# On the switching behavior of sparse optimal controls for the one-dimensional heat equation

## TL;DR

This paper investigates the switching behavior of sparse optimal controls for the 1D heat equation, analyzing how control switches occur and converge as regularization diminishes, with implications for control sparsity and structure.

## Contribution

The study characterizes the countability and accumulation points of switching points in sparse controls and examines their convergence as the regularization parameter approaches zero.

## Key findings

- Switching points are countable with the final time as the only accumulation point.
- Convergence of switching points is established as the regularization parameter tends to zero.
- The control structure exhibits specific sparsity and switching behavior under natural assumptions.

## Abstract

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $\nu$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $\nu \ge 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $\nu \searrow 0$.