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Spatially Controlled Evolution of Composite Materials via Stochastic Partial Differential Equations

Nacira Agram, Isabelle Turpin, Eya Zougar

TL;DR

The paper studies controlled parabolic SPDEs modeling composite materials with spatially varying diffusivity, formulating a mild-solution framework for the state $Y(t,x)$ under a control $u(t)$. A stochastic maximum principle is derived, yielding a Hamiltonian $H$ and a backward adjoint BSPDE for $(p,q)$, which together characterize optimal controls through convexity and variational conditions. Existence and uniqueness of mild solutions are established for both linear and nonlinear cases, with explicit fundamental solutions $G$ for the deterministic part and careful treatment of piecewise coefficients. Two solvable examples with piecewise constant diffusivity illustrate explicit optimal controls and the coupling between state, adjoint, and control, highlighting applications to temperature regulation and heat storage in heterogeneous composites.

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.

Spatially Controlled Evolution of Composite Materials via Stochastic Partial Differential Equations

TL;DR

The paper studies controlled parabolic SPDEs modeling composite materials with spatially varying diffusivity, formulating a mild-solution framework for the state under a control . A stochastic maximum principle is derived, yielding a Hamiltonian and a backward adjoint BSPDE for , which together characterize optimal controls through convexity and variational conditions. Existence and uniqueness of mild solutions are established for both linear and nonlinear cases, with explicit fundamental solutions for the deterministic part and careful treatment of piecewise coefficients. Two solvable examples with piecewise constant diffusivity illustrate explicit optimal controls and the coupling between state, adjoint, and control, highlighting applications to temperature regulation and heat storage in heterogeneous composites.

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.

Paper Structure

This paper contains 12 sections, 10 theorems, 114 equations, 2 figures.

Table of Contents

  1. Introduction
  2. SPDE representation
  3. Some properties of the operator ${\mathcal{A}}$
  4. Existence and uniqueness of the mild solution to Equation (\ref{['equation eq:1']})
  5. Linear case
  6. Non-linear case
  7. Optimization problem
  8. Sufficient Optimality Condition
  9. Necessary Optimality Condition
  10. Applications
  11. Optimal Temperature Control in Composite Materials
  12. Heat Storage Control in Composite Materials

Key Result

Lemma 3

Let $Y\in L^2([0,T]\times { \mathbb{R}} ).$ Then, for any $\varphi \in C^{\infty}_c([0,T]\times { \mathbb{R}} ),$ we have with respect to the measure $\rho(x)\,dx.$ Here, the adjoint operator ${\mathcal{A}}^*$ is given by: where, $\delta_0$ refers to the Dirac measure and $\langle{.,.\rangle}$ is the scalar production of $L^2([0,T]\times { \mathbb{R}}).$

Figures (2)

  • Figure 1: An example of the composite material.
  • Figure 2: Dynamics of a composite material with two regions: Material 1 ($x \leqslant 0$) and Material 2 ($x > 0$). The control $u$ dynamically adjusts thermal diffusivity to ensure uniform temperature distribution.

Theorems & Definitions (19)

  • Lemma 3
  • Remark 4
  • Definition 5
  • Proposition 6
  • Proposition 7
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • Corollary 10
  • ...and 9 more