Absolute stability of a heterogeneous semilinear dissipative parabolic PDE

We analyze absolute stability of the equilibrium solution of a semilinear dissipative parabolic PDE with a spatially varying nonlinearity that satisfies a given sector condition. Stability is shown based on Lyapunov analysis of the infinite-dimensional dynamics. The pertinent linear operators are expressed in terms of their infinite-dimensional matrix representations, some of which have a Toeplitz structure due to the spatial heterogeneity of the nonlinearity. The time derivative of the Lyapunov function is expressed as a sum of finite-dimensional expressions. The analysis is described in terms of finite-dimensional linear matrix inequalities (LMI). Sufficient conditions, in terms of a finite set of finite-dimensional LMI, are given to establish absolute stability. Numerical simulations are presented for a system with Dirichlet boundary conditions with spatially varying saturation nonlinearities.