Vol. 10, No. 2, 2017

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ISSN: 1948-206X (e-only)
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Exact controllability for quasilinear perturbations of KdV

Pietro Baldi, Giuseppe Floridia and Emanuele Haus

Vol. 10 (2017), No. 2, 281–322
Abstract

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in the presence of quasilinear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero (adapting a result of Baldi, Berti and Montalto (2014)), the classical Ingham inequality and the Hilbert uniqueness method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash–Moser implicit function theorems by Hörmander (1976, 1985).

Keywords
control of PDEs, exact controllability, internal controllability, KdV equation, quasilinear PDEs, observability of PDEs, HUM, Nash–Moser theorem
Mathematical Subject Classification 2010
Primary: 35Q53, 35Q93
Milestones
Received: 23 November 2015
Revised: 20 September 2016
Accepted: 12 December 2016
Published: 23 February 2017
Authors
Pietro Baldi
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Università di Napoli Federico II
Via Cintia
80126 Napoli
Italy
Giuseppe Floridia
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Università di Napoli Federico II
Via Cintia
80126 Napoli
Italy
Emanuele Haus
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Università di Napoli Federico II
Via Cintia
80126 Napoli
Italy