A.R. Danilin, O.O. Kovrizhnykh. Asymptotics of a solution to an optimal control problem with a terminal convex performance index and a perturbation of the initial data... P. 41-53

In this paper, we investigate a problem of optimal control over a finite time interval for a linear system with constant coefficients and a small parameter in the initial data in the class of piecewise continuous controls with smooth geometric constraints. We consider a terminal convex performance index. We substantiate the limit relations as the small parameter tends to zero for the optimal value of the performance index and for the vector determining the optimal control in the problem. We show that the asymptotics of the solution can be of complicated nature. In particular, it may have no expansion in the Poincare sense in any asymptotic sequence of rational functions of the small parameter or its logarithms.

Keywords: optimal control, terminal convex performance index, asymptotic expansion, small parameter

Received March 9, 2023

Revised April 13, 2023

Accepted April 17, 2023

Aleksei Rufimovich Danilin, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: dar@imm.uran.ru

Ol’ga Olegovna Kovrizhnykh, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: koo@imm.uran.ru

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Cite this article as: A.R. Danilin, O.O. Kovrizhnykh. Asymptotics of a solution to an optimal control problem with a terminal convex performance index and a perturbation of the initial data. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2023, vol. 29, no. 2, pp. 41–53; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S85–S97.