Polynomial Stability for Weakly Coupled System with Partial Controls

TL;DR

This paper establishes polynomial stability for weakly coupled systems with fewer controls, showing exponential stability of scalar equations implies polynomial decay in the coupled system, regardless of the number of equations.

Contribution

It extends stability results to weakly coupled systems under Kalman's rank condition, demonstrating decay rates are unaffected by the number of equations and applying the theory to various models.

Findings

Polynomial stability holds under Kalman's rank condition.

Decay rate remains unchanged regardless of system size.

Optimal decay rates are confirmed via spectral analysis.

Abstract

We study the stability of general weakly coupled systems subject to a reduced number of local or boundary controls. We show that, under Kalman's rank condition, the exponential stability of the underlying scalar equation implies polynomial stability of the full coupled system. Moreover, the decay rate remains unchanged regardless of the number of equations in the system. The proof relies on resolvent estimates and a clever exploitation of Kalman's rank condition to ensure effective transmission of damping across the coupled equations. The abstract result is applied to several concrete models, including systems of wave equations with local viscous, local viscoelastic, or boundary damping; systems of plate equations with internal damping; and thermoelastic systems of type III. Moreover, the optimality of the decay rate is established via spectral analysis.