2023.2

The paper considers the problem of feedback control for a group of several quadrocopters. The main goal is to transfer this group from a given initial position to a given target position for a fixed period of time, provided that at each intermediate time instant the quadrocopters must be in a small neighborhood of a fixed smooth curve in space. A centralized group control scheme is used. The main difficulty here is the complex, nonlinear mathematical model, describing the movement of each individual aircraft. It is necessary to develop computationally efficient algorithms for approximate search for feedback control, which allow to cope simultaneously with nonlinear dynamics, point-to-point constraints on control parameters, as well as state constraints arising in connection with group motion (in particular, due to the requirements of pairwise collision avoidance). Modifications of ellipsoidal calculus methods are used here to construct such algorithms.

It is considered the question about continuity of optimal time, as the function of initial state, for linear control objects. We obtained some generalization of the Theorems 21, 22 from the second Chapter of monograph E.B.Lee, L.Markus “Foundations of Optimal Control Theory,” J.Wiley and Sons, 1967, connected with continuity of optimal time, as the function of initial state, for linear controlled objects. In the article we broadly use the apparatus of support functions from Convex Analysis. Our results are obtained in more constructive form than another general results In this direction. In the first part we consider stationary case and in the second part it is considered non-stationary case.

Approximate bilinear algorithms for multiplication of 2 × 3 and 3 × 4 matrices (complexity 18), 2 × 4 and 4×4 matrices (complexity 24), 2×5 and 5×4 matrices (complexity 30) are presented, and approximate bilinear algorithms for multiplication of 2×n and n×4 matrices (complexity 6n) over field of characteristic zero gotten via them.

Dirichlet boundary value problem for the elliptic equation in a rectangle with a noninteger degeneration and analytic coefficients is considered. Applying the method of spectral isolation of singularities, the formal solution to the problem is constructed as a series which exhibits its non-analytic dependence with respect to y at the origin. Green’s function method verifies the convergence of the series to the classical solution of the considered problem.

The time optimal control problem with a state constraint investigates in the article. The object’s behavior is described by a system of second-order differential equations. The coefficient matrix for state variables has zero eigenvalues. The state constraint is linear. An admissible control is a piecewise continuous function that takes values from a given compact set. Sets of controllability to the origin are constructed for time intervals of various lengths.

It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l±1. In the paper we prove that there is no universal polynomials for the class of linear functions depending on two arguments for any even k.