Vol 212 (2022)

In this paper, we examine a linear optimal-control problem for a first-order hyperbolic system in which a boundary condition at one of the ends is determined from a controlled system of ordinary differential equations with constant state lag. The approach proposed is based on the use of an exact formula for the increment of the cost functional. The reduced problem can be solved by various effective methods used for optimization problems in systems of ordinary differential equations.

The boundary control problem is considered for the equation of string vibration with given initial and final conditions, with given values of the deflection function and velocities of points at different intermediate times. The control is carried out by displacement at the two string ends. We propose a constructive approach for constructing boundary control of string vibrations by displacement at two ends with given initial and final conditions and values of the deflection function and velocities of points given at different intermediate times. A computational experiment was carried out with the construction of the corresponding graphs and their comparative analysis, which confirmed the results obtained.

The paper describes possible implementation of the general theory of asymptotic integration of singularly perturbed differential equations developed by S. A. Lomov and his disciples to constructing asymptotic solutions for singularly perturbed differential equations with a power boundary layer.

In this work, we consider problems in which two types of controls (distributed and starting control functions) are used simultaneously. The main results concern the solvability of a class of optimal control problems for systems whose states are described by equations in Banach spaces that are resolved with respect to the Gerasimov-Caputo fractional derivative and nonlinear in the lowest fractional derivatives. We consider convex lower semicontinuous, coercive functionals, which are compromise or control-independent. Abstract results are demonstrated by an example of a control problem for a fractional model of metastable states in semiconductors.

In this paper, a convex linear-quadratic problem is considered within the class of nonlocal descent methods. The uniqueness of solutions of the phase and conjugate systems is established. The convergence of iterative methods with respect to the cost functional is proved.

The paper considers the initial value problem for a differential equation with the derivatives of the functionals in Banach spaces. The operator of the elder derivative has the structure of projector, i.e. its core is infinite-dimensional. The solution is constructed in the space of generalized functions with the support bounded on the left in the form of convolution of the fundamental solution of the differential operator with the right-hand side of the equation, which includes a free function and some initial conditions of the initial problem. The process of construction of the fundamental solution is realized with the aid of a fundamental operator function for a specially constructed matrix differential operator with an irreversible (generally speaking) matrix in the derivative, i.e. with the operator of finite index. Analysis of the generalized solution constructed by this technique allows one to obtain the sufficient conditions of solvability for our initial-value problem in the classes of finite smoothness functions, and also propose constructive formulas needed to restore such a solution. An illustrative example is given.

In many problems of dynamics, systems arise whose position spaces are four-dimensional manifolds. Naturally, the phase spaces of such systems are the tangent bundles of the corresponding manifolds. Dynamical systems considered have variable dissipation, and the complete list of first integrals consists of transcendental functions expressed in terms of finite combinations of elementary functions. In this paper, we prove the integrability of more general classes of homogeneous dynamical systems with variable dissipation on tangent bundles of four-dimensional manifolds. The first part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. I. Equations of geodesic lines// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 210 (2022), pp. 77-95. The second part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. II. Potential force fields// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 211 (2022), pp. 29-40.