Strict Lyapunov functions for semilinear parabolic partial differential equations

For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.