Stochastic Maximum Principle for Optimal Control of a Class of Nonlinear SPDEs with Dissipative Drift

Abstract

We prove a version of the stochastic maximum principle, in the sense of Pontryagin, for the finite horizon optimal control of a stochastic partial differential equation driven by an infinite dimensional additive noise. In particular we treat the case in which the non-linear term is of Nemytskii type, dissipative and with polynomial growth. The performance functional to be optimized is fairly general and may depend on point evaluation of the controlled equation. The results can be applied to a large class of non-linear parabolic equations such as reaction-diffusion equations. * Financial support from the grant MIUR-PRIN 2010-11 "Evolution differential problems: deterministic and stochastic approaches and their interactions" is gratefully acknowledged. The second author have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). †

Stochastic evolution equations of jump type with random coefficients: existence, uniqueness and optimal control

“…Proof. The proof can be obtained by Galerkin approximations in the same way as the proof of Theorem 3.2 in [11] with minor change.…”

“…For optimal control problems of stochastic evolution equation or stochastic partial differential equation, many works are concerned with systems driven only by Wiener process and the corresponding stochastic maximum principles are establish, see e.g. [3], [7], [9], [11], [12], [14], [15], [16], [17], [18], [20], [30]. In contrast, there have not been very much results on the optimal control for stochastic partial differential equations driven by jump processes.…”

Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case

“…The key idea is to use the duality arguments to reduce the analysis of (2.13) to that of an associated variational equation (see (5.1) below). See also [30,44] for duality arguments in analyzing backward stochastic equations arising form control problems.…”

“…In that case, one usually needs to deal with a second adjoint equation (see e.g. [30,44,47] and references therein). In the present work, we will not treat this issue.…”

Optimal control of stochastic phase-field models related to tumor growth

“…This permit to get uniform estimates on the adjoint variables through a (more easier) continuous dependence of the linearized system. For an application of this strategy to stochatic reaction-diffusion equations we refer to [26].…”