# zbMATH — the first resource for mathematics

Uniform exponential energy decay of wave equations in a bounded region with $$L_ 2(0,\infty;L_ 2(\Gamma))$$-feedback control in the Dirichlet boundary conditions. (English) Zbl 0629.93047
The authors consider the much studied problem of the wave equation in dimension $$n\geq 2$$ with boundary control. Here the control enters as the nonhomogeneous boundary condition of Dirichlet type and they seek a suitable linear feedback operator depending directly on the velocity such that the resulting closed-loop system becomes exponentially stable. They prove that for “smooth” boundaries, the control law $$u(t)=(\partial /\partial \gamma)(A^{-1}\omega (t))$$ produces asymptotically stable closed-loop solutions; $$\omega$$ (t) is the velocity and A is the Laplacian with homogeneous boundary conditions. To obtain the desired exponential stability extra convexity assumptions must be made on the domain. This result has as an immediate consequence a sharper result or exact boundary controllability for the wave equation in higher dimensions considered in the natural state space.
Reviewer: R.Curtain

##### MSC:
 93D15 Stabilization of systems by feedback 35L05 Wave equation 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 35B37 PDE in connection with control problems (MSC2000)
Full Text:
##### References:
 [1] Balakrishnan, A.V, Applied functional analysis, (1981), Springer-Verlag New York/Berlin · Zbl 0459.46014 [2] Chen, G, Energy decay estimates and exact boundary valued controllability of the wave equation in a bounded domain, J. math. pures appl., 58, 249-274, (1979), (9) [3] Chen, G, A note on boundary stabilization of the wave equation, SIAM J. control optim., 19, 106-113, (1981) · Zbl 0461.93036 [4] Caffarelli, L; Nirenberg, L; Spruck, J, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampere equation, Comm. pure appl. math., 37, 369-402, (1984) · Zbl 0598.35047 [5] Datko, R, Extending a theorem of Liapunov to Hilbert spaces, J. math. anal. appl., 32, 610-613, (1970) · Zbl 0211.16802 [6] Fujiwera, D, Concrete characterizations of domains of fractional powers of some elliptic differential operators of the second order, (), 82-86 [7] Kato, T, Perturbations theory for linear operators, (1976), Springer-Verlag New York/Berlin [8] Kellog, B, Properties of elliptic B. V. P., (), Chapter 3 · Zbl 0225.15014 [9] Lagnese, J, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. differential equations, 50, 2, 163-182, (1983) · Zbl 0536.35043 [10] Lasiecka, I, Unified theory for abstract parabolic boundary problems—A semigroup approach, Appl. math. optim., 6, 31-62, (1980) [11] Levan, N, The stabilization problem: a Hilbert space operator decomposition approach, IEEE trans. circuits and systems, CAS-25, 9, 721-727, (1978) · Zbl 0402.93036 [12] Lions, J.L, Lectures at college de France, (Fall 1984) [13] Lagnese, J, Exact boundary value controllability of a class of hyperbolic equations, SIAM J. control optim., 16, (1978) · Zbl 0398.93012 [14] Littman, W, Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients, Ann. scuola norm. sup. Pisa, cl. sci., 5, 3, 567-580, (1978) · Zbl 0395.35007 [15] Lasiecka, I; Triggiani, R, A cosine operator approach to modeling L2 (0, T;L2(γ))—boundary input hyperbolic equations, Appl. math. optim., 7, 1, 35-93, (1981) · Zbl 0473.35022 [16] Lasiecka, I; Triggiani, R, Regularity of hyperbolic equation under L2(0, T;L2(γ))—dirichlet boundary terms, Appl. math. optim., 10, 275-286, (1983) · Zbl 0526.35049 [17] Lasiecka, I; Triggiani, R, Riccati equation for hyperbolic partial differential equations with L2(σ)—dirichlet boundary terms, SIAM J. control optim., 24, 5, 884-926, (1986) · Zbl 0788.49031 [18] Lasiecka, I; Triggiani, R, Dirichlet boundary stabilization of the wave equation with damping feedback of finite range, J. math. anal. appl., 97, 1, 112-130, (1983) · Zbl 0562.93061 [19] Lasiecka, I; Triggiani, R, Nondissipative boundary stabilization of the wave equation via boundary observation, J. math. pures appl., 63, 59-80, (1984) [20] {\scI.Lasiecka, J. L. Lions and R. Triggiani}, Nonhomogeneous boundary value problems for second-order hyperbolic operations, J. Math. Pures Appl., in press. · Zbl 0631.35051 [21] Lions, J.L; Magenes, E, Nonhomogeneous boundary valued problems and applications. I, (1972), Springer-Verlag New York/Berlin · Zbl 0223.35039 [22] Necas, J, Le méthodes directes en théorie de equations elliptiques, (1967), Messon et cie Paris [23] Quin, J.P; Russell, D.L, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping, (), 97-127 · Zbl 0357.35006 [24] Russell, D.L, Exact boundary value controllability theorems for wave and heat processes in star complemented regions, () · Zbl 0308.93007 [25] Russell, D.L, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. appl. math., 3, 189-211, (1973) · Zbl 0274.35041 [26] Sakamoto, R; Sakamoto, R, Mixed problems for hyperbolic equations I, II, J. math. Kyoto univ., J. math. Kyoto univ., 10, 3, 403-417, (1970) · Zbl 0206.40101 [27] Slemrod, M, Stabilization of boundary control systems, J. differential equations, 22, 402-415, (1976) · Zbl 0304.93022 [28] Zabczyk, J, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.