Regional Control Problem for Distributed Bilinear Systems: Approach and Simulations

where A is the generator of a strongly continuous semigroup (S(t))t≥0 on the state space Z =: L(Ω) endowed with its natural inner product 〈·, ·〉, and the corresponding norm ‖ · ‖, B : Z → Z is a linear bounded operator, while u ∈ L[0, T ] is a control. The main result on the controllability of the system (1) is due to the pioneering work by Ball et al. (1982), which shows that, under the above-mentioned conditions, a mild solution zu of (1) associated with the control u exists and the set of reachable states from an initial state z0 is of dense complement in the state space. This makes exact controllability difficult to be achieved. Most results are established for particular bilinear systems (Ball et al., 1982; Joshi, 2005; Lenhart and Liang, 2000; Khapalov, 2002a; 2002b). Later the concept of regional controllability for linear distributed systems has been introduced and developed by El Jai and Zerrik and concerns the transfer of such a system to a desired state only on a region of the system spacial domain. The system (1) is said to be exactly (respectively, approximately) controllable in ω ⊂ Ω if for all zd ∈ L(ω) there exists a control u ∈ L[0, T ] such that χωzu(T ) = zd (respectively, ||χωzu(T ) − zd||L2(ω) ≤ ε, ε > 0), where zd is a desired state in the space L(ω), χω : Z −→ L(ω) is the restriction operator to ω. Many results for linear and semi linear systems have been developed (see El Jai et al., 1995; Zerrik and Kamal, 2007; Zerrik et al., 2007).