Boundary controllability for the quasilinear wave equation. (English) Zbl 1185.93018

Summary: We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability can be geometrically characterized by a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, basing on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.


93B05 Controllability
93B27 Geometric methods
35B35 Stability in context of PDEs
35L65 Hyperbolic conservation laws
35L70 Second-order nonlinear hyperbolic equations
Full Text: DOI arXiv


[1] Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992) · Zbl 0786.93009
[2] Berger, M.: Nonlinearity and Functional Analysis. Academic Press, San Diego (1977) · Zbl 0368.47001
[3] Castro, C., Zuazua, E.: Concentration and lack observability of waves in highly heterogeneous media. Arch. Ration. Anal. 164(1), 39–72 (2002) · Zbl 1016.35003
[4] Chai, S.: Boundary feedback stabilization of Naghdi’s model. Acta Math. Sin. (Engl. Ser.) 21(1), 169–184 (2005) · Zbl 1084.35012
[5] Chai, S.: Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control. Indian J. Pure Appl. Math. 36(5), 227–249 (2005) · Zbl 1084.74026
[6] Chai, S., Liu, K.: Observability inequalities for the transmission of shallow shells. Syst. Control Lett. 55(9), 726–735 (2006) · Zbl 1100.93004
[7] Chai, S., Liu, K.: Boundary feedback stabilization of the transmission problem of Naghdi’s model. J. Math. Anal. Appl. 319(1), 199–214 (2006) · Zbl 1176.93061
[8] Chai, S., Yao, P.F.: Observability inequalities for thin shells. Sci. China (Ser. A) 46(3), 300–311 · Zbl 1217.74079
[9] Chai, S., Guo, Y., Yao, P.F.: Boundary feedback stabilization of shallow shells. SIAM J. Control Optim. 42(1), 239–259 (2003) · Zbl 1055.93065
[10] Cirina, M.: Boundary controllability of nonlinear hyperbolic systems. SIAM J. Control 7, 198–212 (1969) · Zbl 0182.20203
[11] Cirina, M.: Nonlinear hyperbolic problems with solutions on preassigned sets. Mich. Math. J. 17, 193–209 (1970) · Zbl 0201.42702
[12] Dafermos, C.M., Hrusa, W.J.: Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics. Arch. Ration. Mech. Anal. 87, 267–292 (1985) · Zbl 0586.35065
[13] Egorov, Yu.V.: Some problems in the theory of optimal control. Z. Vycisl. Mat. Mat. Fiz. (5), 887–904 (1963) · Zbl 0156.31804
[14] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1998). Revised third printing
[15] Greenberg, J.M., Li, T.T.: The effect of boundary damping for the quasilinear wave equation. J. Differ. Equ. 52(1), 66–75 (1984) · Zbl 0576.35080
[16] Gulliver, R., Lasiecka, I., Littman, W., Triggiani, R.: The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber. In: Geometric Methods in Inverse Problems and PDE Control. IMA Vol. Math. Appl., vol. 137, pp. 73–181. Springer, New York (2004) · Zbl 1067.35014
[17] Fattorini, H.O.: Boundary control of temperature distributions in a parallelepipedon. SIAM J. Control 13(1), 1–13 (1975) · Zbl 0311.93028
[18] Ho, L.F.: Observabilité frontiére de l’e’quation des ondes. C. R. Acad. Sci. Paris Sér. I Math. 302, 443–446 (1986)
[19] Lasiecka, I., Ong, J.: Global solvability and uniform decays of solutions to quasilinear equations with nonlinear boundary dissipation. Commun. PDEs 24, 2069–2107 (1999) · Zbl 0936.35031
[20] Lasiecka, I., Triggiani, R.: Exact controllability of the wave equation with Neumann boundary control. Appl. Math. Optim. 19, 243–209 (1989) · Zbl 0666.49012
[21] Lasiecka, I., Triggiani, R.: Exact controllability of semilinear abstract systems with applications to waves and plates boundary control problems. Appl. Math. Optim. 23, 109–145 (1991) · Zbl 0729.93023
[22] Lasiecka, I., Triggiani, R.: Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks. J. Math. Anal. Appl. 269(2), 642–688 (2002) · Zbl 1067.74044
[23] Lasiecka, I., Triggiani, R., Yao, P.F.: Inverse/observability estimates for second-order hyperbolic equations with variable systems. J. Math. Anal. Appl. 235, 13–57 (1999) · Zbl 0931.35022
[24] Li, T.T.: Global Classical Solutions for Quasilinear Hyperbolic Systems. Masson/Wiley, Paris/New York (1994) · Zbl 0841.35064
[25] Li, T.T.: Exact boundary controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals. Math. Meth. Appl. Sci. 27, 1089–1114 (2004) · Zbl 1151.93316
[26] Li, T.T.: Exact boundary controllability of unsteady flows in a network of open canals. Math. Nachr. 278, 278–289 (2005) · Zbl 1066.93005
[27] Li, T.T.: Controllability and Observability for Quasilinear Hyperbolic Systems. Springer, Berlin (2008) · Zbl 1147.93009
[28] Li, T.T., Rao, B.P.: Exact boundary controllability for quasilinear hyperbolic systems. SIAM J. Control Optim. 41(6), 1748–1755 (2003) · Zbl 1032.35124
[29] Li, T.T., Yu, L.: Exact boundary controllability for 1-D quasilinear waves equations. SIAM J. Control Optim. 45, 1074–1083 (2006) · Zbl 1116.93021
[30] Li, T.T., Zhang, B.Y.: Global exact boundary controllability of a class of quasilinear hyperbolic systems. J. Math. Anal. Appl. 225, 289–311 (1998) · Zbl 0915.93007
[31] Lions, J.L.: Exact controllability, stabilization and perturbations for distributed system. SIAM Rev. 30, 1–68 (1988) · Zbl 0644.49028
[32] Qin, T.: The global smooth solutions of second order quasilinear hyperbolic equations with dissipative conditions. Chin. Ann. Math. Ser. B 9, 251–269 (1988) · Zbl 0664.35059
[33] Russell, D.L.: Controllability and stability theory for linear partial differential equations, Reccent progress and open questions. SIAM Rev. 20(4), 639–739 (1978) · Zbl 0397.93001
[34] Schmidt, E.J.P.G.: On a non-linear wave equation and th e control of an elastic string from one equilibrium location to another. J. Math. Anal. Appl. 272, 536–554 (2002) · Zbl 1013.35055
[35] Seidman, T.I.: Two results on exact boundary controllability of parabolic equations. Appl. Math. Optim. 11(2), 145–152 (1984) · Zbl 0562.49003
[36] Tataru, D.: A priori estimate of Carleman’s type in domains with boundary. J. Math. Pure Appl. 73, 355–357 (1994) · Zbl 0835.35031
[37] Tataru, D.: Boundary controllability for conservative PDEs. Appl. Math. Optim. 31, 257–295 (1995) · Zbl 0836.35085
[38] Triggiani, R., Yao, P.F.: Carleman estimate with no lower-order terms for general Riemann wave equation. Global uniqueness and observability in one shot. Appl. Math. Optim. 46, 331–375 (2002) · Zbl 1030.35018
[39] Yao, P.F.: On the observability inequalities for the exact controllability of the wave equation with variable coefficients. SIAM J. Control Optim. 37(6), 1568–1599 (1999) · Zbl 0951.35069
[40] Yao, P.F.: Observability inequalities for the shallow shell. SIAM J. Control Optim. 38(6), 1729–1756 (2000) · Zbl 0974.35013
[41] Yao, P.F.: Global smooth solutions for the quasilinear wave equation with boundary dissipation. J. Differ. Equ. 241, 62–93 (2007) · Zbl 1214.35037
[42] Yong, J., Zhang, X.: Exact controllability of the heat equation with hyperbolic memory kernel. In: Control Theory of Partial Differential Equations. Lect. Notes Pure Appl. Math., vol. 424, pp. 387–401 (2005) · Zbl 1085.35038
[43] Zhang, Z.F., Yao, P.F.: Global smooth solutions of the quasilinear wave equation with internal velocity feedback. SIAM J. Control Optim. 47, 2044–2077 (2008) · Zbl 1357.35225
[44] Zhou, Y., Lei, Z.: Local exact boundary controllability for nonlinear wave equations. SIAM J. Control Optim. 46, 1022–1051 (2007) · Zbl 1147.93012
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