1,076 research outputs found
Internal observability for coupled systems of linear partial differential equations
First published in Journal on Control and Optimization in 57.2 (2019): 832-853, published by the Society for Industrial and Applied Mathematics (SIAM)We deal with the internal observability for some coupled systems of partial differential equations with constant or time-dependent coupling terms by means of a reduced number of observed components. We prove new general observability inequalities under some Kalman-like or Silverman-Meadows-like condition. Our proofs combine the observability properties of the underlying scalar equation with algebraic manipulations. In the more specific case of systems of heat equations with constant coefficients and nondiagonalizable diffusion matrices, we also give a new necessary and sufficient condition for observability in the natural L2-setting. The proof relies on the use of the Lebeau-Robbiano strategy together with a precise study of the cost of controllability for linear ordinary differential equations, and allows us to treat the case where each component of the system is observed in a different subdomainPierre Lissy is partially supported by the project IFSMACS (ANR-15-CE40-0010) funded by the french Agence Nationale de la Recherche, 2015-2019.
Enrique Zuazua is partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, FA9550-14-1-0214 of the EOARD-AFOSR, the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and ICON (ANR-16-ACHN-0014) of the French Agence Nationale de la Recherch
Exact controllability for quasi-linear perturbations of KdV
We prove that the KdV equation on the circle remains exactly controllable in
arbitrary time with localized control, for sufficiently small data, also in
presence of quasi-linear 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,
classical Ingham inequality and HUM 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 H\"ormander.Comment: 39 page
Controllability of quasi-linear Hamiltonian NLS equations
We prove internal controllability in arbitrary time, for small data, for
quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of
reduction to constant coefficients up to order zero and HUM method to prove the
controllability of the linearized problem. Then we apply a
Nash-Moser-H\"ormander implicit function theorem as a black box
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