Internal Controllability of First Order Quasi-linear Hyperbolic Systems with a Reduced Number of Controls
Abstract
In this paper we investigate the exact controllability of $n \times n$ first order one-dimensional quasi-linear hyperbolic systems by $m<n$ internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations is not the same, we see that we encounter the problem of loss of derivatives if we try to control quasi-linear systems with a reduced number of controls. To solve this problem, as in a prior article by Coron and Lissy on a Navier--Stokes control system, we first use the notion of algebraic solvability due to Gromov. However, in contrast with this prior article where a standard fixed point argument could be used to treat the nonlinearities, we use here a fixed point theorem of Nash--Moser type due to Gromov in order to handle the problem of loss of derivatives.
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Information & Authors
Information
Published In
SIAM Journal on Control and Optimization
Pages: 300 - 323
DOI: 10.1137/15M1015765
ISSN (online): 1095-7138
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 7 April 2015
Accepted: 24 October 2016
Published online: 8 February 2017
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European Research Council http://dx.doi.org/10.13039/501100000781 : 266907
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