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DISCO - 2011

Section: New Results

Partial Differential Equations

Participants : Frédéric Mazenc, Christophe Prieur [GIPSA Lab, CNRS] .

In [33] and [65] , for families of partial differential equations (PDEs) with particular stabilizing boundary conditions, we have constructed strict Lyapunov functions. The PDEs under consideration were parabolic and, in addition to the diffusion term, might contain a nonlinear source term plus a convection term. The boundary conditions were the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions relied on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions were 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 made it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

In [64] , we have considered a family of time-varying hyperbolic systems of balance laws. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides with a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach which gives a time-varying strict Lyapunov function which allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of Input-to-State Stability (ISS) type.