Modified Sobolev Spaces in Controllability Problems for the Wave Equation on a Half-Plane
DOI:
https://doi.org/10.15407/mag11.01.018Abstract
The 2-d wave equation $w_{tt}=\Delta(w), t\in (0,T)$, on the half-plane $x_1>0$ controlled by the Neumann boundary condition $w_{x_1}(0,x_2,t)=\delta(x_2)u(t)$ is considered in Sobolev spaces, where $T>0$ is a constant and $u\in L^\infty(0,T)$ is a control. This control system is transformed into a control system for the 1-d wave equation in modified Sobolev spaces introduced and studied in the paper, and they play the main role in the study. The necessary and sufficient conditions of (approximate) $L^\infty$-controllability are obtained for the 1-d control problem. It is also proved that the 2-d control system replicates the controllability properties of the 1-d control system and vise versa. Finally, the necessary and sufficient conditions of (approximate) $L^\infty$-controllability are obtained for the 2-d control problem.
Mathematics Subject Classification: 93B05, 35B37, 35L05.
Keywords:
modified Sobolev space, wave equation, half-plane, controllability problem, Neumann boundary controlReferences
P. Antosik, J. Mikusiński, and R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier, Amsterdam, 1973.
M.I. Belishev and A.F. Vakulenko, On a Control Problem for the Wave Equation in R3 . — Zapiski Nauchnykh Seminarov POMI 332 (2006), 19–37. (Russian). (Engl. transl.: J. Math. Sci. 142 (2007), 2528–2539.)
L.V. Fardigola, On Controllability Problems for the Wave Equation on a Half-Plane. — J. Math. Phys., Anal., Geom. 1 (2005), 93–115.
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. https://doi.org/10.1137/070684057
L.V. Fardigola, Neumann Boundary Control Problem for the String Equation on a Half-Axis. — Dopovidi Natsionalnoi Akademii Nauk Ukrainy (2009), No. 10, 36–41. (Ukrainian)
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. https://doi.org/10.1051/cocv/2011169
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. https://doi.org/10.1137/110858318
L.V. Fardigola and K.S. Khalina, Controllability Problems for the Wave Equation. — Ukr. Mat. Zh. 59 (2007), 939–952. (Ukrainian). (Engl. transl.: Ukr. Math. J. 59 (2007), 1040–1058.)
I.M. Gelfand and G.E. Shilov, Generalized Functions. Vol. 2. Fismatgiz, Moscow, 1958. (Russian)
S.G. Gindikin and L.R. Volevich, Distributions and Convolution Equations. Gordon and Breach, Philadelphia, 1992.
M. Gugat, Optimal Switching Boundary Control of a String to Rest in Finite Time. — ZAMM Angew. Math. Mech. 88 (2008), 283–305. https://doi.org/10.1002/zamm.200700154
M. Gugat and G. Leugering, L∞ -norm Minimal Control of the Wave Equation: on the Weakness of the Bang-Bang Principle. — ESAIM: Control, Optim. Calc. Var. 14 (2008), 254–283. https://doi.org/10.1051/cocv:2007044
M. Gugat, G. Leugering, and G. Sklyar, Lp -optimal Boundary Control for the Wave Equation. — SIAM J. Control Optim. 44 (2005), 49–74. https://doi.org/10.1137/S0363012903419212
V.A. Il’in and E.I. Moiseev, A Boundary Control at Two Ends by a Process Described by the Telegraph Equation. — Dokl. Akad. Nauk, Ross. Akad. Nauk 394 (2004) No. 2, 154–158. (Russian). (Engl. transl.: Doklady Mathematics, 69 (2004) No. 1, 33–37.)
K.S. Khalina Boundary Control Problems for the Equation of Vibrating of Nonhomogeneous String on a Half-Axis. — Ukr. Mat. Zh. 64 (2012) No. 4, 525–541.
K.S. Khalina On the Neumann Boundary Controllability for the Non-homogeneous String on a Half-Axis. — J. Math. Phys., Anal., Geom. 8 (2012) No. 4, 307–335.
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 HalfAxis. — J. Math. Anal. Appl. 276 (2002), 109–134. https://doi.org/10.1016/S0022-247X(02)00380-3
J. Vancostenoble and E. Zuazua, Hardy Inequalities, Observability, and Control for the Wave and Schrödinger Equations with Singular Potentials. — SIAM J. Math. Anal. 41 (2009), 1508–1532. https://doi.org/10.1137/080731396
V.S. Vladimirov, Equations of Mathematical Physics. Mir, Moscow, 1984.
