# American Institute of Mathematical Sciences

March  2015, 5(1): 31-53. doi: 10.3934/mcrf.2015.5.31

## Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition

 1 Mathematical Division, Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., Kharkiv 61103

Received  March 2014 Revised  August 2014 Published  January 2015

In this paper necessary and sufficient conditions of $L^\infty$-controllability and approximate $L^\infty$-controllability are obtained for the control system $w_{tt}=\frac{1}{\rho} (k w_x)_x+\gamma w$, $w(0,t)=u(t)$, $x>0$, $t\in(0,T)$. Here $\rho$, $k$, and $\gamma$ are given functions on $[0,+\infty)$; $u\in L^\infty(0,\infty)$ is a control; $T>0$ is a constant. These problems are considered in special modified spaces of the Sobolev type introduced and studied in the paper. The growth of distributions from these spaces is associated with the equation data $\rho$ and $k$. Using some transformation operator introduced and studied in the paper, we see that this control system replicates the controllability properties of the auxiliary system $z_{tt}=z_{xx}-q^2z$, $z(0,t)=v(t)$, $x>0$, $t\in(0,T)$, and vise versa. Here $q\ge0$ is a constant and $v\in L^\infty(0,\infty)$ is a control. Necessary and sufficient conditions of controllability for the main system are obtained from the ones for the auxiliary system.
Citation: Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
##### References:
 [1] P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam, 1973. [2] M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbb{R}^3$ (Russian), Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19-37; English translation: J. Math. Sci., 142 (2007), 2528-2539. doi: 10.1007/s10958-007-0140-3. [3] F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for papabolic problems, ESAIM: Proceedings, 41 (2013), 15-58. doi: 10.1051/proc/201341002. [4] C. Castro, Exact controllability of the 1-D wave equation from a moving interior point, ESAIM: Control, Optim. Calc. Var., 19 (2013), 301-316. doi: 10.1051/cocv/2012009. [5] S. Dolecki and D. R. Russel, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220. doi: 10.1137/0315015. [6] L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys., Anal., Geom., 1 (2005), 93-115. [7] L. V. Fardigola, Controllability problems for the string equation on a half-axis with a boundary control bounded by a hard constant, SIAM J. Control Optim., 47 (2008), 2179-2199. doi: 10.1137/070684057. [8] L. V. Fardigola, Neumann boundary control problem for the string equation on a half-axis (Ukrainian), Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36-41. [9] L. V. Fardigola, Controllability problems for the 1-d wave equation on a half-axis with the Dirichlet boundary control, ESAIM: Control, Optim. Calc. Var., 18 (2012), 748-773. doi: 10.1051/cocv/2011169. [10] L. V. Fardigola, Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control, MCRF, 3 (2013), 161-183. doi: 10.3934/mcrf.2013.3.161. [11] L. V. Fardigola, Transformation operators of the Sturm-Liouville problem in controllability problems for the wave equation on a half-axis, SIAM J. Control Optim., 51 (2013), 1781-1801. doi: 10.1137/110858318. [12] L. V. Fardigola and K. S. Khalina, Controllability problems for the wave equation (Ukrainian), Ukr. Mat. Zh., 59 (2007), 939-952, English translation: Ukr. Math. J., 59 (2007), 1040-1058. doi: 10.1007/s11253-007-0068-2. [13] S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. [14] M. Gugat, A. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, ZAMM Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196. [15] M. Gugat and G. Leugering, $L^\infty$-norm minimal control of the wave equation: On the weakness of the bang-bang principle, ESAIM: Control Optim. Calc. Var., 14 (2008), 254-283. doi: 10.1051/cocv:2007044. [16] M. Gugat and J. Sokolowski, A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains, Applicable Analyis, 92 (2013), 2200-2214. doi: 10.1080/00036811.2012.724404. [17] M. Jaulent and C. Jean, One-dimensional inverse Schrödinger scattering problem with energy-dependent potential, I, Ann. Inst. H. Poincaré Sect. A (N.S.), 25 (1976), 105-118. [18] M. Jaulent and C. Jean, Solution of a Schrödinger inverse scattering problem with a polynomial spectral dependence in the potential, J. Math. Phys, 23 (1982), 258-266. doi: 10.1063/1.525347. [19] V. A. Il'in and A. A. Kuleshov, Generalized solutions of the wave equation in the classes $L_p$ and $W_p^1$, $p\ge1$ (Russian), Dokl. Akad. Nauk, Ross. Akad. Nauk, 446 (2012), 374-377; English translation: Doklady Mathematics, 86 (2012), 657-660. doi: 10.1134/S106456241205016X. [20] F. A. Khalilov and E. Ya. Khruslov, Matrix generalisation of the modified Korteweg-de Vries equation, Inverse Problems, 6 (1990), 193-204. doi: 10.1088/0266-5611/6/2/004. [21] K. S. Khalina, Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a half-axis (Ukrainian), Ukr. Mat. Zh., 64 (2012), 525-541; English translation: Ukr. Math. J., 64 (2012), 594-615. doi: 10.1007/s11253-012-0666-5. [22] K. S. Khalina, On the Neumann boundary controllability for a non-homogeneous string on a half-axis, J. Math. Phys., Anal., Geom., 8 (2012), 307-335. [23] K. S. Khalina, On Dirichlet boundary controllability for a non-homogeneous string on a half-axis (Ukrainian), Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2012), 24-29. [24] E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian), Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561. [25] J.-L. Lions, Contrôlabilité exacte des systèmes distribués (French) [Exact controllability of distributed systems], C. R. Acad. Sci. Paris. Sér I Math., 302 (1986), 471-475. doi: 10.1007/BFb0007542. [26] Y. Liu, Some sufficient conditions for the controllability of the wave equation with variable coefficients, Acta Appl. Math., 128 (2013), 181-191. doi: 10.1007/s10440-013-9825-4. [27] V. A. Marchenko, Sturm-Liouville Operators and Applications, American Mathematical Society, Providence, R.I., 2011. [28] Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Ann. Inst. Poincaré Anal Non Linéaire, 30 (2013), 1097-1126. doi: 10.1016/j.anihpc.2012.11.005. [29] Ch. Seck, G. Bayili, A. Séne and M. T. Niane, Contrôlabilité exacte de l'équation des ondes dans des espaces de Sobolev non-réguliers pour un ouvert polygonal (French) [Exact controllability of the wave equation in Sobolev spaces non-regular for an open polygon], Afr. Mat., 23 (2012), 1-9. doi: 10.1007/s13370-011-0001-6. [30] G. M. Sklyar and L. V. Fardigola, The Markov power moment problem in problems of controllability and frequency extinguishing for the wave equation on a half-axis, J. Math. Anal. Appl., 276 (2002), 109-134. doi: 10.1016/S0022-247X(02)00380-3. [31] J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinder equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532. doi: 10.1137/080731396. [32] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, in Proceedings of the International Congress of Mathematicians. Vol. IV, Hindustan Book Agency, New Delhi, 2010, 3008-3034. doi: 10.1007/978-0-387-89488-1. [33] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some Results and Open Problems, in Handbook of Differential Equations: Evolutionary Equations. Vol. III, Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.

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##### References:
 [1] P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam, 1973. [2] M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbb{R}^3$ (Russian), Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19-37; English translation: J. Math. Sci., 142 (2007), 2528-2539. doi: 10.1007/s10958-007-0140-3. [3] F. Boyer, On the penalised HUM approach and its applications to the numerical approximation of null-controls for papabolic problems, ESAIM: Proceedings, 41 (2013), 15-58. doi: 10.1051/proc/201341002. [4] C. Castro, Exact controllability of the 1-D wave equation from a moving interior point, ESAIM: Control, Optim. Calc. Var., 19 (2013), 301-316. doi: 10.1051/cocv/2012009. [5] S. Dolecki and D. R. Russel, A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220. doi: 10.1137/0315015. [6] L. V. Fardigola, On controllability problems for the wave equation on a half-plane, J. Math. Phys., Anal., Geom., 1 (2005), 93-115. [7] L. V. Fardigola, Controllability problems for the string equation on a half-axis with a boundary control bounded by a hard constant, SIAM J. Control Optim., 47 (2008), 2179-2199. doi: 10.1137/070684057. [8] L. V. Fardigola, Neumann boundary control problem for the string equation on a half-axis (Ukrainian), Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36-41. [9] L. V. Fardigola, Controllability problems for the 1-d wave equation on a half-axis with the Dirichlet boundary control, ESAIM: Control, Optim. Calc. Var., 18 (2012), 748-773. doi: 10.1051/cocv/2011169. [10] L. V. Fardigola, Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control, MCRF, 3 (2013), 161-183. doi: 10.3934/mcrf.2013.3.161. [11] L. V. Fardigola, Transformation operators of the Sturm-Liouville problem in controllability problems for the wave equation on a half-axis, SIAM J. Control Optim., 51 (2013), 1781-1801. doi: 10.1137/110858318. [12] L. V. Fardigola and K. S. Khalina, Controllability problems for the wave equation (Ukrainian), Ukr. Mat. Zh., 59 (2007), 939-952, English translation: Ukr. Math. J., 59 (2007), 1040-1058. doi: 10.1007/s11253-007-0068-2. [13] S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. [14] M. Gugat, A. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, ZAMM Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196. [15] M. Gugat and G. Leugering, $L^\infty$-norm minimal control of the wave equation: On the weakness of the bang-bang principle, ESAIM: Control Optim. Calc. Var., 14 (2008), 254-283. doi: 10.1051/cocv:2007044. [16] M. Gugat and J. Sokolowski, A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains, Applicable Analyis, 92 (2013), 2200-2214. doi: 10.1080/00036811.2012.724404. [17] M. Jaulent and C. Jean, One-dimensional inverse Schrödinger scattering problem with energy-dependent potential, I, Ann. Inst. H. Poincaré Sect. A (N.S.), 25 (1976), 105-118. [18] M. Jaulent and C. Jean, Solution of a Schrödinger inverse scattering problem with a polynomial spectral dependence in the potential, J. Math. Phys, 23 (1982), 258-266. doi: 10.1063/1.525347. [19] V. A. Il'in and A. A. Kuleshov, Generalized solutions of the wave equation in the classes $L_p$ and $W_p^1$, $p\ge1$ (Russian), Dokl. Akad. Nauk, Ross. Akad. Nauk, 446 (2012), 374-377; English translation: Doklady Mathematics, 86 (2012), 657-660. doi: 10.1134/S106456241205016X. [20] F. A. Khalilov and E. Ya. Khruslov, Matrix generalisation of the modified Korteweg-de Vries equation, Inverse Problems, 6 (1990), 193-204. doi: 10.1088/0266-5611/6/2/004. [21] K. S. Khalina, Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a half-axis (Ukrainian), Ukr. Mat. Zh., 64 (2012), 525-541; English translation: Ukr. Math. J., 64 (2012), 594-615. doi: 10.1007/s11253-012-0666-5. [22] K. S. Khalina, On the Neumann boundary controllability for a non-homogeneous string on a half-axis, J. Math. Phys., Anal., Geom., 8 (2012), 307-335. [23] K. S. Khalina, On Dirichlet boundary controllability for a non-homogeneous string on a half-axis (Ukrainian), Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2012), 24-29. [24] E. Ya. Khruslov, One-dimensional inverse problems of electrodynamics (Russian), Zh. Vychisl. Mat. i Mat. Fiz., 25 (1985), 548-561. [25] J.-L. Lions, Contrôlabilité exacte des systèmes distribués (French) [Exact controllability of distributed systems], C. R. Acad. Sci. Paris. Sér I Math., 302 (1986), 471-475. doi: 10.1007/BFb0007542. [26] Y. Liu, Some sufficient conditions for the controllability of the wave equation with variable coefficients, Acta Appl. Math., 128 (2013), 181-191. doi: 10.1007/s10440-013-9825-4. [27] V. A. Marchenko, Sturm-Liouville Operators and Applications, American Mathematical Society, Providence, R.I., 2011. [28] Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, Ann. Inst. Poincaré Anal Non Linéaire, 30 (2013), 1097-1126. doi: 10.1016/j.anihpc.2012.11.005. [29] Ch. Seck, G. Bayili, A. Séne and M. T. Niane, Contrôlabilité exacte de l'équation des ondes dans des espaces de Sobolev non-réguliers pour un ouvert polygonal (French) [Exact controllability of the wave equation in Sobolev spaces non-regular for an open polygon], Afr. Mat., 23 (2012), 1-9. doi: 10.1007/s13370-011-0001-6. [30] G. M. Sklyar and L. V. Fardigola, The Markov power moment problem in problems of controllability and frequency extinguishing for the wave equation on a half-axis, J. Math. Anal. Appl., 276 (2002), 109-134. doi: 10.1016/S0022-247X(02)00380-3. [31] J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinder equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532. doi: 10.1137/080731396. [32] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, in Proceedings of the International Congress of Mathematicians. Vol. IV, Hindustan Book Agency, New Delhi, 2010, 3008-3034. doi: 10.1007/978-0-387-89488-1. [33] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some Results and Open Problems, in Handbook of Differential Equations: Evolutionary Equations. Vol. III, Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.
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