Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact Cost-Increment Formula for Optimal Control of Semilinear Evolution Equations

Roman Chertovskih, Nikolay Pogodaev, Maxim Staritsyn, A. Pedro Aguiar

Abstract

We address optimal control of semilinear evolution equations on Banach spaces with finitely many control channels, a framework encompassing a broad class of infinite-dimensional dynamical systems, arising in many applications. For this setting, we derive an exact and global formula quantifying the increment of the cost functional with respect to an arbitrary reference control. This identity enables the design of monotone descent algorithms that require no linearization or step-size tuning. We further establish the existence of optimal controls and propose a practical sample-and-hold realization of the descent step suitable for numerical implementation. The effectiveness of the method is demonstrated on a controlled reaction-diffusion equation.

Exact Cost-Increment Formula for Optimal Control of Semilinear Evolution Equations

Abstract

We address optimal control of semilinear evolution equations on Banach spaces with finitely many control channels, a framework encompassing a broad class of infinite-dimensional dynamical systems, arising in many applications. For this setting, we derive an exact and global formula quantifying the increment of the cost functional with respect to an arbitrary reference control. This identity enables the design of monotone descent algorithms that require no linearization or step-size tuning. We further establish the existence of optimal controls and propose a practical sample-and-hold realization of the descent step suitable for numerical implementation. The effectiveness of the method is demonstrated on a controlled reaction-diffusion equation.
Paper Structure (12 sections, 5 theorems, 55 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 5 theorems, 55 equations, 1 figure, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem Statement and Analytical Foundations
  3. Notation, Standing Assumptions & Preliminaries
  4. Differentiability of the Mild Flow
  5. Existence of Optimal Controls
  6. Exact Cost-Increment Formula and Monotone Descent Method
  7. Implementation via Sample‑and‑Hold Updates
  8. Numerical Illustration: Reaction–Diffusion Control
  9. CONCLUSION
  10. Proof of Lemma \ref{['lem:diff-mild-flow']}
  11. Proof of Corollary
  12. Proof of Proposition \ref{['pro0']}

Key Result

Lemma 1

Assume $({\bf A})$ and $({\bf A}_+)$. Then the following statements hold.

Figures (1)

  • Figure 1: Controls generated by Alg. 1 (upper panel) and terminal distributions $\rho_T$ (lower panel) --- optimized by Alg. 1 (green) and the uncontrolled one (red) vs the target profile (dotted).

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Proposition 1