Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. (English) Zbl 0931.35022

The authors consider the Dirichlet and respectively the Neumann mixed second-order hyperbolic problems in the unknown \(w(t,x)\) and their dual homogeneous problems in \(\psi(t,x)\), as it follows: \[ \begin{cases} w_{tt}+{\mathcal A}w= F_1(w)\text{ in }\Omega,\quad & \psi_{tt}+{\mathcal A}_\psi=F(\psi)\text{ in }Q,\\ w(0,\cdot)= w_0,\;w_t(0, \cdot)= w_1\text{ in }\Omega, \quad & \psi(T, \cdot)= \psi_0,\;\psi_t(T,\cdot)= \psi_1\text{ in }\Omega,\\ w|_{\Sigma_0} =0;\;w |_{\Sigma_1}=u,\quad & \psi|_\Sigma=0, \end{cases} \tag{1} \]
\[ \begin{cases} w_{tt} +{\mathcal A}_w=F_1(w);\quad & \psi_{tt}+{\mathcal A}_\psi=F(\psi)\text{ in }Q;\\ w(0,\cdot)=w_0,\;w_t(0,\cdot)=w_1;\quad & \psi(T, \cdot)= \psi_0,\;\psi_t(T, \cdot)= \psi_1\text{ in }\Omega; \\ w|_{\Sigma_0}=0; \quad & \psi |_{ \Sigma_0} =0\text{ in }\Sigma_0; \\ {\partial w\over\partial \nu_{\mathcal A}} |_{\Sigma_1}=u; \quad & \left[{\partial\psi \over\partial \nu_{\mathcal A}}+ B \psi \right]_{\Sigma_1}= 0\text{ in }\Sigma_1; \end{cases} \tag{2} \] where the boundary \(\Gamma= \partial\Omega\) is of class \(C^2\), \(\Gamma=\Gamma_0 \cup \Gamma_1\), \({\mathcal A}\) is the second-order differential operator \[ {\mathcal A}w \equiv -\sum^n_{i,j=1}{\partial\over\partial x_i}\left(a_{ij}(x){\partial w \over\partial x_j} \right) \] satisfying the uniform ellipticity condition, \(F\) is a suitable first-order differential operator depending on the original operator \(F_1\) and \(u\in L_2(0,T;L_2(\Gamma_1))\).
The first goal of this paper is to establish an a priori inequality of the homogeneous Dirichlet \(\psi\)-problem (1): There exists a constant \(T_0>0\), depending upon the triple \((\Omega,\Gamma_0, \Gamma_1)\) and the coefficients \(a_{ij}\) such that for all \(T>T_0\), there is a constant \(c_T>0\) for which \[ \int^T_0 \int_{\Gamma_1} \left( {\partial \psi\over \partial\nu_{\mathcal A}} \right)^2 d\Sigma_1\geq c_T\bigl \|(\psi_0, \psi_1) \bigr\|^2_{H^1_0 (\Omega) \times L_2(\Omega)}.\tag{3} \] The inequality (3) is the continuous observability inequality for the \(\psi\)-problem (1) and it is, by duality, equivalent to the exact controllability property of the nonhomogeneous \(w\)-problem (1) at time \(T\), on the space \(L_2(\Omega)\times H^{-1}(\Omega)\), within the class of \(L_2 (0,T; L_2(\Gamma_1))\)-controls.
The second goal is to establish the continuous observability inequality for the \(\psi\)-problem (2): There exists a constant \(T_0>0\), depending upon the triple \((\Omega, \Gamma_0,\Gamma_1)\) and the coefficients \(a_{ij}\), such that for all \(T>T_0\) there is a constant \(c_T>0\) for which \[ \int^T_0 \int_{\Gamma_1}\psi^2_t d\Sigma_1\geq c_T \bigl\|(\psi_0,\psi_1) \bigr \|^2_{H^1_{\Gamma_0}(\Omega)\times L_2(\Omega)}, \] where \(H^1_{\Gamma_0} (\Omega)= \{f\in H^1(\Omega);f|_{\Gamma_0}=0\}\).
The method used has three main steps: the first step employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in \(\Omega\subset\mathbb{R}^n\) to a problem on an appropriate Riemann manifold; the second step employs explicit Carleman estimates at the differential level to take care of the variable first-order terms; the third step employs the micro-local analysis yielding a sharp trace estimate, in the Neumann case.


35B45 A priori estimates in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
93B05 Controllability
35L70 Second-order nonlinear hyperbolic equations
Full Text: DOI


[1] Bardos, C.; Lebeau, G.; Rauch, J., Sharp efficient conditions for the observation, control, and stabilization of wave from the boundary, SIAM J. control optim., 30, 1024-1065, (1992) · Zbl 0786.93009
[2] Dolecki, S.; Russell, D., A general theory of observation and control, SIAM J. control, 15, 185-220, (1977) · Zbl 0353.93012
[3] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Vol. 34, pp. 151-742, Research Institute of Mathematics, Seoul National University, Seoul, Korea.
[4] Greene, R.E.; Wu, H., C∞ convex functions and manifolds of positive curvature, Acta math., 137, 209-245, (1976) · Zbl 0372.53019
[5] Ho, F.L., Observabilité frontière de l’equation des ondes, C. R. acad. sci. Paris, ser. I math, 302, 443-446, (1986) · Zbl 0598.35060
[6] Hörmander, L., The analysis of linear partial differential operators, (1985), Springer-Verlag Berlin/New York
[7] Hörmander, L., On the uniqueness of the Cauchy problem under partial analyticity assumptions, () · Zbl 0907.35002
[8] Imanuvilov, O.Yu., Exact controllability of hyperbolic equations, parts I & II, Automatika, 3, 10-13, (1990)
[9] Isakov, V., Inverse problems for partial differential equations, Applied mathematical sciences, 127, (1998), Springer-Verlag Berlin/New York
[10] Lagnese, J., (), 158-181
[11] Lasiecka, I.; Lions, J.L.; Triggiani, R., Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. math. pures appl., 65, 149-192, (1981) · Zbl 0631.35051
[12] Lasiecka, I.; Triggiani, R., Regularity of hyperbolic equations under L2(0,T;L2(γ))-Dirichlet boundary terms, App. math. optimiz., 10, 275-286, (1983) · Zbl 0526.35049
[13] Lasiecka, I.; Triggiani, R., Uniform exponential energy decay of wave equations in a bounded region with L2(0,∞;L2(γ))-feedback control in the Dirichlet boundary conditions, J. differential equations, 66, 340-390, (1987) · Zbl 0629.93047
[14] Lasiecka, I.; Triggiani, R., Exact controllability of the wave equation with Neumann boundary control, Appl. math. optimiz., 19, 243-290, (1989) · Zbl 0666.49012
[15] Lasiecka, I.; Triggiani, R., Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. math. optimiz., 25, 189-244, (1992) · Zbl 0780.93082
[16] Lasiecka, I.; Triggiani, R., Sharp regularity for mixed second-order hyperbolic equations of Neumann type, part I: the L2-boundary case, Annal. di matemat. pura e appl. (IV), 157, 285-367, (1990) · Zbl 0742.35015
[17] Lasiecka, I.; Triggiani, R., Carleman estimates and exact boundary controllability for a system of coupled, non-conservative second order hyperbolic equations, Lecture notes in pure and applied mathematics, (1994), Marcel Dekker New York, p. 215-243 · Zbl 0881.35015
[18] Lasiecka, I.; Triggiani, R.; Yao, P.F., Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms, Nonlinear analysis theory, methods appl., 30, 111-122, (1997) · Zbl 0904.35045
[19] Lasiecka, I.; Triggiani, R.; Yao, P.F., An observability estimate in L2(ω)×H−1(ω) for second-order hyperbolic equations with variable coefficients, (), 71-78 · Zbl 0979.93049
[20] I. Lasiecka, R. Triggiani, and, X. Zhang, Carleman estimates with no interior lower-order term for non conservative wave equations: uniqueness and observability in one, University of Virginia, preprint, January 1999.
[21] Lions, J.L., Contrôlabilité exacte, stabilisation et perturbations des systèmes distribués, (1988), Masson Paris · Zbl 0653.93002
[22] Littman, W., Near optimal time boundary controllability for a class of hyperbolic equations, Lecture notes in control and information, (1987), Springer-Verlag Berlin/New York, p. 307-312
[23] Littman, W., Boundary controllability for polyhedral domains, Lecture noes in control and information, (1992), Springer-Verlag Berlin/New York, p. 272-284 · Zbl 0776.93051
[24] Littman, W.; Taylor, S., Smoothing evolution equations and boundary control theory, J. d’analyse mathematique, 59, 117-131, (1992) · Zbl 0802.35059
[25] Russell, D., Controllability and stabilizability theory for linear partial differential equations recent progress and open questions, SIAM rev., 20, 431-639, (1978) · Zbl 0397.93001
[26] Taylor, A.E.; Lay, D.C., Introduction to functional analysis, (1980), John Wiley New York
[27] Tataru, D., Boundary controllability of conservative pdes, Appl. math. optim., 31, 257-295, (1995) · Zbl 0836.35085
[28] Tataru, D., Unique continuation for solutions to P.D.E.s, between Hörmander’s theorem and Holmgren’s theorem, Comm. partial differential equations, 20, 855-884, (1995) · Zbl 0846.35021
[29] Triggiani, R., Exact boundary controllability on L2(ω)×H−1(ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems, Appl. math. optimiz., 18, 241-277, (1988) · Zbl 0656.93011
[30] P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, Siam. J. Control & Optimiz, to appear. · Zbl 0951.35069
[31] Wu, H.; Shen, C.L.; Yu, Y.L., An introduction to Riemann geometry, (1989), University of Beijing
[32] X. Zhang, Observability estimates: a direct approach and its applications, preprint, 1998.
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.