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

We investigate the exact controllability of a nonlinear plant described by the equation (Formula presented.), where t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group 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 the function 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 (Formula presented.) in the plant is in general not Lipschitz in x. We assume that there exist linear operators F and F _b such that the triples (A, [B B _N ], F) and (−A, [B B _N ], F _b ) are regular and A + BF _Λ and −A + BF _{b, Λ} are generators of operator semigroups T ^f and T ^b and ǁT ^f _t ǁǁT ^b _t ǁ on X such that (Formula presented.) decays to zero exponentially. We prove that if L _N 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 x ₀ ∈ X and a final state x _τ ∈ X, we propose an approach for constructing a control signal u of class L ² for the nonlinear plant which ensures that if x(0) = x ₀ , then x(τ) = x _τ . 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.