Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

Abstract

We consider stochastic optimal control problems in Banach spaces, related to nonlinear controlled equations with dissipative non linearities: on the nonlinear term we do not impose any growth condition. The problems are treated via the backward stochastic differential equations approach, that allows also to solve in mild sense Hamilton Jacobi Bellman equations in Banach spaces. We apply the results to controlled stochastic heat equation, in space dimension 1, with control and noise acting on a subdomain.

HJB equations in infinite dimensions under weak regularizing properties

“…Stochastic optimal control problems for reaction diffusion equations have been extensively studied in the literature. We cite in particular the papers [2] and [3], where equations with a more general structure than equation (2.6) below are treated, but some more smoothing properties on the transition semigroup are required, and the paper [16] where the case of a non linear term in the heat equation is treated and the problem is solved in the Banach space of continuous functions, on the contrary all the coefficients are asked to be Gâteaux differentiable. In the present paper we are able to remove differentiability assumptions on the cost.…”

“…([0, 1]). We refer also to [16] for more details in the reformulation. We consider the Hamilton Jacobi Bellman equation relative to (6.15)…”

“…In this Section we revisit the results contained in the previous sections in the case when the problem is considered in a Banach space E instead of the Hilbert space H. For the Banach space framework, we mainly refer to the papers [14], [15] and [16]. From now we assume that E admits a countable Schauder basis.…”

“…No regularizing assumption on the transition semigroup is imposed, on the contrary ψ, l and φ are assumed differentiable. The papers [15] and [16] are the extension of [8] to the Banach space framework, with some restrictions on B, which is asked to be constant. We notice that [8] is the infinite dimensional extension of results in [19].…”

Optimal Relaxed Control of Dissipative Stochastic Partial Differential Equations in Banach Spaces

“…Remark 3.6. Existence of weak optimal feedback controls for a similar cost functional and a similar class of dissipative stochastic PDEs has been recently proved in [Mas08a] and [Mas08b] using Backward SDEs and the associated Hamilton-Jacobi-Bellman equation. However, only the case of additive noise is considered and the nonlinear term is assumed to be bounded with respect to the control variable.…”