Abstract
Consideration is given to the problem of optimal control of a system of semilinear hyperbolic equations at finite-dimensional (pointwise) relations between initial boundary states of the system and control actions. In this case, the optimality condition in the form of a pointwise maximum principle is invalid. A nonclassical optimality condition of a variational type is proved. The sense of the obtained result is in that an optimal boundary or start control nearly at each point is a solution to a special problem of control of initial conditions for the system of ordinary differential equations constructed on the family of characteristics of the initial hyperbolic system. Constructions of the indicated optimality condition are illustrated by two examples. An iterative method based on the obtained result is stated. The method does not require additional assumptions about the differentiability of parameters of the control problem and convexity of the set of admissible controls necessary in using gradient methods.
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References
Brokate, M., Necessary Optimality Conditions for the Control of Semilinear Hyperbolic Boundary Value Problems, SIAM J. Control Optim., 1987, vol. 25, no. 5, pp. 1353–1369.
Arguchintsev, A.V., The Search for Optimal Boundary Controls in Two-dimensional Semilinear Hyperbolic Equations, in Modeli i metody issledovaniya operatsii (Models and Methods of Operations Research), Novosibirsk: Nauka, 1988, pp. 50–58.
Wolfersdorf, L., A Counter Example to the Maximum Principle of Pontryagin for a Class of Distributed Parameter Systems, Z. Angew. Math. Mech., 1980, vol. 6, p. 204.
Srochko, V.A., Optimality Conditions of the Type of the Maximum Principle in Goursat-Darboux Systems, Sib. Mat. Zh., 1984, vol. 25. no. 1, pp. 126–133.
Rozhdestvenskii, B.L. and Yanenko, N.N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike (Systems of Quasi-linear Equations and Their Applications to Gas Dynamics), Moscow: Nauka, 1978.
Potapov, M.M., A Generalized Solution to a Mixed Problem for a Semilinear Hyperbolic First-Order System, Diff. Uravn., 1983, vol. 19, no. 10, pp. 1826–1828.
Sobolev, S.L., Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike (Some Applications of Functional Analysis in Mathematical Physics), Moscow: Nauka, 1988.
Morozov, S.F., Initial Boundary-Value Problem for Almost Linear Hyperbolic System in Plane, in Kraevye zadachi (Boundary-Value Problems), Perm, 1989, pp. 141–149.
Vasil’ev, O.V., Srochko, V.A., and Terletskii, V.A., Metody optimizatsii i ikh prilozheniya. Ch. 2. Optimal’noe upravlenie (Methods of Optimization and Their Applications. Vol. 2. Optimal Control), Novosibirsk: Nauka, 1990.
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Original Russian Text © A.V. Arguchintsev, V.P. Poplevko, 2008, published in Avtomatika i Telemekhanika, 2008, No. 4, pp. 17–28.
This work was supported by the Russian Foundation for Basic Research, projects nos. 05-01-00187 and 06-01-81016 and by the grant of President of the Russian Federation, project no. MD 989.2005.1.
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Arguchintsev, A.V., Poplevko, V.P. Variational optimality condition in the problem of control of initial boundary conditions for semilinear hyperbolic systems. Autom Remote Control 69, 559–569 (2008). https://doi.org/10.1134/S0005117908040024
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DOI: https://doi.org/10.1134/S0005117908040024