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

Abstract

We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrizable control sets.

Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise

“…For (a) if condition 1 holds, the result follows from Theorem 2.1.3-Part G in Castaing et al [19] and for condition 2, the result follows using Proposition 2.1.12-Part (d) in Castaing et al [19]. Also see Brzeźniak and Serrano [18]. Proof of (b) is from Lemma 10 of Sritharan [69], Haussmann and Lepeltier [38], Jacod and Memin [39].…”

“…Proof. The proof follows from Lemma 2.18 of Brzeźniak and Serrano [18]. Let us provide a sketch of the proof.…”

“…Cutland et al in [25,26] combined relaxed controls and non-standard analysis techniques to study existence of an optimal control for stochastic 2D and 3D Navier-Stokes equations in a bounded domain with multiplicative Gaussian noise. Brzeźniak and Serrano in [18] have analyzed the existence of a weak optimal relaxed control for semilinear dissipative stochastic evolution equations influenced by cylindrical Wiener process in unconditional martingale difference (UMD) type-2 Banach spaces implementing semigroup approach.…”

“…This section is devoted to the class of Young measures on metrizable Suslin control sets and the ideas are borrowed mainly from Brzeźniak and Serrano [18], Castaing et al [19] and Sritharan [69].…”

Existence of an optimal control for fractional stochastic partial neutral integro-differential equations with infinite delay

“…The existence of optimal controls for backward stochastic partial evolution differential systems in the abstract space; see Meng and Shi [16], Zhou and Liu [27]. Brzeźniak and Serrano [8] discussed the existence of optimal relaxed controls for a class of semilinear stochastic evolution equation on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process.…”