2018/03/20 by Matthew M. Peet, Peet, Matthew M. · 1 citation
Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Electrical engineering #FOS: Mathematics #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1803.07290
openalex publication_date 2018/03/20 · openalex created_date 2018/04/13 · openalex updated_date 2026/08/01
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 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.