Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation

TL;DR

This paper investigates a penalized optimal control approach for the heat equation with a terminal state constraint, providing explicit convergence rates and illustrating the projection structure through numerical examples.

Contribution

It introduces a quantitative convergence analysis for penalized controls in heat equation terminal constraints, including explicit rates and modal assumptions.

Findings

Optimal controls converge to the constrained solution as penalty parameter increases.

Explicit convergence rate of O(1/α) under stronger modal assumptions.

Numerical illustrations demonstrate the projection structure and convergence behavior.

Abstract

We study an optimal control problem for the heat equation with a prescribed terminal state. To circumvent the difficulty of enforcing a hard terminal constraint, we analyze a penalized formulation and prove that the corresponding optimal controls and terminal states converge to the exact constrained solution as the penalty parameter \(\alpha \to \infty\). We establish explicit quantitative convergence estimates of order \(O(\alpha^{-\theta})\), including the sharp \(O(1/\alpha)\) rate under stronger modal summability assumptions on the terminal mismatch. A finite-dimensional prototype is used to illustrate the underlying projection structure, while numerical illustrations are reported in a companion study.