American Institute of Mathematical Sciences

June  2013, 3(2): 161-183. doi: 10.3934/mcrf.2013.3.161

Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control

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

Received  February 2012 Revised  November 2012 Published  March 2013

In this paper necessary and sufficient conditions of approximate $L^\infty$-controllability at a free time are obtained for the control system $w_{tt}=w_{xx}-q^2w$, $w_x(0,t)=u(t)$, $x>0$, $t\in(0,T)$, where $q>0$, $T>0$, $u\in L^\infty(0,T)$ is a control. This system is considered in the Sobolev spaces.
Citation: Larissa V. Fardigola. Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control. Mathematical Control & Related Fields, 2013, 3 (2) : 161-183. doi: 10.3934/mcrf.2013.3.161
References:
 [1] M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$,, (in Russian) Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19.  doi: 10.1007/s10958-007-0140-3.  Google Scholar [2] I. Erdélyi, A generalized inverse for arbitrary operators between Hilbert spaces,, Proc. Camb. Phil. Soc., 71 (1972), 43.   Google Scholar [3] L. V. Fardigola, On controllability problems for the wave equation on a half-plane,, J. Math. Phys. Anal. Geom., 1 (2005), 93.   Google Scholar [4] 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.  doi: 10.1137/070684057.  Google Scholar [5] L. V.Fardigola, Neumann boundary control problem for the string equation on a half-axis,, (in Ukrainian) Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36.   Google Scholar [6] L. V.Fardigola, Controllability problems for the 1-d wave equation on a half-axis with the Dirichlet boundary control,, ESAIM: Control, 18 (2012), 748.  doi: 10.1051/cocv/2011169.  Google Scholar [7] L. V. Fardigola and K. S. Khalina, Controllability problems for the string equation,, (in Ukrainian) Ukr. Mat. Zh., 59 (2007), 939.  doi: 10.1007/s11253-007-0068-2.  Google Scholar [8] I. M. Gelfand and G. E. Shilov, "Generalized Functions,", Vol. 2, (1958).   Google Scholar [9] S. G. Gindikin and L. R. Volevich, "Distributions and Convolution Equations,", Gordon and Breach Sci. Publ., (1992).   Google Scholar [10] 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.  doi: 10.1002/zamm.200800196.  Google Scholar [11] 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.  doi: 10.1051/cocv:2007044.  Google Scholar [12] M. Gugat, G. Leugering and G. M. Sklyar, $L^p$-optimal boundary control for the wave equation,, SIAM J. Control Optim., 44 (2005), 49.  doi: 10.1137/S0363012903419212.  Google Scholar [13] V. A. Il'in and E. I. Moiseev, A boundary control at two ends by a process described by the telegraph equation,, (in Russian) Dokl. Akad. Nauk, 394 (2004), 154.   Google Scholar [14] E. H. Moore, On the reciprocal of the general algebraic matrix,, Bull. Amer. Math. Soc., 26 (1920), 394.   Google Scholar [15] R. Penrose, A generalized inverse for matrices,, Proc. Camb. Phil. Soc., 51 (1955), 406.   Google Scholar [16] L. Schwartz, "Théorie des Distributions," Vol. 1, 2,, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).   Google Scholar [17] 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.  doi: 10.1016/S0022-247X(02)00380-3.  Google Scholar [18] 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.  doi: 10.1137/080731396.  Google Scholar [19] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations,, in, (2010), 3008.   Google Scholar [20] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems,, in, (2006), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
 [1] M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$,, (in Russian) Zapiski Nauchnykh Seminarov POMI, 332 (2006), 19.  doi: 10.1007/s10958-007-0140-3.  Google Scholar [2] I. Erdélyi, A generalized inverse for arbitrary operators between Hilbert spaces,, Proc. Camb. Phil. Soc., 71 (1972), 43.   Google Scholar [3] L. V. Fardigola, On controllability problems for the wave equation on a half-plane,, J. Math. Phys. Anal. Geom., 1 (2005), 93.   Google Scholar [4] 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.  doi: 10.1137/070684057.  Google Scholar [5] L. V.Fardigola, Neumann boundary control problem for the string equation on a half-axis,, (in Ukrainian) Dopovidi Natsionalnoi Akademii Nauk Ukrainy, (2009), 36.   Google Scholar [6] L. V.Fardigola, Controllability problems for the 1-d wave equation on a half-axis with the Dirichlet boundary control,, ESAIM: Control, 18 (2012), 748.  doi: 10.1051/cocv/2011169.  Google Scholar [7] L. V. Fardigola and K. S. Khalina, Controllability problems for the string equation,, (in Ukrainian) Ukr. Mat. Zh., 59 (2007), 939.  doi: 10.1007/s11253-007-0068-2.  Google Scholar [8] I. M. Gelfand and G. E. Shilov, "Generalized Functions,", Vol. 2, (1958).   Google Scholar [9] S. G. Gindikin and L. R. Volevich, "Distributions and Convolution Equations,", Gordon and Breach Sci. Publ., (1992).   Google Scholar [10] 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.  doi: 10.1002/zamm.200800196.  Google Scholar [11] 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.  doi: 10.1051/cocv:2007044.  Google Scholar [12] M. Gugat, G. Leugering and G. M. Sklyar, $L^p$-optimal boundary control for the wave equation,, SIAM J. Control Optim., 44 (2005), 49.  doi: 10.1137/S0363012903419212.  Google Scholar [13] V. A. Il'in and E. I. Moiseev, A boundary control at two ends by a process described by the telegraph equation,, (in Russian) Dokl. Akad. Nauk, 394 (2004), 154.   Google Scholar [14] E. H. Moore, On the reciprocal of the general algebraic matrix,, Bull. Amer. Math. Soc., 26 (1920), 394.   Google Scholar [15] R. Penrose, A generalized inverse for matrices,, Proc. Camb. Phil. Soc., 51 (1955), 406.   Google Scholar [16] L. Schwartz, "Théorie des Distributions," Vol. 1, 2,, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1966).   Google Scholar [17] 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.  doi: 10.1016/S0022-247X(02)00380-3.  Google Scholar [18] 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.  doi: 10.1137/080731396.  Google Scholar [19] X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations,, in, (2010), 3008.   Google Scholar [20] E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems,, in, (2006), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar
 [1] Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463 [2] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [3] Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 [4] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [5] Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 [6] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [7] Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355 [8] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [9] Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 [10] Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106 [11] Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104 [12] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [13] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [14] Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055 [15] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [16] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [17] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [18] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [19] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 [20] Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

2019 Impact Factor: 0.857