Internal stabilization of an underactuated linear parabolic system via modal decomposition (extended version)

TL;DR

This paper develops an explicit internal stabilization method for underactuated linear heat equation systems using modal decomposition and state transformation, enabling stabilization with limited actuators and distinct diffusion coefficients.

Contribution

It introduces a novel stabilization approach combining modal decomposition with a state transformation, applicable to systems with partial actuation and different diffusion coefficients.

Findings

Explicit stabilizing control law derived

Applicable to systems with actuators in only some states

Stabilization achieved without boundary control methods

Abstract

This work concerns the internal stabilization of underactuated linear systems of $m$ heat equations in cascade, where the control is placed internally in the first equation only and the diffusion coefficients are distinct. Combining the modal decomposition method with a recently introduced state-transformation approach for observation problems, a proportional-type stabilizing control is given explicitly. It is based on a transformation for the ODE system corresponding to the comparatively unstable modes into a target one, where calculation of the stabilization law is independent of the arbitrarily large number of them and it is achieved by solving generalized Sylvester equations recursively. This provides a finite-dimensional counterpart of a recently introduced infinite-dimensional one, which led to Lyapunov stabilization. The present approach answers to the problem of stabilization with actuators not appearing in all the states and when boundary control results do not apply.