Stabilization of the wave equation by on-off and positive-negative feedbacks

Martinez, Patrick; Vancostenoble, Judith

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 335-377.

Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback $a\left(t\right)u_t$. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions $a$: typically $a$ is equal to $1$ on $(0,T)$, equal to $0$ on $(T,qT)$ and is $qT$-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of $T$ for which the energy of some solutions remains constant with time. If $T$ is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: $a\left(t\right)=a_0>0$ on $(0,T)$, and $a\left(t\right)=-b_0<0$ on $(T,qT)$, and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual “optimal time condition” in the locally distributed case. These new inequalities provide also new exact controllability results.

[1] Z. Artstein and E.F. Infante, On the asymptotic stability of oscillators with unbounded damping. Quart. Appl. Math. 34 (1976) 195-199. | MR | Zbl

[2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR | Zbl

[3] C. Bardos, G. Lebeau and J. Rauch, Un exemple d'utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino 1988, Special Issue (1989) 11-31. | Zbl

[4] A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations. Comm. Pure Appl. Math. 33 (1980) 707-725. | MR | Zbl

[5] A. Benaddi and B. Rao, Energy decay rate of wave equations with indefinite damping. J. Differential Equations 161 (2000) 337-357. | MR | Zbl

[6] C. Castro and S.J. Cox, Achieving arbitrarily large decay in the damped wave equation. SIAM J. Control Optim. 39 (2001) 1748-1755. | MR | Zbl

[7] P. Freitas and E. Zuazua, Stability results for the wave equation with indefinite damping. J. Differential Equations 132 (1996) 338-352. | MR | Zbl

[8] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Publications du Laboratoire d'Analyse numérique. Université Pierre et Marie Curie (1988).

[9] A. Haraux, A generalized internal control for the wave equation in a rectangle. J. Math. Anal. Appl. 153 (1990) 190-216. | MR | Zbl

[10] W.A. Harris Jr., P. Pucci and J. Serrin, Asymptotic behavior of solutions of a nonstandard second order differential equation. Differential Integral Equations 6 (1993) 1201-1215. | MR | Zbl

[11] L. Hatvani, On the stability of the zero solution of second order nonlinear differential equations. Acta Sci. Math. 32 (1971) 1-9. | MR | Zbl

[12] L. Hatvani and V. Totik, Asymptotic stability of the equilibrium of the damped oscillator. Differential Integral Equation 6 (1993) 835-848. | MR | Zbl

[13] L. Hatvani, T. Krisztin, V. Totik and Vilmos, A necessary and sufficient condition for the asymptotic stability of the damped oscillator. J. Differential Equations 119 (1995) 209-223. | MR | Zbl

[14] S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. | MR | Zbl

[15] V. Komornik and E. Zuazua A direct method for boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54. | MR | Zbl

[16] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. John Wiley, Chicester and Masson, Paris (1994). | MR | Zbl

[17] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations 50 (1983) 163-182. | MR | Zbl

[18] J. Lagnese, Note on boundary stabilization of wave equation. SIAM J. Control Optim. 26 (1988) 1250-1256. | MR | Zbl

[19] I. Lasiecka and R. Triggiani, Uniform exponential decay in a bounded region with $L_2(0,T;L_2\left(\Sigma \right))$-feedback control in the Dirichlet boundary condition. J. Differential Equations 66 (1987) 340-390. | MR | Zbl

[20] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25 (1992) 189-224. | MR | Zbl

[21] J.-L. Lions, Contrôlabilité exacte de systèmes distribués. C. R. Acad. Sci. Paris 302 (1986) 471-475. | MR | Zbl

[22] J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. Masson, RMA 8 (1988). | Zbl

[23] J.-L. Lions, Exact controllability, stabilization adn perturbations for distributd systems. SIAM Rev. 30 (1988) 1-68. | MR | Zbl

[24] S. Marakuni, Asymptotic behavior of solutions of one-dimensional damped wave equations. Comm. Appl. Nonlin. Anal. 1 (1999) 99-116. | Zbl

[25] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems. Israël J. Math. 119 (2000) 291-324. | MR | Zbl

[26] P. Martinez and J. Vancostenoble, Exact controllability in “arbitrarily short time” of the semilinear wave equation. Discrete Contin. Dynam. Systems (to appear). | Zbl

[27] M. Nakao, On the time decay of solutions of the wave equation with a local time-dependent nonlinear dissipation. Adv. Math. Sci. Appl. 7 (1997) 317-331. | MR | Zbl

[28] K. Petersen, Ergodic Theory. Cambridge University Press, Cambridge, Studies in Adv. Math. 2 (1983). | MR | Zbl

[29] P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems. Acta Math. 170 (1993) 275-307. | MR | Zbl

[30] P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, II. J. Differential Equations 113 (1994) 505-534. | MR | Zbl

[31] P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators. SIAM J. Math. Anal. 25 (1994) 815-835. | MR | Zbl

[32] P. Pucci and J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems. Comm. Pure Appl. Math. XLIX (1996) 177-216. | MR | Zbl

[33] P. Pucci and J. Serrin, Local asymptotic stability for dissipative wave systems. Israël J. Math. 104 (1998) 29-50. | MR | Zbl

[34] R.A. Smith, Asymptotic stability of $x^{\text{'}\text{'}}+a\left(t\right)x^{\text{'}}+x=0$. Quart. J. Math. Oxford (2) 12 (1961) 123-126. | MR | Zbl

[35] L.H. Thurston and J.W. Wong, On global asymptotic stability of certain second order differential equations with integrable forcing terms. SIAM J. Appl. Math. 24 (1973) 50-61. | MR | Zbl

[36] J. Vancostenoble and P. Martinez, Optimality of energy estimates for the wave equation with nonlinear boundary velocity feedbacks. SIAM J. Control Optim. 39 (2000) 776-797. | MR | Zbl

[37] E. Zuazua, An introduction to the exact controllability for distributed systems, Textos et Notas 44, CMAF. Universidades de Lisboa (1990). | MR