We consider linear control systems of the form y′(t) = Ay(t) + Bu(t) on a Hilbert space Y . We suppose that the control operator B is bounded from the control space U to a larger extrapolation space containing Y . The aim is to study the null controllability in the case where the control u is constrained to lie in a bounded subset Γ ⊂ U. We obtain local constrained controllability properties. When ($$)$$ is a group of isometries, we establish necessary conditions and sufficient ones for global constrained controllability. Moreover, when the constraint set Γ contains the origin in its interior, the local constrained property turns out to be equivalent to a dual observability inequality of L1 type with respect to the time variable. In this setting, the study is focused on hyperbolic-like systems which can be reduced to a second order evolution equation. Furthermore, we treat the problem of determining a steering control for general constraint set Γ in nonsmooth convex analysis context. In the case where Γ contains the origin in its interior, a steering control can be obtained by minimizing a convenient smooth convex functional. Applications to the wave equation and Euler-Bernoulli beams are presented.
Accepté le :
DOI : 10.1051/cocv/2018018
Keywords: Admissible control operator, admissible observation operator, constrained null controllability, hyperbolic-like systems, steering control
Berrahmoune, Larbi 1
@article{COCV_2019__25__A32_0,
author = {Berrahmoune, Larbi},
title = {Constrained null controllability for distributed systems and applications to hyperbolic-like equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018018},
zbl = {1447.93023},
mrnumber = {4001034},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018018/}
}
TY - JOUR AU - Berrahmoune, Larbi TI - Constrained null controllability for distributed systems and applications to hyperbolic-like equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018018/ DO - 10.1051/cocv/2018018 LA - en ID - COCV_2019__25__A32_0 ER -
%0 Journal Article %A Berrahmoune, Larbi %T Constrained null controllability for distributed systems and applications to hyperbolic-like equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018018/ %R 10.1051/cocv/2018018 %G en %F COCV_2019__25__A32_0
Berrahmoune, Larbi. Constrained null controllability for distributed systems and applications to hyperbolic-like equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 32. doi: 10.1051/cocv/2018018
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