Vol 212 (2022)
Статьи
Variational optimality condition in a control problem of a linear first-order hyperbolic system with boundary delay
Abstract
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.



Lie algebras of projective motions of five-dimensional pseudo-Riemannian spaces. I. Preliminaries
Abstract
This work is devoted to the problem of studying multidimensional pseudo-Riemannian manifolds that admit Lie algebras of infinitesimal projective (in particular, affine) transformations, wider than Lie algebras of infinitesimal homotheties. Such manifolds have numerous geometric and physical applications. This paper is the first part of the work; continuation will be published in future issues.



The problem of boundary control of vibrations of a string by displacements at two ends with given states at intermediate time moments
Abstract
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.



On strongly coupled multidimensional elliptic systems
Abstract
The article investigates the Dirichlet problem in a half-space for an elliptic system as per Petrovsky of three equations with three unknown functions depending on three independent variables. Conditions for violation of the Noetherian property of the problem in a concrete half-space are found. It is shown that the strong connectedness of the system does not imply the violation of the Noetherian property of the Dirichlet problem in any half-space.



Construction of asymptotic solutions of some degenerate differential equations with a small parameter
Abstract
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.



On small solutions of nonlinear operator equations with noninvertible operator in the principal term
Abstract
In this paper, we examine a nonlinear operator equation with vector parameter, which does not satisfy the implicit operator theorem since the operator in the principal term is not continuously invertible at a given point. We prove a sufficient conditions of existing small continuous solution and propose an algorithm of constructing such solution in some domain.



Mixed control for semilinear fractional equations
Abstract
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.



On the unique solvability of a problem of identifying lower coefficients in a multidimensional system of composite type
Abstract
In this paper, we prove the existence and uniqueness theorem for a solution of the problem of determining four lower coefficients in a composite multidimensional system in the case of the Cauchy data.



Numerical solution of a linear-quadratic optimal control problem based on nonlocal methods
Abstract
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.



Variational statement of a coefficient inverse problem for a multidimensional parabolic equation
Abstract
In this paper, we consider the variational statement of an inverse problem of determining the leading coefficient of a multidimensional parabolic equation with nonlocal conditions. The leading coefficient of the equation playing the role of a control function is an element of the Sobolev space. The objective functional is based on the overdetermination condition, which can be interpreted as setting the weighted average value of the solution of the equation considered with respect to the time variable. The well-posedness of the problem in the weak topology of the control space is examined, the Frechet differentiability of the objective functional is proved, and a necessary optimality condition is obtained.



On the solvability in the class of distributions of differential equations with derivatives of functionals in Banach spaces
Abstract
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.



On hyperbolic approximation of the problem of determining a source function
Abstract
The paper considers the unique solvability of the problem of determining source function in a hyperbolic heat equation with a small parameter as a coefficient to the second time derivative.



Integrable homogeneous dynamical systems with dissipation on the tangent bundles of four-dimensional manifolds. III. Force fields with dissipation
Abstract
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.



Systems with dissipation with a finite number of degrees of freedom: analysis and integrability. II. General class of dynamical systems on the tangent bundle of a multidimensional sphere
Abstract
This paper is the second part of a survey on the integrability of systems with a large number n of degrees of freedom (the first part: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 211 (2022), pp. 41-74). The review consists of three parts. In the first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. In this second part, we consider more general dynamical systems on the tangent bundles to the n-dimensional sphere. In the third part, which will be published in the next issue, we will consider dynamical systems on the tangent bundles to smooth manifolds of a sufficiently wide class.
Theorems on sufficient conditions for the integrability of the considered dynamical systems in the class of transcendental functions are proved.


