We consider an optimal control problem in the class of piecewise continuous controls with smooth geometric constraints for a linear system with constant coefficients and an integral convex performance criterion containing two small parameters (the first of them multiplying the integral term, and the second in the initial data). Such problems are called cheap control problems. It is shown that a problem with a terminal performance index will be the limit one. It is established that if the limit problem is actually one-dimensional whereas the initial problem is not, then the asymptotics of the solution can be more complicated. In particular, the asymptotics of the solution 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, cheap control, asymptotic expansion, small parameter
Received January 4, 2023
Revised February 3, 2023
Accepted February 6, 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
Aleksandr Aleksandrovich Shaburov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: alexandershaburov@mail.ru
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Cite this article as: A.R.Danilin, A.A.Shaburov. Asymptotics of a solution to an optimal control problem with integral convex performance index, cheap control, and initial data perturbations. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 67–76; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S69–S77.
