Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems
Abstract
This paper addresses exact controllability and rapid stabilization issues for linear difference delay systems with finitely many pointwise delays. We first establish a criterion for exact controllability in Lr, r ∈ [1, +∞), characterized by a necessary and sufficient Hautus-type frequency-domain condition. The sufficiency of this condition in Lr was already known [https://hal.science/hal-04830757], whereas its necessity had previously been established only in L1 [https://hal.science/hal-03827918]. We answer [https://hal.science/hal-03827918, Conjecture 3.8] by proving that this condition is also necessary for r ∈ (1, +∞). The analysis is based on rewriting the system as an evolution equation in a suitable Banach space, which allows one to apply duality arguments in the spirit of [https://doi.org/10.1137/S036301299119795X]. Building on this controllability analysis and the abstract stabilization framework developed in [https://doi.org/10.1137/S0363012996301609, https://hal.science/hal-01272024], we prove that our exact controllability criterion in Lr is also a sufficient condition for rapid stabilization in L2, i.e., that the system of interest can be stabilized by a family of state feedback control laws ensuring that the closed-loop system is exponentially stable in L2 with an arbitrarily large decay rate. As an application, we extend our results to linear one-dimensional hyperbolic partial differential equations, which can be reduced to linear difference delay systems via the method of characteristics, obtaining also a necessary and sufficient frequency-domain criterion for exact controllability in Lr, r∈ [1, +∞), which, as before, is a sufficient condition for rapid stabilization in L2.
Related
- Boundary stabilization of multi-dimensional first-order quasilinear hyperbolic systems
- Finite dimensional reductions of integrable differential-difference equations
- Internal controllability of first order quasilinear hyperbolic systems with a reduced number of controls
- Some remarks on the moment method for null controllability of parabolic equations in any dimension
- Small-time approximate controllability of bilinear Schrödinger equations on compact normal homogeneous spaces
BibTeX
@misc{chitour2026exact,
title = {{Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems}},
author = {Yacine Chitour and Sébastien Fueyo and Pierre Lissy and Guilherme Mazanti},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31225},
eprint = {2610.31225},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05835233v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems
AU - Yacine Chitour
AU - Sébastien Fueyo
AU - Pierre Lissy
AU - Guilherme Mazanti
PY - 2026
DA - 2026/10/22
PB - HAL
DO - 10.48550/arxiv.2610.31225
UR - https://hal.science/hal-05835233v1
AB - This paper addresses exact controllability and rapid stabilization issues for linear difference delay systems with finitely many pointwise delays. We first establish a criterion for exact controllability in L^r, r ∈ [1, +∞), characterized by a necessary and sufficient Hautus-type frequency-domain condition. The sufficiency of this condition in L^r was already known [https://hal.science/hal-04830757], whereas its necessity had previously been established only in L^1 [https://hal.science/hal-03827918]. We answer [https://hal.science/hal-03827918, Conjecture 3.8] by proving that this condition is also necessary for r ∈ (1, +∞). The analysis is based on rewriting the system as an evolution equation in a suitable Banach space, which allows one to apply duality arguments in the spirit of [https://doi.org/10.1137/S036301299119795X]. Building on this controllability analysis and the abstract stabilization framework developed in [https://doi.org/10.1137/S0363012996301609, https://hal.science/hal-01272024], we prove that our exact controllability criterion in L^r is also a sufficient condition for rapid stabilization in L^2, i.e., that the system of interest can be stabilized by a family of state feedback control laws ensuring that the closed-loop system is exponentially stable in L^2 with an arbitrarily large decay rate. As an application, we extend our results to linear one-dimensional hyperbolic partial differential equations, which can be reduced to linear difference delay systems via the method of characteristics, obtaining also a necessary and sufficient frequency-domain criterion for exact controllability in L^r, r∈ [1, +∞), which, as before, is a sufficient condition for rapid stabilization in L^2.
ER -