Geodesic
Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints
Ural mathematical journal, Tome 4 (2018) no. 1, pp. 63-73 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 smooth control constraints. In a general case, to solve such a problem, the Pontryagin maximum principle is applied as the necessary and suficient optimum condition. The main difference from the preceding article [10] is that the terminal part of the convex integral quality index depends not only on slow, but also on fast variables. In a particular case, we derive an equation that is satisfied by an initial vector of the conjugate system. Then this equation is extended to the optimal control problem with the convex integral quality index for a linear system with the fast and slow variables. It is shown that the solution of the corresponding equation as $\varepsilon\to0$ tends to the solution of an equation corresponding to the limit problem. The results obtained are applied to study a problem which describes the motion of a material point in Rnfor a fixed interval 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.
Keywords: Optimal control, Singularly perturbed problems, Asymptotic expansion, Small parameter. 
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Alexander A. Shaburov. Asymptotic expansion of a solution for the singularly perturbed optimal control problem with a convex integral quality index and smooth control constraints. Ural mathematical journal, Tome 4 (2018) no. 1, pp. 63-73. http://geodesic.mathdoc.fr/item/UMJ_2018_4_1_a5/

[1] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F., The Mathematical Theory of Optimal Processes, John Wiley Sons Inc., New York, 1962, 360 pp. | MR | Zbl

[2] Krasovskii N.N., Theory of Control of Motion. Linear Systems, Nauka, Moscow, 1968, 476 pp. (in Russian) | MR

[3] Lee E.B., Markus L., Foundations of Optimal Control Theory, John Wiley Sons Inc., New York, 1967, 576 pp. | MR | Zbl

[4] Vasil'eva A.B., Dmitriev M.G., “Singular perturbations in optimal control problems”, J. of Soviet Mathematics, 34:3 (1986), 1579–1629 | DOI | Zbl

[5] Kokotović P.V., Haddad A.H., “Singular perturbations in optimal control problems”, Controllability and time-optimal control of systems with slow and fast models, 20:1 (1975), 111–113 | DOI | MR | Zbl

[6] Dontchev A.L., Perturbations, approximations and sensitivity analisis of optimal control systems, Springer-Verlag, Berlin–Heidelberg–New York–Tokio, 1983, 161 pp. | DOI | MR

[7] Kalinin A.I., Semenov K.V., “The asymptotic optimization method for linear singularly perturbed systems with the multidimensional control”, Computational Mathematics and Mathematical Physics, 44:3 (2004), 407–417 | MR | Zbl

[8] Danilin A.R., Parysheva Y.V., “Asymptotics of the optimal cost functional in a linear optimal control problem”, Doklady Mathematics, 80:1 (2009), 478–481 | DOI | MR | Zbl

[9] Danilin A.R., Kovrizhnykh O.O., “Time-optimal control of a small mass point without environmental resistance”, Doklady Mathematics, 88:1 (2013), 465–467 | DOI | MR | Zbl

[10] Shaburov A.A., “Asymptotic expansion of a solution of a singularly perturbed optimal control problem in the space $\mathbb{R}^n$ with an integral convex performance index”, Ural. Math. J., 3:1 (2017), 68–75 | DOI | MR