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A Semi-Definite Programming Approach to Stability Analysis of Linear Partial Differential Equations
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
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