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.

• The paper demonstrates that if the scalar control system is exponentially stable and the coupling matrices satisfy Kalman’s rank condition, the full weakly coupled system attains polynomial stability.
• It employs resolvent analysis and semigroup theory to derive energy decay rates that remain independent of the number of coupled equations.
• Concrete PDE models, including wave, plate, and thermoelastic systems, validate the approach and highlight the effect of partial damping on overall system performance.

Polynomial Stability for Weakly Coupled Systems with Partial Controls

Introduction

This paper addresses the stability properties of weakly coupled systems of evolution equations, particularly when only a partial subset of the system components are subject to control—referred to as partial or reduced control. The central result establishes a direct link between exponential stability of the underlying scalar (single-component) equation and polynomial stability of the entire weakly coupled system under Kalman's rank condition on the coupling and control matrices. Furthermore, the polynomial decay rate is shown to be independent of the number of equations in the coupled system, and several concrete PDE models are analyzed as examples.

Abstract Coupled System and Main Result

Consider a set of $N$ weakly coupled second-order evolution equations in a Hilbert space framework, where the coupling between equations is through displacement terms, and the controls are applied to only a subset of the components. The abstract formulation is:

$$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$

where $A$ and $D$ are symmetric, positive semi-definite coupling and control matrices, respectively. The operator $\mathcal{G}$ may represent local, boundary, or more general damping mechanisms. The paper imposes the Kalman rank condition:

$\mathrm{rank}[D, AD, ..., A^{N-1}D] = N,$

and a small non-commutativity condition $\|AD - DA\|$.

Main Theorem:

• If the underlying scalar control system is exponentially stable,
• $A$ and $D$ satisfy the Kalman rank condition and are nearly commutative,
• and a damping operator regularity holds,

then the full system is polynomially stable: the energy decays as $O(t^{-1/2(1+r)})$ where $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$0 is determined by the regularity (interpolation) of the damping term.

This is formally proved via resolvent analysis, exploiting the abstract semigroup theory and the fine algebraic structure provided by the Kalman condition.

Notably:

• The decay rate and its optimality are independent of $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$1, i.e., the number of coupled equations.
• Exponential stability is not typically possible with partial control—polynomial rates are the best achievable unless more components are damped or the coupling is strong.

Resolvent Approach to Polynomial Stability

A central technical ingredient is the estimate of the resolvent operator of the infinitesimal generator of the evolution, along the imaginary axis. Polynomial decay of the semigroup is characterized by the growth bound of the resolvent:

$$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$2

implies solution energy decays as $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$3. Sharp conditions and rates are obtained by spectral analysis—if the generator has eigenvalues approaching the imaginary axis with real parts decaying like $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$4, this gives optimality of the rate.

The proof relies on reducing the original resolvent problem to the scalar equation, utilizing the nearly simultaneous diagonalization of $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$5, and tracing how the damping applied to a subset of variables disperses (or fails to disperse) across the coupled system.

Concrete Models

The theory is instantiated for several canonical PDE systems:

1. Coupled Wave Equations with Local or Boundary Damping

• Exponential stability of the scalar damped wave is classical under the geometric control condition.
• When coupling $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$6 and control $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$7 satisfy the Kalman condition, polynomial decay holds even if only a single component is damped.

2. Coupled Plate Equations with Local Damping

• Employs biharmonic operators, with similar algebraic structure to the wave model.
• The polynomial rate is again governed by the regularity of the damping operator.

3. Kelvin-Voigt and Viscoelastic Damping

• Regularity of the operator becomes lower; thus, the polynomial rate worsens (e.g., $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$8 for Kelvin-Voigt).

4. Boundary Damping: Dirichlet/Neumann Trace

• Here, trace regularity yields decay rates of $$U'' + \mathcal{L} U + AU + D\mathcal{G}^* \mathcal{G} U' = 0,$$9 (multidimensional) or $A$0 (1D).

5. Thermoelasticity Models (Type III)

• Involves both displacement and temperature variables, coupled with the heat flux damping.
• Analysis extends to $A$1-component systems.

For each, the framework gives a direct method to verify polynomial stability by checking the algebraic (Kalman) condition and the regularity of the scalar damping.

Spectral Analysis and Optimality

Sharpness/optimality of the decay rates is rigorously demonstrated by explicit spectral computation for specific $A$2, $A$3, and underlying domain/operator choices. For instance:

• Two-component wave and plate systems with specially constructed $A$4 and $A$5 matrices exhibit eigenvalues converging to the imaginary axis at rates matching the predicted decay.
• For boundary-damped problems, eigenvalue expansion confirms that $A$6 decay is indeed optimal in 1D, in contrast with $A$7 in higher dimensions.

Implications and Open Questions

Practical Implications

• The result allows for stabilization of large-scale coupled systems in applications (e.g., structural vibration, thermoelasticity) by damping only a minimal number of components, provided the coupling is sufficient and satisfies the Kalman algebraic condition.
• The technique reduces the complexity of stability analysis for multiphysics and multivariate PDE systems, important for controllability and observer design.

Theoretical Implications and Future Directions

• The general method establishes a blueprint for analyzing polynomial stability in other indirectly damped or partially controlled PDE systems.
• The independence of decay rates from the system size (number of equations) is analytically significant.

Open Questions for Future Research:

• Necessary and sufficient conditions for polynomial/exponential stability in strongly (velocity- or higher-derivative) coupled systems.
• Analysis of systems with non-symmetric or non-positive-definite coupling.
• Extension to systems with variable wave speeds or non-uniform coefficients.
• Complete characterization of resolvent behavior for non-selfadjoint or degenerate settings.

Conclusion

The paper develops a rigorous, general polynomial stability theory for abstract weakly coupled linear systems under partial damping, employing algebraic control-theoretic criteria. The results unify and extend prior results for indirectly damped PDEs, providing explicit rates and demonstrating optimality via resolvent spectral analysis. The approach is robust and broadly applicable, setting a foundation for future work on stabilization and control of complex multi-component PDE systems.