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Polynomial Stability for Weakly Coupled System with Partial Controls

Bopeng Rao, Qiong Zhang

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.

Polynomial Stability for Weakly Coupled System with Partial Controls

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.

Paper Structure

This paper contains 7 sections, 16 theorems, 195 equations.

Table of Contents

  1. Introduction and the main result
  2. Polynomial stability and preliminaries
  3. Proof of the main result
  4. Examples of application
  5. Optimality by spectral analysis
  6. Optimality
  7. Conclusion

Key Result

Theorem 2.1

\newlabelth1.1 Assume that the following conditions hold: Then there exists a constant $M>0$, such that for any initial data $(U_0, U_1) \in \frak D(\mathcal{A})$, the solution $U$ to system 1.3 satisfies the following polynomial decay estimate: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (39)

  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Proposition 3.1
  • ...and 29 more