On boundary control of a hyperbolic system with a vanishing characteristic speed
Hu, Long1, 2, 3 ; Wang, Zhiqiang4
1 School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China
2 Sorbonne Universités, UPMC University Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
3 School of Mathematical Sciences, Fudan University, Shanghai 200433. P.R. China
4 School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P.R. China
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 134-147.

Despite of the fact that distributed (internal) controls are usually used to obtain controllability for a hyperbolic system with vanishing characteristic speeds, this paper is, however, devoted to study the case where only boundary controls are considered. We first prove that the system is not (null) controllable in finite time. Next, we give a sufficient and necessary condition for the asymptotic stabilization of the system under a natural feedback.

Reçu le :
DOI : 10.1051/cocv/2015031
Classification : 93B05, 93D15, 35L04
Mots-clés : Hyperbolic systems, controllability, stabilization, vanishing characteristic speed, zero eigenvalue
Hu, Long 1, 2, 3 ; Wang, Zhiqiang 4

1 School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China
2 Sorbonne Universités, UPMC University Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
3 School of Mathematical Sciences, Fudan University, Shanghai 200433. P.R. China
4 School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, P.R. China 
@article{COCV_2016__22_1_134_0,
     author = {Hu, Long and Wang, Zhiqiang},
     title = {On boundary control of a hyperbolic system with a vanishing characteristic speed},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {134--147},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {1},
     year = {2016},
     doi = {10.1051/cocv/2015031},
     zbl = {1336.93031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2015031/}
}
TY  - JOUR
AU  - Hu, Long
AU  - Wang, Zhiqiang
TI  - On boundary control of a hyperbolic system with a vanishing characteristic speed
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 134
EP  - 147
VL  - 22
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2015031/
DO  - 10.1051/cocv/2015031
LA  - en
ID  - COCV_2016__22_1_134_0
ER  - 
%0 Journal Article
%A Hu, Long
%A Wang, Zhiqiang
%T On boundary control of a hyperbolic system with a vanishing characteristic speed
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 134-147
%V 22
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2015031/
%R 10.1051/cocv/2015031
%G en
%F COCV_2016__22_1_134_0
Hu, Long; Wang, Zhiqiang. On boundary control of a hyperbolic system with a vanishing characteristic speed. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 134-147. doi : 10.1051/cocv/2015031. http://archive.numdam.org/articles/10.1051/cocv/2015031/

F. Ancona and A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36 (1998) 290–312. | DOI | Zbl

F. Ancona and A. Marson, Asymptotic Stabilization of Systems of Conservation Laws by Controls Acting at a Single Boundary Point. In Control methods in PDE-dynamical systems. Vol. 426 of Contemp. Math. Amer. Math. Soc., Providence, RI (2007) 1–43. | Zbl

K. Beauchard, J.-M. Coron and P. Rouchon, Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations. Comm. Math. Phys. 296 (2010) 525–557. | DOI | Zbl

A. Bressan and G.M. Coclite, On the boundary control of systems of conservation laws. SIAM J. Control Optim. 41 (2002) 607–622. | DOI | Zbl

J.-M. Coron, Control and Nonlinearity. Vol. 136 of Math. Surv. Monogr. American Mathematical Society, Providence, RI (2007). | Zbl

J.-M. Coron, O. Glass and Z. Wang, Exact boundary controllability for 1-D quasilinear hyperbolic, systems with a vanishing characteristic speed. SIAM J. Control Optim. 48 (2009/10) 3105–3122. | DOI | Zbl

O. Glass, On the controllability of the 1-D isentropic Euler equation. JEMS J. Eur. Math. Soc. 9 (2007) 427–486. | DOI | Zbl

O. Glass, On the controllability of the non-isentropic 1-D Euler equation. J. Differ. Equ. 257 (2014) 638–719. | DOI | Zbl

M. Gugat, Boundary controllability between sub- and supercritical flow. SIAM J. Control Optim. 42 (2003) 1056–1070. | DOI | Zbl

T. Horsin, On the controllability of the Burgers equation. ESAIM: COCV 3 (1998) 83–95. | Numdam | Zbl

V. Komornik, Exact Controllability and Stabilization. The multiplier method. RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester (1994). | Zbl

P. Koosis, The Logarithmic Integral. I. Vol. 12 of Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge (1988). | Zbl

J.P. LaSalle, Some extensions of Liapunov’s second method. IRE Trans. CT-7 (1960) 520–527.

T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems. Vol. 3 of AIMS Ser. Appl. Math. American Institute of Mathematical Sciences AIMS, Springfield, MO; Higher Education Press, Beijing (2010). | Zbl

T. Li and B. Rao, Local exact boundary controllability for a class of quasilinear hyperbolic systems. Dedicated to the memory of Jacques-Louis Lions. Chinese Ann. Math. Ser. B 23 (2002) 209–218. | DOI | Zbl

T. Li and B. Rao, Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optim. 41 (2003) 1748–1755. | DOI | Zbl

T. Li and B. Rao, Exact controllability for first order quasilinear hyperbolic systems with vertical characteristics. Acta Math. Sci. Ser. B Engl. Ed. 29 (2009) 980–990. | Zbl

T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems. Duke Univ. Math. Series. V. Duke University, Mathematics Department, Durham, NC (1985). | Zbl

T. Li and L. Yu, Exact controllability for first order quasilinear hyperbolic systems with zero eigenvalues. Chinese Ann. Math. Ser. B 24 (2003) 415–422. | DOI | Zbl

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. | DOI | MR | Zbl

S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control Optim. 39 (2001) 1677–1696. | DOI | MR | Zbl

A. Pazy, Semigroups of linear operators and applications to partial differential equations. Vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983). | MR | Zbl

V. Perrollaz, Exact controllability of scalar conservation laws with an additional control in the context of entropy solutions. SIAM J. Control Optim. 50 (2012) 2025–2045. | DOI | MR | Zbl

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat. 5 (2007) 79–84. | MR

D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. | DOI | MR | Zbl

Z. Wang, Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chinese Ann. Math. Ser. B 27 (2006) 643–656. | DOI | MR | Zbl

Z. Wang, Global exact controllability for quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics. Nonlinear Anal. 69 (2008) 510–522. | DOI | MR | Zbl

Z. Wang and L. Yu, Exact boundary controllability for a one-dimensional adiabatic flow system. Appl. Math. J. Chinese Univ. Ser. A 23 (2008) 35–40. | MR | Zbl

Cité par Sources :