This paper considers a diagonal semilinear system of hyperbolic partial differential equations with positive and constant velocities. The boundary condition is composed of an unstable linear term and a saturated feedback control. Weak solutions with initial data in L2([0, 1]) are considered and well-posedness of the system is proven using nonlinear semigroup techniques. Local L$$ exponential stability is tackled by a Lyapunov analysis and convergence of semigroups. Moreover, an explicit estimation of the region of attraction is given.
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DOI : 10.1051/cocv/2019069
Keywords: Diagonal semilinear hyperbolic systems, saturation, Lyapunov theory
@article{COCV_2020__26_1_A23_0,
author = {Dus, Mathias and Ferrante, Francesco and Prieur, Christophe},
title = {On {\protect\emph{L}\protect\textsuperscript{\ensuremath{\infty}}} stabilization of diagonal semilinear hyperbolic systems by saturated boundary control},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019069},
mrnumber = {4070783},
zbl = {1441.93241},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019069/}
}
TY - JOUR AU - Dus, Mathias AU - Ferrante, Francesco AU - Prieur, Christophe TI - On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019069/ DO - 10.1051/cocv/2019069 LA - en ID - COCV_2020__26_1_A23_0 ER -
%0 Journal Article %A Dus, Mathias %A Ferrante, Francesco %A Prieur, Christophe %T On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019069/ %R 10.1051/cocv/2019069 %G en %F COCV_2020__26_1_A23_0
Dus, Mathias; Ferrante, Francesco; Prieur, Christophe. On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 23. doi: 10.1051/cocv/2019069
[1] and , Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169–179. | MR | Zbl | DOI
[2] , and , First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4 (1979) 1017–1034. | MR | Zbl | DOI
[3] and , Stability And Boundary Stabilization Of 1-D Hyperbolic Systems. Progress in Nonlinear Differential Equations and Their Applications. Springer International Publishing (2016). | MR | Zbl | DOI
[4] and , Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C1 norm. SIAM J. Control Optim. 53 (2015) 1464–1483. | MR | Zbl | DOI
[5] , , , and , Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws. IEEE Trans. Autom. Control 62 (2017) 1620–1635. | MR | Zbl | DOI
[6] , Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010). | MR | Zbl
[7] and , Robust control of processes subject to saturation nonlinearities. Comput. Chem. Eng. 14 (1990) 343–358. | DOI
[8] and , Output feedback stabilization for a scalar conservation law with a nonlocal velocity. SIAM J. Math. Anal. 45 (2013) 2646–2665. | MR | Zbl | DOI
[9] , and , A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Autom. Control 52 (2004) 2–11. | MR | Zbl | DOI
[10] , and , Stabilization of boundary controlled hyperbolic PDEs via Lyapunov-based event triggered sampling and quantization. In Proceedings of the 56th IEEE Conference on Decision and Control (2017) 1266–1271.
[11] , Exponential stability of general 1-D quasilinear systems with source terms for the C1 norm under boundaryconditions (2017) submitted . | arXiv
[12] and , Boundary Control of PDEs: A Course on Backstepping Designs. Advances in Design and Control. Society for Industrial and Applied Mathematics (2008). | Zbl | MR
[13] and , Strong stability of elastic control systems with dissipative saturating feedback. Syst. Control Lett. 48 (2003) 243–252. | MR | Zbl | DOI
[14] , and , Semi-global C1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems. ESAIM: M2AN 34 (2000) 399–408. | MR | Zbl | Numdam | DOI
[15] , Nonlinear Semigroups. Translations of mathematical monographs. American Mathematical Society (1992). | MR | Zbl | DOI
[16] and , Boundary control design for linear conservation laws in the presence of energy-bounded measurement noise, in Proceedings of the 57th IEEE Conference on Decision and Control (2018) 6550–6555.
[17] , and , Wave equation with cone-bounded control laws. IEEE Trans. Autom. Control 61 (2016) 3452–3463. | MR | Zbl | DOI
[18] , Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Math. Control Signals Syst. 2 (1989) 265–285. | MR | Zbl | DOI
[19] , , and , Stability and Stabilization of Linear Systems with Saturating Actuators. Springer, London (2011). | MR | Zbl | DOI
[20] , Existence of solutions to first-order periodic boundary value problems. J. Math. Anal. Appl. 323 (2006) 1325–1332. | MR | Zbl | DOI
Cité par Sources :
Research by F. Ferrante has been partially supported by the CNRS-INS2I under the JCJC grant CoBrA and by the Grenoble Institute of Technology under the grant CrYStAL.
