## Abstract

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.

Original language | English |
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Pages (from-to) | 145-162 |

Number of pages | 18 |

Journal | International Journal of Control |

Volume | 92 |

Issue number | 1 |

DOIs | |

State | Published - 2 Jan 2019 |

## Keywords

- Back-and-forth iterations
- Russell's principle
- exact controllability
- nonlinear perturbation
- regular linear system
- stabilisability