In this paper, we study the local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws with characteristics of constant multiplicity. We prove the two-sided boundary controllability, the one-sided boundary controllability and the two-sided boundary controllability with fewer controls, by applying the strategy used in [T. Li and L. Yu, J. Math. Pures et Appl. 107 (2017) 1–40; L. Yu, Chinese Ann. Math., Ser. B (To appear)]. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions, and on some further properties of both ε-approximate front tracking solutions and limit solutions.
Accepted:
DOI: 10.1051/cocv/2017072
Keywords: Linearly degenerate quasilinear hyperbolic systems of conservation laws, local exact boundary controllability, semi-global entropy solutions, ε-approximate front tracking solutions
@article{COCV_2018__24_2_793_0, author = {Li, Tatsien and Yu, Lei}, title = {Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws\protect\textsuperscript{,}}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {793--810}, publisher = {EDP-Sciences}, volume = {24}, number = {2}, year = {2018}, doi = {10.1051/cocv/2017072}, zbl = {1403.93042}, mrnumber = {3816415}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017072/} }
TY - JOUR AU - Li, Tatsien AU - Yu, Lei TI - Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws, JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 793 EP - 810 VL - 24 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017072/ DO - 10.1051/cocv/2017072 LA - en ID - COCV_2018__24_2_793_0 ER -
%0 Journal Article %A Li, Tatsien %A Yu, Lei %T Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws, %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 793-810 %V 24 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017072/ %R 10.1051/cocv/2017072 %G en %F COCV_2018__24_2_793_0
Li, Tatsien; Yu, Lei. Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws,. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 2, pp. 793-810. doi : 10.1051/cocv/2017072. http://archive.numdam.org/articles/10.1051/cocv/2017072/
[1] Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA 4 (1997) 1–42 | DOI | MR | Zbl
,[2] On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optimiz. 43 (2005) 2166–2190 | DOI | MR | Zbl
and ,[3] On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optimiz. 36 (1998) 290–312 | DOI | MR | Zbl
and ,[4] Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point. Contemporary Math. 426 (2007) 1–43 | DOI | MR | Zbl
and ,[5] Chocs caractéristiques. C. R. Acad. Sci. Paris, Ser. A 274 (1972) 1018–1021 | MR | Zbl
,[6] Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem. Oxford University Press (2000) | DOI | MR | Zbl
,[7] On the boundary control of systems of conservation laws. SIAM J. Control Optimiz. 41 (2002) 607–622 | DOI | MR | Zbl
and ,[8] Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws. NoDEA 14 (2007) 569–592 | DOI | MR | Zbl
and ,[9] Linear degeneracy and shock waves. Mathematische Zeitschrift 207 (1991) 583–596 | DOI | MR | Zbl
,[10] On the uniform controllability of the Burgers’ equation. SIAM J. Control Optimiz. 46 (2007) 1211–1238 | DOI | MR | Zbl
and ,[11] On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc. 9 (2007) 427–486 | DOI | MR | Zbl
,[12] On the controllability of the non-isentropic 1-D Euler equation. J. Differ. Equ. 257 (2014) 638–719 | MR | Zbl
,[13] Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws. Syst. Control Lett. 47 (2002) 287–298 | MR | Zbl
,[14] Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws II. SIAM J. Control Optimiz. 44 (2005) 140–158 | DOI | MR | Zbl
and ,[15] Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10 (1957) 537–566 | DOI | MR | Zbl
,[16] Uniform controllability of scalar conservation laws in the vanishing viscosity limit. SIAM J. Control Optimiz. 50 (2012) 1661–1699 | DOI | MR | Zbl
,[17] Controllability and Observability for Quasilinear Hyperbolic Systems. AIMS and Higher Education Press (2010) | MR | Zbl
,[18] Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chinese Ann. Math., Series B 23 (2002) 209–218 | DOI | MR | Zbl
and ,[19] Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optimiz. 41 (2003) 1748–1755 | DOI | MR | Zbl
and ,[20] Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems. Chinese Ann. Math., Ser. B 31 (2010) 723–742 | DOI | MR | Zbl
and ,[21] One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws. J. Math. Pures et Appl. 107 (2017) 1–40 | DOI | MR | Zbl
and ,[22] Boundary Value Problems for Quasilinear Hyperbolic Systems. Duke University (1985) | MR | Zbl
and ,[23] On the controllability of the Burger’s equation. ESAIM: COCV 3 (1998) 83–95 | Numdam | MR | Zbl
,[24] L. Yu, One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws with constant multiplicity. Chinese Ann. Math., Ser. B (To appear) | MR
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