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

Abstract

Consider a nonlinear distributed parameter system (DPS) described by ẋ (t) = Ax(t)+Bu(t)+B N N(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 N N(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) ∈ 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 ∈ 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 𝕋f and 𝕋b 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.