Résumé:
This thesis explores, on one hand, the minimum energy control problem for an infinite-dimensional degenerate Cauchy problem with variable operator coefficients, skew-hermitian pencil, and bounded input and for a finite dimensional singular dynamical systems with rectangular inputs, and on the other hand, the problem of stability and stabilization for the infinite-dimensional dynamical systems. For the first problem, the investigation follows a set of methods and techniques, among them, the Weierstrass theorem and some concepts of controllability in the case of finite dimension, and the orthogonal decomposition, the Gramian operator, and a suitable expression of the input in the case of infinite dimension. More than that, a procedure for computing the optimal input and minimizing the performance index is proposed for both cases. Then, for the second problem, some existing results on stability and stabilization have been extended for infinite-dimensional dynamical systems with bounded operator coefficients. The promising results that we have obtained encouraged us to study the problem of minimum energy control for infinite-dimensional dynamical systems and to analyse their stabilities and stabilizations.
