# Time optimal sampled-data controls for heat equations

## TL;DR

This paper develops a framework for time optimal sampled-data controls for heat equations, providing error estimates and connections to distributed controls, advancing numerical methods for such control problems.

## Contribution

It introduces a novel approach to approximate time optimal distributed controls using sampled-data controls, with proven error bounds and theoretical properties.

## Key findings

- Error estimates for control and time between sampled-data and distributed controls
- Connections established among sampled-data control problems, minimal norm, and minimization problems
- Proven optimality of the error bounds in terms of sampling period

## Abstract

In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls. Our design is reasonable from perspective of sampled-data controls. And it might provide a right way for the numerical approach of a time optimal distributed control problem, via the corresponding semi-discretized (in time variable) time optimal control problem. The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem. And obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but also prove that such estimates are optimal in some sense.