# Time-varying Bang-bang Property of Minimal Controls for Approximately Null-controllable Heat Equations

## TL;DR

This paper investigates the time-varying bang-bang property of minimal controls in heat equations that are approximately null-controllable, introducing new boundary conditions for control variables and establishing key equivalence results.

## Contribution

It extends existing control theory by analyzing time-varying control boundaries and proving a bang-bang property for optimal controls in heat equations.

## Key findings

- Established the time-varying bang-bang property for optimal controls.
- Proved an equivalence theorem between optimal control and target problems.
- Extended control boundary conditions from constants to functions.

## Abstract

In this paper, optimal time control problems and optimal target control problems are studied for the approximately null-controllable heat equations. Compared with the existed results on these problems, the boundary of control variables are not constants but time varying functions. The time-varying bang-bang property for optimal time control problem, and an equivalence theorem for optimal control problem and optimal target problem are obtained.