Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations

Abstract

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary (null) controllability and the local boundary (weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.

Keywords

Coupled system of quasilinear wave equations / Exact boundary controllability / Exact boundary observability

References

[1] Emanuilov O Y. Boundary control by semilinear evolution equations. Russian Math. Surveys., 1989, 44: 183-184

[2] Dáger R, Zuazua E. Wave Propagation, Observation and Control in 1-D Flexible Multistructures, 2000, Berlin, Heidelberg, New York: Springer-Verlag

[3] Gu Q L, Li T T. Exact boundary controllability for quasilinear wave equations in a planar tree-like network of strings. Ann. I. H. Poincaré-AN., 2009, 26: 2373-2384

[4] Guo L N, Wang Z Q. Exact boundary observability for nonautonomous quasilinear hyperbolic systems. Math. Methods Appl. Sci., 2008, 31: 1956-1971

[5] Lasiecka I, Triggiani R. Exact controllability of semilinear abstract systems with applications to waves and plates boundary control problems. Appl. Math. Optim., 1991, 23: 109-154

[6] Lasiecka I, Triggiani R, Yao P F. Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl., 1999, 235: 13-57

[7] Li T T, Jin Y. Semi-global C ¹ solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math., 2001, 22B(3): 325-336

[8] Li T T, Rao B P. Local exact boundary controllability for a class of quasilinear hyperbolic systems. Chin. Ann. Math., 2002, 23B(2): 209-218

[9] Li T T, Rao B P. Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control. Optim., 2003, 41: 1748-1755

[10] Li T T, Rao B P. Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems. Chin. Ann. Math., 2010, 31B(5): 723-742

[11] Li T T. Exact boundary controllability for quasilinear wave equations. J. Comput. Appl. Math., 2006, 190: 127-135

[12] Li T T. Controllability and Observability for Quasilinear Hyperbolic Systems, 2010, Beijing: AIMS and Higher Education Press

[13] Li T T, Yu L X. Exact boundary controllability for 1-D quasilinear wave equations. SIAM J. Control. Optim., 2006, 45: 1074-1083

[14] Li T T. Exact boundary observability for 1-D quasilinear wave equations. Math. Meth. Appl. Sci., 2006, 29: 1543-1553

[15] Li T T, Rao B P, Hu L. Exact boundary synchronization for a coupled system of 1-D wave equations, 2012

[16] Li T T, Yu W C. Boundary Value Problems for Quasilinear Hyperbolic Systems, 1985, Durham: Duke University Mathematics Series V

[17] Lions J L. Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués, Vol. I, 1988

[18] Lions J L. Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev., 1988, 30: 1-68

[19] Russell D L. Controllability and stabilization theory for linear partial differential equations, recent progress and open questions. SIAM Rev., 1978, 20: 639-739

[20] Wang K. Exact boundary controllability of nodal profile for 1-D quasilinear wave equations. Front. Math. China., 2011, 6: 545-555

[21] Wang K. Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems. Chin. Ann. Math., 2011, 32B(6): 803-822

[22] Yu L X. Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems and its applications. Math. Meth. Appl. Sci., 2010, 33: 273-286

[23] Yu L X. Exact boundary observability for a kind of second-order quasilinear hyperbolic systems and its applications. Nonlinear Anal.-Theory, Methods Appl., 2010, 72: 4452-4465

[24] Yu L X. Semi-global C1 solutions to the mixed initial-boundary value problem for a general class of quasilinear hyperbolic systems (in Chinese). Chin. Ann. Math., 2004, 25A(5): 549-560

[25] Zuazua E. Exact controllability for the semilinear wave equation. J. Math. Pures et Appl., 1990, 69: 1-31

[26] Zuazua E. Exact controllability for semilinear wave equations in one space dimension. Ann. I. H. Poincaré-AN., 1993, 10: 109-129