How to establish the exact controllability of nonlinear DPS using iterations back and forth in time

Consider a nonlinear distributed parameter system (DPS) described by (t) = Ax(t)+Bu(t)+B_NN(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators T on a Hilbert space X, B and B_N, defined on Hilbert spaces U and U_N, respectively, are admissible control operators for T and N : X × [0,∞) → U_N is continuous in t and Lipschitz in x, with Lipschitz constant L_N independent of t. Thus B and B_N can be unbounded as operators from U and U_N to X, in which case the nonlinear term B_NN(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ϵ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ϵ L²([0, τ];U). We suppose that there exist linear operators F and F_b such that (A, [B B_N], F) and (-A; [B B_N], F_b) are regular triples and A + BF_Λ and -A + BF_b,Λ are generators of strongly continuous semigroups T^f and T^b on X such that ∥T_t^f∥ · ∥T_t^b∥ decays to zero exponentially. We prove that if L_N is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.