A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework

Abstract

We present a framework for stability analysis of systems of coupled linear Partial-Differential Equations (PDEs). 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 L2 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 and can analyze systems of up to 20 coupled PDEs. Abstract Proofs of the Lemmas in "A New State-Space Representation for Coupled PDEs and Scalable Lyapunov StabilityAnalysis in the SOS Framework". These proofs are not compressed in order to allow for easier review and verification.The proofs are relatively straightforward and almost all manipulation in all proofs is based on three basic ways to use indicator functions to change the order of integration.

Computing Input-Ouput Properties of Coupled Linear PDE systems

“…We restate main result from Lemmas 4, 5 and 6 from [12]. These results are used in the proof of Lemma 4.2 in Section V.…”

“…Consider the system shown below. In [12], it was shown to be stable for λ < 4.65. At at mesh size of 600, we had an L 2 gain of 14.82 (LMI bound was 14.99).…”

“…In our approach, we use linear operators parameterized by matrix-valued polynomials to parameterize quadratic Lyapunov functionals for infinite-dimensional systems. This is an extension of the work on stability analysis in [12]. Note that such an approach was previously used for time-delay systems (e.g.…”

“…From (9), Notation: In the following Lemma, L 1 , L 2 and L 3 are linear maps between matrix valued polynomials that satisfy Lemmas 4, 5 and 6 of [12], respectively. Detailed definition of these maps can be found in the appendix.…”

A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

“…Lemma 3.3 can be proved by using fundamental theorem of calculus; refer [20]. Now, clearly, primal state, z p , is dependent on the boundary conditions and can be directly embedded into the dynamics.…”