ISSN 0005-1179, Automation and Remote Control, 2013, Vol. 74, No. 1, pp. 36–45. © Pleiades Publishing, Ltd., 2013. Original Russian Text © A.I. Kalinin, 2013, published in Avtomatika i Telemekhanika, 2013, No. 1, pp. 47–58. NONLINEAR SYSTEMS Construction of Suboptimal Solution of the Singularly Perturbed Problem of Minimal-intensity Control A. I. Kalinin Belarus State University, Minsk, Belarus Received April 2, 2012 Abstract—For the linear singularly perturbed system, consideration was given to the problem of transient process optimization which resides in determining a multidimensional minimal- intensity control. It was demonstrated that solution of two nonperturbed problems of optimal control of lower dimensionality suffices to construct its asymptotically approximate solution. DOI: 10.1134/S0005117913010049 1. INTRODUCTION The systems of differential equations with small parameters at part of the derivatives are usually called the singularly perturbed systems. The mathematical theory of optimal processes pays much attention to their optimization (see [1]) which is due to the efficiency of the asymptotic methods for their solution. The numerical solution of the optimal control problems is known to require multiple integrations of the direct and conjugate systems. In the problems with singular pertur- bations, these dynamic systems are rigid [2], and as the result their calculations encounter serious difficulties manifested in an inadmissibly long time of calculations and inevitable accumulation of the calculation errors. The asymptotic methods not only enable one to do without integrating the singularly perturbed systems, but also to reduce the original problem of optimal control to problems of lower dimensionality. The present paper is devoted to the construction of an asymptotic approximation to the solution of the problem of transient process optimization in a linear singularly perturbed system with multidimensional controls. This problem lies in determining a minimal-intensity control. In this case, by the intensity is meant the minimum of the Euclidean norm of the values of the control action. In the applied problems, the control often has the sense of a generalized force, and intensity then estimates the greatest value of this force. Therefore, the problem under consideration is referred to as that of minimal-force control [3]. Owing to the nonsmoothness of the performance functional, such problems occupy a special place among the typical problems of optimization of dynamic systems. They arise in the applications where great values of the control actions are either technically nonrealizable or undesirable because of the excessive overloads caused by accelerations. It deserves noting that a similar problem was already considered in [4]. The assumptions made in this reference are substantially relaxed below. 2. FORMULATION OF THE PROBLEM In the class of r-dimensional control actions u(t)=(u 1 (t),...,u r (t)), t ∈ T = [0,t ∗ ], r ≥ 2, with sectionally continuous components we consider the problem of optimal control of the stationary 36