Internal controllability of first order quasilinear hyperbolic systems with a reduced number of controls
Abstract
In this paper we investigate the exact controllability of n × n first order one-dimensional quasilinear 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 are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due to M. 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 M. Gromov in order to handle the problem of loss of derivatives.
Related
- Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary
- Boundary stabilization of multi-dimensional first-order quasilinear hyperbolic systems
- Minimally-coupled Quasilinear Symmetric Hyperbolic Systems, Naturally
- Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems
- A conditional gradient-flow framework for edge-length optimization of quasilinear hyperbolic networks
BibTeX
@misc{alabauboussouira2026internal,
title = {{Internal controllability of first order quasilinear hyperbolic systems with a reduced number of controls}},
author = {Fatiha Alabau-Boussouira and Jean-Michel Coron and Guillaume Olive},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12490},
eprint = {2609.12490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12490},
}
RIS
TY - UNPB
TI - Internal controllability of first order quasilinear hyperbolic systems with a reduced number of controls
AU - Fatiha Alabau-Boussouira
AU - Jean-Michel Coron
AU - Guillaume Olive
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12490
UR - https://arxiv.org/abs/2609.12490
AB - In this paper we investigate the exact controllability of n × n first order one-dimensional quasilinear 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 are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due to M. 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 M. Gromov in order to handle the problem of loss of derivatives.
ER -