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New cost terms through the homogenization of an optimal control problem under dynamic boundary conditions on the microscopic particles

J. I. Díaz, T. A. Shaposhnikova, A. V. Podolskiy

TL;DR

This work analyzes the homogenization of an optimal control problem in a periodically perforated domain with dynamic boundary conditions on a subset of micro-particles. By focusing on a critical scaling that relates the structure period, particle size, and the boundary-growth coefficient, it reveals the emergence of non-local in time "strange terms" in both the limit parabolic equation and the limit cost functional. A novel set of non-local in time operators $G$, $H$, and $M$ (and their adjoints) are introduced to capture boundary-control effects, and the authors derive a homogenized system and a limit cost functional $J_0$ that incorporate these terms. They prove convergence of microscopic optimal controls to a macroscopic control $v_0=-N^{-1}H(G^{\ast}(p_0))$ and establish the corresponding limit optimality conditions, highlighting how localized boundary actuation can generate nonlocal-in-time phenomena with practical implications for control design in porous media and related applications.

Abstract

Given an optimal control problem on a heterogeneous body with a periodical structure of particles depending on a small parameter e, we study the asymptotic behavior, as e converges to zero, of the optimal control functional and the optimal state when the initial problem is of parabolic type, and when on the particles' boundary, we assume a dynamic condition and the actuation of some controls for some subset of the particles. We show, in the so-called "critical case" (concerning a certain relation between the structure's period, the diameter of the balls, and the growth coefficient of the particles boundary condition), the appearance of some new non-local in time "strange terms", not only in the limit parabolic equation but also in the limit cost functional. Microscopic localized controls generate peculiar terms in both the limit equation and the cost functional that do not appear in the case of controls applied to the entire set of particles or when the boundary condition on the particles is of Robin type.

New cost terms through the homogenization of an optimal control problem under dynamic boundary conditions on the microscopic particles

TL;DR

This work analyzes the homogenization of an optimal control problem in a periodically perforated domain with dynamic boundary conditions on a subset of micro-particles. By focusing on a critical scaling that relates the structure period, particle size, and the boundary-growth coefficient, it reveals the emergence of non-local in time "strange terms" in both the limit parabolic equation and the limit cost functional. A novel set of non-local in time operators , , and (and their adjoints) are introduced to capture boundary-control effects, and the authors derive a homogenized system and a limit cost functional that incorporate these terms. They prove convergence of microscopic optimal controls to a macroscopic control and establish the corresponding limit optimality conditions, highlighting how localized boundary actuation can generate nonlocal-in-time phenomena with practical implications for control design in porous media and related applications.

Abstract

Given an optimal control problem on a heterogeneous body with a periodical structure of particles depending on a small parameter e, we study the asymptotic behavior, as e converges to zero, of the optimal control functional and the optimal state when the initial problem is of parabolic type, and when on the particles' boundary, we assume a dynamic condition and the actuation of some controls for some subset of the particles. We show, in the so-called "critical case" (concerning a certain relation between the structure's period, the diameter of the balls, and the growth coefficient of the particles boundary condition), the appearance of some new non-local in time "strange terms", not only in the limit parabolic equation but also in the limit cost functional. Microscopic localized controls generate peculiar terms in both the limit equation and the cost functional that do not appear in the case of controls applied to the entire set of particles or when the boundary condition on the particles is of Robin type.
Paper Structure (7 sections, 5 theorems, 166 equations, 1 figure)

This paper contains 7 sections, 5 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem statement and the adjoint problem
  3. A priori estimates
  4. Auxiliary non-local in time operators $G,$ $H$ and $M$
  5. Statement of the homogenization theorems
  6. Proof of the homogenization theorem
  7. Proof of the limit cost functional and controls convergence theorems

Key Result

Theorem 1

Let the pair of functions $(u_{ \if@compatibility \mathchar"0122 {} \mathchar"0122 }(v_{ \if@compatibility \mathchar"0122 {} \mathchar"0122 }), v_{ \if@compatibility \mathchar"0122 {} \mathchar"0122 })$ be an optimal solution of the problem optim control def, then $v_{ \if@compa

Figures (1)

  • Figure 1: Perforated domain. The control is implemented only on $S_{{ \if@compatibility \mathchar"0122 {} \mathchar"0122 } }^{2}$, the boundary of some of the internal balls: the ones collected under the $G^{2}_ { \if@compatibility \mathchar"0122 {} \mathchar"0122 }$.

Theorems & Definitions (12)

  • Remark 1
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof
  • proof
  • ...and 2 more