Boundary control of heat-heat cascades

TL;DR

This paper develops a spectral reduction approach for boundary control and stabilization of coupled heat equations in a cascade, providing explicit feedback strategies and extending results to dual problems.

Contribution

It introduces a novel spectral reduction method directly for the PDE cascade, enabling explicit feedback control design for stabilization.

Findings

Spectral analysis characterizes controllability of PDE cascade modes.

Explicit state-feedback control strategies achieve exponential stabilization.

Results extend to dual problems and include output-feedback control in different measurement settings.

Abstract

This paper addresses the problem of feedback stabilization of a cascade of two heat equations that are coupled in the boundary conditions, the input being a boundary control for the first component of the cascade. Two distinct control input settings are studied: one being collocated with the coupling condition of the two heat equations, and the other being noncollocated. These two different configurations induce different controllability properties. The key idea developed in this paper is to carry out spectral reductions, not for each of the two components of the cascade separately, but instead, directly for the PDE cascade viewed as one single system. A detailed study of the eigenelements of the PDE cascade yields a complete characterization of the spectral mode controllability and allows us to derive an explicit state-feedback control strategy for the exponential stabilization of the plant. This approach is extended to a systematic output-feedback control strategy either with a distributed output operator or with a pointwise measurement done on the second heat equation of the PDE cascade. In both state-feedback and output-feedback scenarios, stabilization results are established in L^2 and H^1 norms. Finally, we show how the results developed in this paper for the two studied heat-heat cascades extend to their dual problems.