The paper considers the problem of optimizing a long-term transient process in a linear singularly perturbed system, which consists of finding a multidimensional admissible control with the minimal value of an integral quadratic performance index. This problem can be interpreted as a control problem with minimal energy consumption. By passing to “slow time”, the problem is reduced to that of optimizing a singularly perturbed system with two groups of fast variables at different rates in the absence of slow variables. Using an approach based on the maximum principle and the method of boundary functions, an asymptotic approximation to the solution of this problem is constructed as a sum of optimal controls in two unperturbed problems of lower dimension with infinite process duration.

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