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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.

MSC:

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
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[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.
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