A New State-Space Representation of Lyapunov Stability for Coupled PDEs and Scalable Stability Analysis in the SOS Framework - with Lemma Proofs
This work provides a scalable and less conservative framework for stability analysis of coupled PDEs, which is important for control theory and applied mathematics.
The paper introduces a new state-space representation for Lyapunov stability analysis of coupled linear PDEs, enabling direct formulation as a linear operator inequality on L2 without integration by parts. Numerical tests show little or no conservatism for a broad class of systems.
In this paper, we present a framework for Stability Analysis of Systems of Coupled Linear Partial-Differential Equations. The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable and assume existence and continuity of solutions except in such cases when existence and continuity can be inferred from existence of a Lyapunov function. Our approach is based on a new concept of state for PDE systems which allows us to express the derivative of the Lyapunov function as a Linear Operator Inequality directly on $L_2$ and allows for any type of suitably well-posed boundary conditions. This approach obviates the need for integration by parts, spacing functions or similar mathematical encumbrances. The resulting algorithms are implemented in Matlab, tested on several motivating examples, and the codes have been posted online. Numerical testing indicates the approach has little or no conservatism for a large class of systems.