Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimality Conditions for Control Systems Governed by Monotone Stochastic Evolution Equations

Ioana Ciotir, Nicolas Forcadel, Piero Visconti, Hasnaa Zidani

TL;DR

The article develops a general framework for open-loop optimal control of nonlinear monotone SPDEs, deriving first-order necessary conditions via linearization and spike perturbations. It introduces costate (p,q) through backward stochastic evolution equations and establishes a stochastic Pontryagin maximum principle under diffusion that may or may not depend on the control. The results are instantiated for nonlinear divergence-form, Burgers-type, and porous media-type equations, with Dirichlet, Neumann, and Robin boundary conditions, demonstrating the method’s breadth and potential for diffusion-type models under uncertainty. This advances open-loop control theory for SPDEs with nonlinear principal operators and provides concrete PMP forms for several physically relevant models.

Abstract

We study a class of optimal control problems governed by nonlinear stochastic equations of monotone type under certain coercivity and linear growth conditions. We give first order necessary conditions of optimality. A stochastic Pontryagin principle can be recovered in the case that the diffusion doesn't depend on the control. We give several applications, most notably for stochastic porous media equations in the Lipschitz case.

Optimality Conditions for Control Systems Governed by Monotone Stochastic Evolution Equations

TL;DR

The article develops a general framework for open-loop optimal control of nonlinear monotone SPDEs, deriving first-order necessary conditions via linearization and spike perturbations. It introduces costate (p,q) through backward stochastic evolution equations and establishes a stochastic Pontryagin maximum principle under diffusion that may or may not depend on the control. The results are instantiated for nonlinear divergence-form, Burgers-type, and porous media-type equations, with Dirichlet, Neumann, and Robin boundary conditions, demonstrating the method’s breadth and potential for diffusion-type models under uncertainty. This advances open-loop control theory for SPDEs with nonlinear principal operators and provides concrete PMP forms for several physically relevant models.

Abstract

We study a class of optimal control problems governed by nonlinear stochastic equations of monotone type under certain coercivity and linear growth conditions. We give first order necessary conditions of optimality. A stochastic Pontryagin principle can be recovered in the case that the diffusion doesn't depend on the control. We give several applications, most notably for stochastic porous media equations in the Lipschitz case.
Paper Structure (20 sections, 8 theorems, 121 equations)

This paper contains 20 sections, 8 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem. Main results
  3. Notations and Preliminaries
  4. Optimal control problem
  5. Main result
  6. Comments on the main result.
  7. State Equation and Adjoint Equation
  8. State equation
  9. Linearized state equation
  10. Linear perturbation
  11. Spike perturbation
  12. Adjoint equation
  13. Optimality conditions
  14. Applications
  15. Nonlinear divergence form equations
  16. ...and 5 more sections

Key Result

Theorem 2.1

Assume that Hypotheses hyp:dyn:1-hyp:4:Cost hold. Let $\bar{x} \in \mathcal{X}$ and $\bar{u}\in \mathcal{U}^{ad}$. Assume that $(\bar{x},\bar{u})$ is an optimal pair for problem ProbPSPM. Then, there exists a process $(p,q)\in \mathcal{P}\times\mathcal{Q}$, such that

Theorems & Definitions (19)

  • Theorem 2.1
  • Remark 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Definition 3.5
  • ...and 9 more