A Semi-Definite Programming Approach to Stability Analysis of Linear Partial Differential Equations
This work provides a systematic, computational framework for stability analysis of linear PDEs, which is important for control theory and engineering applications involving distributed parameter systems.
The paper proposes a semi-definite programming approach to algorithmically construct Lyapunov certificates for stability analysis of a broad class of linear 1-D PDEs with polynomial data, including parabolic, hyperbolic, and coupled systems. Numerical results demonstrate the method's applicability across various PDE types.
We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled PDEs. Our approach is Lyapunov based which allows us to reduce the stability problem to the verification of integral inequalities on the subspaces of Hilbert spaces. Then, using fundamental theorem of calculus and Green's theorem, we construct a polynomial problem to verify the integral inequalities. Constraining the solution of the polynomial problem to belong to the set of sum-of-squares polynomials subject to affine constraints allows us to use semi-definite programming to algorithmically construct Lyapunov certificates of stability for the systems under consideration. We also provide numerical results of the application of the proposed method on different types of PDEs.