Exact controllability for first order quasilinear hyperbolic systems with internal controls

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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
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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 and Continuous Dynamical Systems, 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), American Institute of Mathematical Sciences & Higher Education Press, Springfield, Beijing.
[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), 325-336. 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), 209-218. doi: 10.1142/S0252959902000201.
[4]	T. Li and B. Rao, Exact boundary controllability for quasilinear hyperbolic systems, SIAM J. Control Optim., 41 (2003), 1748-1755.
[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), 723-742. 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, Duke University Press, Durham, 1985.
[7]	J.-L. Lions, Exact controllability, stabilization and pertubations for distributed systems, SIAM Rev., 30 (1988), 1-68. 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), 639-739. 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), 549-560.
[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), American Institute of Mathematical Sciences & Higher Education Press, Springfield, Beijing.
[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), 325-336. 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), 209-218. doi: 10.1142/S0252959902000201.
[4]	T. Li and B. Rao, Exact boundary controllability for quasilinear hyperbolic systems, SIAM J. Control Optim., 41 (2003), 1748-1755.
[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), 723-742. 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, Duke University Press, Durham, 1985.
[7]	J.-L. Lions, Exact controllability, stabilization and pertubations for distributed systems, SIAM Rev., 30 (1988), 1-68. 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), 639-739. 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), 549-560.
[10]	K. Zhuang, Exact controllability with internal controls for first order quasilinear hyperbolic systems with zero eigenvalues,, to appear in Chin. Ann. Math., ().

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