# Stabilization of Partial Differential Equations by Sequential Action Control

## TL;DR

This paper introduces a sequential action control framework for stabilizing partial differential equations, utilizing variational principles and adjoint information, with applications demonstrated on an unstable heat equation.

## Contribution

The paper develops a novel SAC framework for PDE stabilization, explicitly deriving control actions and analyzing closed-loop systems in Hilbert spaces.

## Key findings

- Control actions can be explicitly obtained from variational principles.
- The framework successfully stabilizes an unstable heat equation.
- Numerical results verify the effectiveness of the approach.

## Abstract

We present a framework of sequential action control (SAC) for stabilization of systems of partial differential equations which can be posed as abstract semilinear control problems in Hilbert spaces. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyze the closed-loop system obtained from the SAC feedback for the linear problem with quadratic stage costs. We apply this theory to a prototypical example of an unstable heat equation and provide numerical results as the verification and demonstration of the framework.