Défense de thèse de doctorat en sciences mathématiques

Invariant stabilization of discretized boundary control systems

Jury

Christophe PRIEUR (Gipsa-Lab Grenoble), Alain VANDE WOUWER (UMons), Anne LEMAITRE, présidente (UNamur), Alexandre MAUROY (UNamur), Joseph WINKIN, promoteur (UNamur)

Résumé

Stabilization and invariance are the two keywords of this work. By invariant stabilization, one should understand the asymptotic stabilization of a system while keeping the state trajectories in a predetermined domain. First, we deal with the positive linear time-invariant (LTI) finite-dimensional systems for which we discuss the relevance of choosing a nonnegative input for the stabilization process, we provide a parameterization of all positively stabilizing feedbacks for a particular class of positive systems, and we extend the concept of invariance to cones, sectors and Lyapunov level sets. Then, we adapt the results to the positive LTI infinite-dimensional systems, we explain how one can switch from an input acting in the boundary conditions to an input acting in the dynamics, we introduce the standard example of the pure diffusion, and we discuss the boundary conditions when discretizing a PDE system. Finally, we deal with the positive nonlinear time-invariant (NTI) infinite-dimensional systems, for which we once again adapt the previous theoretical results and consider a relevant example, namely a biochemical reactor model.

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