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Maximum principle for an optimal control problem associated to a SPDE with nonlinear boundary conditions

Stefano Bonaccorsi, Adrian Zalinescu

TL;DR

This work addresses stochastic optimal control of a nonlinear diffusion with dynamical boundary conditions and boundary control acting on $\Gamma$. It develops a stochastic maximum principle based on an adjoint BSDE and a variational equation to obtain necessary and, under convexity, sufficient optimality conditions via a Hamiltonian $\mathcal{H}$. The paper also establishes the existence of an optimal boundary control under linear boundary interaction and convex costs. The results provide a rigorous variational framework for SPDEs with nonlinear boundary dynamics and boundary control, enabling precise optimality criteria and existence results in high-dimensional settings.

Abstract

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.

Maximum principle for an optimal control problem associated to a SPDE with nonlinear boundary conditions

TL;DR

This work addresses stochastic optimal control of a nonlinear diffusion with dynamical boundary conditions and boundary control acting on . It develops a stochastic maximum principle based on an adjoint BSDE and a variational equation to obtain necessary and, under convexity, sufficient optimality conditions via a Hamiltonian . The paper also establishes the existence of an optimal boundary control under linear boundary interaction and convex costs. The results provide a rigorous variational framework for SPDEs with nonlinear boundary dynamics and boundary control, enabling precise optimality criteria and existence results in high-dimensional settings.

Abstract

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.

Paper Structure

This paper contains 8 sections, 5 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum principle
  4. The adjoint equation
  5. The variational equation
  6. Necessary conditions of optimality
  7. Sufficient conditions of optimality
  8. Existence of an optimal control

Key Result

Theorem 2.1

Under hypotheses (A$_{0}$)--(A$_{2}$), (B), (C$_{0}$)--(C$_{2}$), for every control $u$, there exists a unique solution $\mathbf{Y}^{u}=(Y^{u},\bar{Y}^{u})\in L^{2}\left( \Omega\times \lbrack0,T];V\right)$ of equation (Eq_state) such that $\mathbf{Y}^{u}$ is a continuous, adapted process with values

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • proof