Abstract
In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.
References
[1] F. Alabau-Boussouira, P. Cannarsa and G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim. 55 (2017), no. 3, 2052–2087. 10.1137/15M1020538Search in Google Scholar
[2] B. Allal, A. Moumni and J. Salhi, Boundary controllability for a degenerate and singular wave equation, Math. Methods Appl. Sci. 45 (2022), no. 17, 11526–11544. 10.1002/mma.8464Search in Google Scholar
[3] J. Bai and S. Chai, Exact controllability for some degenerate wave equations, Math. Methods Appl. Sci. 43 (2020), no. 12, 7292–7302. 10.1002/mma.6464Search in Google Scholar
[4] J. Bai and S. Chai, Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary, Appl. Math. Lett. 119 (2021), Paper No. 107235. 10.1016/j.aml.2021.107235Search in Google Scholar
[5] J. Bai and S. Chai, Exact controllability of wave equations with interior degeneracy and one-sided boundary control, J. Syst. Sci. Complex. 36 (2023), no. 2, 656–671. 10.1007/s11424-023-1094-3Search in Google Scholar
[6] 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), no. 5, 1024–1065. 10.1137/0330055Search in Google Scholar
[7] I. Boutaayamou, G. Fragnelli and D. Mugnai, Boundary controllability for a degenerate wave equation in nondivergence form with drift, SIAM J. Control Optim. 61 (2023), no. 4, 1934–1954. 10.1137/22M151491XSearch in Google Scholar
[8] N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal. 14 (1997), no. 2, 157–191. 10.3233/ASY-1997-14203Search in Google Scholar
[9] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997),no. 7, 749–752. 10.1016/S0764-4442(97)80053-5Search in Google Scholar
[10] P. Cannarsa, R. Ferretti and P. Martinez, Null controllability for parabolic operators with interior degeneracy and one-sided control, SIAM J. Control Optim. 57 (2019), no. 2, 900–924. 10.1137/18M1198442Search in Google Scholar
[11] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Ser. Math. Appl. 13, Clarendon Press, New York, 1998. 10.1093/oso/9780198502777.001.0001Search in Google Scholar
[12] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9) 58 (1979), no. 3, 249–273. Search in Google Scholar
[13] L. V. Fardigola, Transformation operators in control problems for a degenerate wave equation with variable coefficients, Ukrainian Math. J 70 (2019), 1300–1318. 10.1007/s11253-018-1570-4Search in Google Scholar
[14] X. Fu and Z. Liao, Observability estimate for the wave equation with variable coefficients, SIAM J. Control Optim. 60 (2022), no. 4, 2344–2372. 10.1137/21M1468231Search in Google Scholar
[15] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim. 46 (2007), no. 5, 1578–1614. 10.1137/040610222Search in Google Scholar
[16] H. Gajewski, K. Gröger and K. Zacharias, Nonlinear Operator Equations and Operator Differential Equations, Mir, Moscow, 1978. Search in Google Scholar
[17] M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim. 52 (2014), no. 4, 2037–2054. 10.1137/120901374Search in Google Scholar
[18] L. F. Ho, Observabilité frontière de l’équation des ondes, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 12, 443–446. Search in Google Scholar
[19] L. F. Ho, Exact controllability of the one-dimensional wave equation with locally distributed control, SIAM J. Control Optim. 28 (1990), no. 3, 733–748. 10.1137/0328043Search in Google Scholar
[20] P. I. Kogut, O. P. Kupenko and G. Leugering, On boundary exact controllability of one-dimensional wave equations with weak and strong interior degeneration, Math. Methods Appl. Sci. 45 (2022), no. 2, 770–792. 10.1002/mma.7811Search in Google Scholar
[21] V. Komornik, Exact Controllability and Stabilization, RAM Res. Appl. Math., Masson, Paris, 1994. Search in Google Scholar
[22] J. Lagnese, Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim. 21 (1983), no. 1, 68–85. 10.1137/0321004Search in Google Scholar
[23] I. Lasiecka, R. Triggiani and P. F. Yao, Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms, Nonlinear Anal. 30 (1997), 111–122. 10.1016/S0362-546X(97)00004-7Search in Google Scholar
[24] I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl. 235 (1999), no. 1, 13–57. 10.1006/jmaa.1999.6348Search in Google Scholar
[25] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Masson, Paris, 1988. Search in Google Scholar
[26] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev. 30 (1988), no. 1, 1–68. 10.1137/1030001Search in Google Scholar
[27] Y. Liu, Some sufficient conditions for the controllability of wave equations with variable coefficients, Acta Appl. Math. 128 (2013), 181–191. 10.1007/s10440-013-9825-4Search in Google Scholar
[28] Y.-X. Liu, Exact controllability of the wave equation with time-dependent and variable coefficients, Nonlinear Anal. Real World Appl. 45 (2019), 226–238. 10.1016/j.nonrwa.2018.07.005Search in Google Scholar
[29] L. Lu, S. Li, G. Chen and P. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems Control Lett. 80 (2015), 30–41. 10.1016/j.sysconle.2015.04.003Search in Google Scholar
[30] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679–698. 10.2140/pjm.1961.11.679Search in Google Scholar
[31] S. Micu and E. Zuazua, An Introduction to the Controllability of Partial Differential Equations, Hermann, Paris, 2004. Search in Google Scholar
[32] A. Moumni and J. Salhi, Exact controllability for a degenerate and singular wave equation with moving boundary, Numer. Algebra Control Optim. 13 (2023), no. 2, 194–209. 10.3934/naco.2022001Search in Google Scholar
[33] J.-P. Puel, Global Carleman inequalities for the wave equations and applications to controllability and inverse problems, Report, Universit´e de Versailles Saint-Quentin, Versailles, 2011. Search in Google Scholar
[34] D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl. 40 (1972), 336–368. 10.1016/0022-247X(72)90055-8Search in Google Scholar
[35] F. Wang, J.-M. Wang and X.-H. Wu, Exact controllability of the interconnected Schrödinger and wave equations with a boundary control at the wave equation, J. Math. Anal. Appl. 519 (2023), no. 2, Article ID 126831. 10.1016/j.jmaa.2022.126831Search in Google Scholar
[36] P.-F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim. 37 (1999), no. 5, 1568–1599. 10.1137/S0363012997331482Search in Google Scholar
[37] X. Zhang, Explicit observability estimate for the wave equation with potential and its application, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), no. 1997, 1101–1115. 10.1098/rspa.2000.0553Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- On the Laplace transform
- An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
- Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces
- Polynomial convergence of iterations of certain random operators in Hilbert space
- Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator
- Almost compactness in neutrosophic topological spaces
- Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations
- On the stochastic elliptic equations involving fractional derivative
- A note on Köthe–Toeplitz duals and multiplier spaces of sequence spaces involving bicomplex numbers
- A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations
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- On random pairwise comparisons matrices and their geometry
- Statistically convergent difference sequences of bi-complex numbers
- On some inequalities concerning polynomials with restricted zeros
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- On a new variant of cyclic (noncyclic) condensing operators with existence of optimal solutions to an FDE
- Exponential stability of non-conformable fractional-order systems
Articles in the same Issue
- Frontmatter
- On the Laplace transform
- An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
- Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces
- Polynomial convergence of iterations of certain random operators in Hilbert space
- Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator
- Almost compactness in neutrosophic topological spaces
- Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations
- On the stochastic elliptic equations involving fractional derivative
- A note on Köthe–Toeplitz duals and multiplier spaces of sequence spaces involving bicomplex numbers
- A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations
- Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy
- On random pairwise comparisons matrices and their geometry
- Statistically convergent difference sequences of bi-complex numbers
- On some inequalities concerning polynomials with restricted zeros
- New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications
- On a new variant of cyclic (noncyclic) condensing operators with existence of optimal solutions to an FDE
- Exponential stability of non-conformable fractional-order systems