for (P ) and for (73) coincide.…”

“…Then we obtain the equation 83) is well-defined on L 2 (I; V ′ ). Hence we can at first apply the same technique as in [19,Proposition 2,pg. 32] to obtain the Lipschitz continuous dependence of δp with respect to…”

“…Proof. The structure of the proof is rather standard, [19,Theorem 5.1]. But due to regularity issues special treatment is required.…”

“…But due to regularity issues special treatment is required. Also compared to [19] we modify some arguments to allow local rather than global solutions. Let ŷ0 ∈ B V (y 0 , δ 4 ) with associated local solution and adjoint state (ŷ, û, p) ∈ Û .…”