Exact controllability of a class of nonlinear distributed parameter systems using back-and-forth iterations

Abstract

ABSTRACTWe investigate the exact controllability of a nonlinear plant described by the equation , where t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group on a Hilbert space X, B and , defined on Hilbert spaces U and , respectively, are admissible control operators for and the function is continuous in t and Lipschitz in x, with Lipschitz constant independent of t. Thus, B and can be unbounded as operators from U and to X, in which case the nonlinear term in the plant is in general not Lipschitz in x. We assume that there exist linear operators F and Fb such that the triples and are regular and A + BFΛ and −A + BFb, Λ are generators of operator semigroups and on X such that decays to zero exponentially. We prove that if is sufficiently small, then the nonlinear plant is exactly controllable in some time τ > 0. Our proof is constructive, i.e. given an initial state x0 ∈ X and a final state xτ ∈ X, we propose an approach for constructing a control signal u of class L2 for the nonlinear plant which ensures that if x(0) = x0, then x(τ) = xτ. We illustrate our approach using two examples: a sine-Gordon equation and a nonlinear wave equation. Our main result can be regarded as an extension of Russell's principle on exact controllability to a class of nonlinear plants.