Geodesic
On a control problem by lumped-parameter at the right-hand side of the semi-linear hyperbolic system
The Bulletin of Irkutsk State University. Series Mathematics, Tome 11 (2015), pp. 3-12 Cet article a éte moissonné depuis la source Math-Net.Ru

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At the article an optimal control problem by a first order system of semi-linear hyperbolic equations in the class of smooth control function is studied. A function at the right-hand side of the system is determined by a control system of ordinary differential equations. The initial-boundary value problem equivalent the system of integral equations on the characteristics of the hyperbolic system [4] (generalized solution). Control functions are satisfied the pointwise constraints. Such problems arise in modeling of chemical technology processes [3]. We obtain necessary optimality conditions of variational type by using the procedure [1] in the class of admissible smooth controls for the first order semi-linear hyperbolic systems [2]. An optimal control methods based on the maximum principle of Pontryagin do not use for such problems. These methods are focused on classes of discontinuous control functions. The proposed approach is based on the use of special variations which provide smooth control function and satisfaction the pointwise constraints. The condition of optimality is proved, and a scheme of iterative methods is proposed. The numerical experiment is carried out. Numerical results are presented by graphics of solutions. The numerical experiments show that the proposed method of improving the smooth control functions can be effectively used to solve this class of problems.
Keywords: hyperbolic systems, optimal control, necessary condition of optimality, smooth control. 
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A. V. Arguchintsev; V. P. Poplevko. On a control problem by lumped-parameter at the right-hand side of the semi-linear hyperbolic system. The Bulletin of Irkutsk State University. Series Mathematics, Tome 11 (2015), pp. 3-12. http://geodesic.mathdoc.fr/item/IIGUM_2015_11_a0/

[1] Arguchintsev A. V., Optimal control by hyperbolic systems, Fizmatlit, M., 2007, 165 pp. (in Russian)

[2] Arguchintsev A. V., Poplevko V. P., “Optimization by a class of hyperbolic system with smooth controls”, Izvestiya Vuzov. Matematika, 2009, no. 7, 71–76 (in Russian)

[3] Demidenko N. D., Potapov V. I., Shokin U. I., Modeling and optimization of systems with distributed parameter, Nauka, Novosibirsk, 1983, 271 pp. (in Russian)

[4] Rozhdestvenskiy B. L., Yanenko N. N., System of quasi-linear equations and their applications to gas dynamics, Nauka, M., 1978, 686 pp. (in Russian)