Spatially Controlled Evolution of Composite Materials via Stochastic Partial Differential Equations

TL;DR

This paper develops a mathematical framework for controlling the spatial evolution of composite materials modeled by stochastic partial differential equations, providing existence results, optimality conditions, and explicit solutions for specific cases.

Contribution

It introduces a novel approach to controlling composite materials via SPDEs, including existence proofs, optimality conditions, and explicit solutions for piecewise constant diffusivity.

Findings

Existence of mild solutions under regularity conditions

Derivation of stochastic maximum principle for optimal control

Explicit solutions for specific composite material models

Abstract

This paper investigates a class of controlled stochastic partial differential equations (SPDEs) arising in the modeling of composite materials with spatially varying properties. The state equation describes the evolution of a material property, influenced by control inputs that adjust the diffusivity in different spatial regions. We establish the existence of mild solutions to the SPDE under appropriate regularity conditions on the coefficients and the control. A derivation of the sufficient and necessary conditions for optimality is provided using the stochastic maximum principle. These conditions connect the state dynamics to adjoint processes, enabling the characterization of the optimal control in terms of the curvature of the state and the sensitivity of the cost. Two explicit solvable examples are presented to illustrate the theoretical results, where the optimal control is computed explicitly for a composite material with piecewise constant diffusivity.