Sur quelques problèmes relatifs aux systèmes non linéaires : elinéarisation statique et singularités, stabilisation globale de certains systèmes

Abstract : This work contains three parts. The first one deals with the static noninteracting control problem with stability for bilinear systems. For this class of nonlinear systems, the decoupling matrix is singular on an algebraic surface which contains the origin, and this relates the noninteracting control with stability to the difficult problem of completeness of trajectories of the closed-loop system and/or boundedness of feedback laws. We consider here two inputs-two outputs bilinear systems without zero dynamics, for which we give sufficient conditions for noninteracting control with stability by linearizing feedback. The second part is devoted to the global stabilization problem for nonlinear systems. First, we consider partially linear systems for which various authors gave global stabilization sufficient conditions by means of a strict lyapunov function. Unfortunately, there is no systematic method to compute such a function. Here we show how to compute the global stabilizing feedback thanks to a weak Lyapunov function satisfying LaSalle's invariance principle instead of a strict one. The advantage of our result is that for a large class of systems it is easier to construct a weak Lyapunov function than a strict one. This is tipically the case for mechanical systems for example. Later on, we give a global stabilization sufficient condition for nonaffine control systems generalizing the well known Jurdjevic-Quinn condition for affine control systems. The last part extends deterministics stabilisation results to nonlinear stochastic systems. We give a sufficient condition for global stabilization of partially linear stochastic systems. We also give, for nonaffine control stochastic systems, a version of Jurdjevic-Quinn condition
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Abdelhak Ferfera. Sur quelques problèmes relatifs aux systèmes non linéaires : elinéarisation statique et singularités, stabilisation globale de certains systèmes. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1997. Français. ⟨NNT : 1997METZ004S⟩. ⟨tel-01777192⟩

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