Bang-bang property of time optimal controls for some semilinear heat equation

TL;DR

This paper proves a bang-bang property for time optimal controls in certain semilinear heat equations, showing controls are at their maximum magnitude almost everywhere in the optimal control process.

Contribution

It establishes the bang-bang property for a class of semilinear heat equations with distributed controls, extending known results to nonlinear settings.

Findings

Controls are at their maximum magnitude in the optimal solution.

The bang-bang property holds under specified boundary and domain conditions.

The results apply to bounded $C^2$ domains with homogeneous Dirichlet boundary conditions.

Abstract

In this paper, we derive a bang-bang property of a kind of time optimal control problem for some semilinear heat equation on bounded $C^2$ domains (of the Euclidean space), with homogeneous Dirichlet boundary condition and controls distributed on an open and non-empty subset of the domain where the equation evolves.