Abstract
We consider the process described in the rectangle Q T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T ] by the equation u tt -u xx -q(x, t)u = 0 with the condition u(l, t) = 0, where the coefficient q(x, t) is only square integrable on Q T . We show that for T = 2l the problem of boundary control of this process by the condition u(0, t) = µ(t) has exactly one solution in the class W 12 (Q T ) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at x = l.
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References
Lions, J.L., Exact Controllability, Stabilization and Perturbations for Distributed Systems, SIAM Rev.,1988, vol. 30, no. 1, pp. 1–68.
Zuazua, E., Exact Controllability for the Semilinear Wave Equation, J. Math. Pures Appl., 1990, vol. 69,pp. 1–31.
Triggiani, R., Exact Boundary Controllability on L2(Ω)×H-1(Ω) for the Wave Equation with DirichletControl Action on the Boundary, and Related Problems, Appl. Math. Optim., 1988, vol. 18, pp. 241–277.
Komornik, V.A., Exact Controllability and Stabilization. The Multiplier Method, Paris; Chichester, 1994.
Butkovskii, A.G., Teoriya optimal’nogo upravleniya sistemami s raspredelennymi parametrami (Theoryof Optimal Control of Systems with Distributed Parameters), Moscow: Nauka, 1965.
Vasil’ev, F.P., On Duality in Linear Problems of Control and Observation, Differ. Uravn., 1995, vol. 31,no. 11, pp. 1893–1900.
Vasil’ev, F.P., Kurzhanskii, M.A., Potapov, M.M., and Razgulin, A.V., Priblizhennoe reshenie dvoistvennykhzadach upravleniya i nablyudeniya (Approximate Solution of Dual Problems of Control and Observation),Moscow, 2010.
Il’in, V.A., Boundary Control of Vibrations at One End with the Other End Fixed in Terms of theGeneralized Solution of the Wave Equation with Finite Energy, Differ. Uravn., 2000, vol. 36, no. 12,pp. 1670–1686.
Il’in, V.A., Izbrannye trudy (Selected Papers), Moscow, 2008, vol. 2.
Il’in, V.A. and Moiseev, E.I., Optimal Boundary Control by Displacement at One End of a String withthe Second End Fixed, and the Corresponding Distribution of Total Energy of the String, Dokl. Akad.Nauk, 2004, vol. 399, no. 6, pp. 727–731.
Moiseev, E.I. and Kholomeeva, A.A., Optimization of the Boundary Control of Vibrations of a Stringwith a Fixed End in the Excitation Problem in the Class W22, Differ. Uravn., 2009, vol. 45, no. 5,pp. 741–745.
Revo, P.A. and Chabakauri, G.D., Boundary Control of Vibrations on One End with the Other EndFree in Terms of the Generalized Solution of the Wave Equation with Finite Energy, Differ. Uravn., 2001, vol. 37, no. 8, pp. 1082–1095.
Znamenskaya, L.N., Upravlenie uprugimi kolebaniyami (Control of Elastic Vibrations), Moscow, 2004.
Il’in, V.A. and Moiseev, E.I., Boundary Control at One Endpoint of a Process Described by a TelegraphEquation, Dokl. Akad. Nauk, 2002, vol. 387, no. 5, pp. 600–603.
Il’in, V.A. and Moiseev, E.I., Boundary Control at Two Endpoints of a Process Described by theTelegraph Equation, Dokl. Akad. Nauk, 2004, vol. 394, no. 2, pp. 154–158.
Smirnov, I.N., On Vibrations Described by the Telegraph Equation in the Case of a System Consistingof Several Parts of Distinct Density and Elasticity, Differ. Uravn., 2013, vol. 49, no. 5, pp. 643–648.
Abdukarimov, M.F. and Kritskov, L.V., Boundary Control Problem for the One-Dimensional Klein–Gordon–Fock Equation with a Variable Coefficient. The Case of Control by Displacement at OneEndpoint with the Other Endpoint Being Fixed, Differ. Uravn., 2013, vol. 49, no. 6, pp. 759–771.
Kritskov, L.V. and Abdukarimov, M.F., Boundary Control by Displacement at One End with the OtherEnd Free for a Process Described by the Telegraph Equation with a Variable Coefficient, Dokl. Akad. Nauk, 2013, vol. 450, no. 6, pp. 640–643.
Abdukarimov, M.F. and Kritskov, L.F., Boundary Control Problem for the One-Dimensional Klein–Gordon–Fock Equation with a Variable Coefficient: the Case of Control by Displacements at TwoEndpoints, Differ. Uravn., 2013, vol. 49, no. 8, pp. 1036–1046.
Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of MathematicalPhysics), Moscow, 2004.
Lions, J.-L., Upravlenie singulyarnymi raspredelennymi sistemami (Control for Singular DistributedSystems), Moscow, 1987.
Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of MathematicalPhysics), Moscow: Nauka, 1973.
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Original Russian Text © L.V. Kritskov, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 5, pp. 688–696.
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Kritskov, L.V. On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient. Diff Equat 51, 701–709 (2015). https://doi.org/10.1134/S0012266115050122
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DOI: https://doi.org/10.1134/S0012266115050122