On the explicit feedback stabilisation of 1D linear nonautonomous parabolic equations via oblique projections

TL;DR

This paper introduces a new explicit feedback stabilization method for 1D linear nonautonomous parabolic equations using oblique projections, proving boundedness of the projections' operator norms for specific actuator sequences and demonstrating numerical effectiveness.

Contribution

It provides a novel explicit feedback stabilization technique based on oblique projections with proven bounded operator norms for certain actuator configurations.

Findings

Operator norms of oblique projections remain bounded for explicit actuator sequences.

Numerical results confirm the effectiveness of the feedback control under different boundary conditions.

Abstract

In recently proposed stabilisation techniques for parabolic equations, a crucial role is played by a suitable sequence of oblique projections in Hilbert spaces, onto the linear span of a suitable set of M actuators, and along the subspace orthogonal to the space spanned by the first M eigenfunctions of the Laplacian operator. This new approach uses an explicit feedback law, which is stabilising provided that the sequence of operator norms of such oblique projections remains bounded. The main result of the paper is the proof that, for suitable explicitly given sequences of sets of actuators, the operator norm of the corresponding oblique projections remains bounded. In the final part of the paper we provide numerical results, showing the performance of the explicit feedback control for both Dirichlet and Neumann homogeneous boundary conditions.