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

Vivek Natarajan*, Hua Cheng Zhou, George Weiss, Emilia Fridman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)145-162
Number of pages18
JournalInternational Journal of Control
Issue number1
StatePublished - 2 Jan 2019


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


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