*Article* **Justification of Direct Scheme for Asymptotic Solving Three-Tempo Linear-Quadratic Control Problems under Weak Nonlinear Perturbations**

**Galina Kurina 1,2,***<sup>∗</sup>* **and Margarita Kalashnikova <sup>3</sup>**


**Abstract:** The paper deals with an application of the direct scheme method, consisting of immediately substituting a postulated asymptotic solution into a problem condition and determining a series of control problems for finding asymptotics terms, for asymptotics construction of a solution of a weakly nonlinearly perturbed linear-quadratic optimal control problem with three-tempo state variables. For the first time, explicit formulas for linear-quadratic optimal control problems, from which all terms of the asymptotic expansion are found, are justified, and the estimates of the proximity between the asymptotic and exact solutions are proved for the control, state trajectory, and minimized functional. Non-increasing of the minimized functional, if a next approximation to the optimal control is used, following from the proposed algorithm of the asymptotics construction, is also established.

**Keywords:** optimal control problems; weak nonlinear perturbations; three-tempo variables; asymptotic solutions; the direct scheme method; estimates of asymptotic solution

**MSC:** 34H05; 34E13
