The paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder's fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.

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Abstract. In this paper, we consider the problem of optimal regional controllability of a distributed bilinear system evolving on Ω. The question is to obtain a control with minimum energy that drives such a system from an initial state to a final state close to a desired one in finite time, only on a subregion ω of Ω. Our purpose is to prove that a regional optimal control exists and characterized in both bounded and unbounded cases. The obtained results are successfully illustrated by simulations.Résumé. Dans cet article nous considérons le problème de la contrôlabilité régionale optimale d'un système distribué bilinéaireévoluant sur un domaine spatial Ω. La question est d'obtenir un contrôleà energie minimale qui conduit un tel système d'unétat initial vers unétat final proche d'unétat désiré uniquement sur sur une région ω de Ω. On montre qu'un tel contrôle existe et ceractérisé dans les cas borné et non borné. Les résultats obtenus sont illustrés avec succès par des simulations.