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The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time interval for a linear steady–state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time–optimal problems. © 2020 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved.

We consider an optimal control problem with a convex integral quality functional for a linear system with fast and slow variables in the class of piecewise continuous controls with smooth constraints on the control, where x ∈ Rn, y ∈ Rm, u ∈ Rr; Aij and Bi, i, j = 1, 2, are constant matrices of corresponding dimension, and the functions ϕ1(·), ϕ2(·) are continuously differentiable in Rn, Rm, strictly convex, and cofinite in the sense of the convex analysis. In the general case, for such problem, the Pontryagin maximum principle is a necessary and sufficient optimality condition and there exist unique vectors le and pe determining an optimal control by the formula, The main difference of our problem from the previous papers is that the terminal part of quality functional depends on the slow and fast variables and the controlled system is a more general form. We prove that in the case of a finite number of control change points, a power asymptotic expansion can be constructed for the initial vector of dual state, which determines the type of the optimal control. © 2019 Danilin A.R., Shaburov A.A.

The paper continues the author's previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball (Formula Presented) The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix J at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter μ. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence "("k + μk), 0 < < 1. Keywords: Optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter. © 2019 Trudy Instituta Matematiki i Mekhaniki UrO RAN. All rights reserved.

This work was supported by the Russian Foundation for Basic Research (project no. 11-01-00679), the program "Leading Scientific Schools" (project no. NSh-2215.2008.1), and the Targeted Federal Program (state contract no. 02.740.11.0612).

This work was supported by the Russian Foundation for Basic Research (project no. 08-01-00260) and by the Program for State Support of Leading Scientific Schools (project no. NSh-2215.2008.1)., A problem of optimal boundary control of solutions to an elliptic-type equation with a small coefficient at the highest derivative and integral constraints on the control is considered. Asymptotic estimates for solutions to a problem that approximates the original problem are obtained.