Partial Complete Controllability of Semilinear Control Systems

This work is devoted to examining the partially complete controllability for deterministic semilinear systems in Hilbert spaces. Besides reviewing briefly some existing results of controllability concepts, two main sets of sufficient conditions for partial controllability concepts are proved. The strategy in both results is based on the contraction mapping principle which has played an effective role as the cornerstone of studying controllability concepts for semilinear system, provided that the corresponding linear system is partially complete controllable. The first one is simply obtained by contraction mapping theorem. However, the second result uses the generalized contraction mapping theorem. In the first part, we study the partially complete controllability of deterministic semilinear systems for any positive time. The benefit of this result is demonstrated on some appropriate examples. In the second part, we deal with the same kind of deterministic semilinear systems but with additional condition on the nonlinear part. By this technique, we can defeat the improper integral which arises when we select a suitable control operator by which a generalized contraction mapping theorem can be applied. Keywords: Contraction mapping principle, complete controllability, partial controllability, semilinear system.