### ASYMPTOTIC EXPANSION OF A SOLUTION FOR ONE SINGULARLY PERTURBED OPTIMAL CONTROL PROBLEM IN \(\mathbb{R}^n\) WITH A CONVEX INTEGRAL QUALITY INDEX

#### Abstract

The paper deals with the problem of optimal control with a convex integral quality index for a linear steady-state control system in the class of piecewise continuous controls with a smooth control constraints. In a general case, for solving such a problem, the Pontryagin maximum principle is applied as the necessary and sufficient optimum condition. In this work, we deduce an equation to which an initial vector of the conjugate system satisfies. Then, this equation is extended to the optimal control problem with the convex integral quality index for a linear system with a fast and slow variables. It is shown that the solution of the corresponding equation as \(\varepsilon\to 0\) tends to the solution of an equation corresponding to the limit problem. The results received are applied to study of the problem which describes the motion of a material point in \(\mathbb{R}^n\) for a fixed period of time. The asymptotics of the initial vector of the conjugate system that defines the type of optimal control is built. It is shown that the asymptotics is a power series of expansion.

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Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The Mathematical Theory of Optimal Processes. NY–London–Sydney: J. Wiley and Sons inc., 1962. 360 pp.

Krasovskii N.N. Theory of Control of Movement. Linear Systems. M.: Nauka, 1968. 476 pp. [in Russian]

Lee E.B., Markus L. Foundations of Optimal Control Theory. NY–London–Sydney: John Wiley and Sons, Inc., 1967. 576 pp.

Vasil'eva A.B., Dmitriev M.G. Mathematical analysis. The results of science and technology. M.:VINITI, 1982. Vol. 20. P. 3–77. [in Russian]

Kokotovic P.V., Haddad A.H. Controllability and time-optimal control of systems with slow and fast models // IEEE Trans. Automat. Control. 1975. Vol. 20, no. 1. P. 111–113. DOI: 10.1109/TAC.1975.1100852

Dontchev A.L. Perturbations, approximations and sensitivity analisis of optimal control systems. Berlin–Heidelberg–New York–Tokio: Springer-Verlag. 1983. 156 pp. DOI: 10.1007/BFb0043612

Kalinin A.I., Semenov K.V. Asymptotic Optimization Method for Linear Singularly Perturbed Systems with Multidimensional Control // Computational Mathematics and Mathematical Physics. 2004. Vol. 44, no. 3. P. 407–418.

Danilin A.R., Parysheva Y.V. Asymptotics of the optimal cost functional in a linear optimal control problem // Doklady Mathematics. 2009. Vol. 80, no. 1. P. 478–481.

Danilin A.R., Kovrizhnykh O.O. Time-optimal control of a small mass point without environmental resistance // Doklady Mathematics. 2013. Vol. 88, no. 1. P. 465–467.

Vasil'eva A.B., Butuzov V.F. Asymptotic expansions of a solutions of singularly perturbed equations. M.: Nauka, 1973. [in Russian]

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