Exact controllability for first order quasilinear hyperbolic systems with internal controls

Previous Article
On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum
DCDS Home
This Issue
Next Article
Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

February 2016, 36(2): 1105-1124. doi: 10.3934/dcds.2016.36.1105

Exact controllability for first order quasilinear hyperbolic systems with internal controls

Kaili Zhuang ^1, , Tatsien Li ^1, and Bopeng Rao ^1,

1.	School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China

Received May 2014 Published August 2015

Abstract
Related Papers

Based on the theory of the local exact boundary controllability for first order quasilinear hyperbolic systems, using an extension method, the authors establish the exact controllability in a shorter time by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.

Keywords: First order quasilinear hyperbolic system, $C^1$ classical solution, local exact controllability, internal control., boundary control.

Mathematics Subject Classification: Primary: 35L04, 49J20; Secondary: 93B0.

Citation: Kaili Zhuang, Tatsien Li, Bopeng Rao. Exact controllability for first order quasilinear hyperbolic systems with internal controls. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1105-1124. doi: 10.3934/dcds.2016.36.1105

References:

[1]	T. Li, Controllability and Observabilty for Quasilinear Hyperbolic Systems,, AIMS Series on Applied Mathematics, 3 (2010).
[2]	T. Li and Y. Jin, Semi-global $C^1$ solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems,, Chin. Ann. Math., 22 (2001), 325. doi: 10.1142/S0252959901000334.
[3]	T. Li and B. Rao, Local exact boundary controllability for a class of quasilinear hyperbolic systems,, Chin. Ann. Math., 23 (2002), 209. doi: 10.1142/S0252959902000201.
[4]	T. Li and B. Rao, Exact boundary controllability for quasilinear hyperbolic systems,, SIAM J. Control Optim., 41 (2003), 1748.
[5]	T. Li and B. Rao, Strong(Weak) exact controllability and Strong(Weak) exact observability for quasilinear hyperbolic systems[J],, Chin. Ann. Math., 31 (2010), 723. doi: 10.1007/s11401-010-0600-9.
[6]	T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985).
[7]	J.-L. Lions, Exact controllability, stabilization and pertubations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001.
[8]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095.
[9]	L. Yu, Semi-global $C^1$ solution to the mixed initial-boundary value problem for a kind of quasilinear hyperbolic systems (in Chinese),, Chin. Ann. Math., 25 (2004), 549.
[10]	K. Zhuang, Exact controllability with internal controls for first order quasilinear hyperbolic systems with zero eigenvalues,, to appear in Chin. Ann. Math., ().

show all references

References:

[1]	T. Li, Controllability and Observabilty for Quasilinear Hyperbolic Systems,, AIMS Series on Applied Mathematics, 3 (2010).
[2]	T. Li and Y. Jin, Semi-global $C^1$ solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems,, Chin. Ann. Math., 22 (2001), 325. doi: 10.1142/S0252959901000334.
[3]	T. Li and B. Rao, Local exact boundary controllability for a class of quasilinear hyperbolic systems,, Chin. Ann. Math., 23 (2002), 209. doi: 10.1142/S0252959902000201.
[4]	T. Li and B. Rao, Exact boundary controllability for quasilinear hyperbolic systems,, SIAM J. Control Optim., 41 (2003), 1748.
[5]	T. Li and B. Rao, Strong(Weak) exact controllability and Strong(Weak) exact observability for quasilinear hyperbolic systems[J],, Chin. Ann. Math., 31 (2010), 723. doi: 10.1007/s11401-010-0600-9.
[6]	T. Li and W. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985).
[7]	J.-L. Lions, Exact controllability, stabilization and pertubations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001.
[8]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095.
[9]	L. Yu, Semi-global $C^1$ solution to the mixed initial-boundary value problem for a kind of quasilinear hyperbolic systems (in Chinese),, Chin. Ann. Math., 25 (2004), 549.
[10]	K. Zhuang, Exact controllability with internal controls for first order quasilinear hyperbolic systems with zero eigenvalues,, to appear in Chin. Ann. Math., ().

[1]	Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[2]	Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243
[3]	Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014
[4]	Tatsien Li, Bopeng Rao, Zhiqiang Wang. A note on the one-side exact boundary controllability for quasilinear hyperbolic systems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 405-418. doi: 10.3934/cpaa.2009.8.405
[5]	Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311
[6]	Nicolás Carreño. Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Mathematical Control & Related Fields, 2012, 2 (4) : 361-382. doi: 10.3934/mcrf.2012.2.361
[7]	Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027
[8]	Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77
[9]	Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665
[10]	Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control & Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125
[11]	Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
[12]	Shu-Guang Shao, Shu Wang, Wen-Qing Xu, Yu-Li Ge. On the local C^{1, α} solution of ideal magneto-hydrodynamical equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090
[13]	Bao-Zhu Guo, Liang Zhang. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Mathematical Control & Related Fields, 2016, 6 (1) : 143-165. doi: 10.3934/mcrf.2016.6.143
[14]	Larissa V. Fardigola. Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control. Mathematical Control & Related Fields, 2013, 3 (2) : 161-183. doi: 10.3934/mcrf.2013.3.161
[15]	Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
[16]	Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743
[17]	Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[18]	Roberto Triggiani. Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space. Evolution Equations & Control Theory, 2016, 5 (4) : 489-514. doi: 10.3934/eect.2016016
[19]	Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305
[20]	Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences