Vol 226 (2023)

We consider boundary-control problems for a distributed inhomogeneous oscillatory system described by a one-dimensional wave equation with piecewise constant characteristics. We assume that the propagation times for all homogeneous sections are the same. The control is performed by shifting one end with the other end fixed. The initial, intermediate, and final conditions on the deflection function and the velocities of the points of the system are given. An approach to the analytical construction of the boundary control is proposed. The results obtained are illustrated by a specific example. A computational experiment and a comparative analysis were performed.

In this paper, we obtain general equations for canonical first-type almost geodesic mappings of affinely connected spaces under which the Riemann tensor is preserved. These equations are reduced to a closed system of Cauchy-type equations in covariant derivatives. The number of essential parameters on which the general solution of the resulting system of equations depends is established. A particular case of such mappings is considered and examples of almost geodesic mappings of the first type of flat space onto flat space are given.

In this paper, we examine a system of singular integral equations with a Carleman shift corresponding to a multielement boundary-value problem for bianalytic functions. The results obtained are applicable to the solution of the main problems of the theory of elasticity in the contact interaction of bodies with various elastic properties.

For the abstract Euler–Poisson–Darboux equation, an inverse problem with a final redefinition of the second kind is considered. A uniqueness criterion for solutions is established. As an application of the criterion established, uniqueness criteria for solutions of inverse problems for degenerate differential equations are given.

In this paper, we consider a class of start control problems for systems whose states are described by equations in Banach spaces that are not solvable with respect to the highest Gerasimov–Caputo fractional derivative and depend nonlinearly on lower-order fractional derivatives. A theorem on the existence of an optimal control is obtained. Abstract results are applies to the study of the start control problem for the modified Sobolev equation with a fractional derivative in time.

In this paper, we study a model for the distribution of a resource flow in a resource network with dynamic durations of passage along arcs. A feature of such networks is the dependence of the duration of passage along arcs on discrete time. This feature significantly affects the process of redistribution of resources. It is shown that in the networks considered, the total resource is preserved, while the total resource can be distributed not only over vertices, but also over some arcs. A relation is obtained for the conservation of the total resource in the network. A method for finding the threshold value in a resource network with dynamic durations of passage along arcs is proposed. It is shown that if the total resource is not less than the threshold value in the original network, then in a network with dynamic durations of passage along arcs, there is a unique limiting flow.

We study initial-value problems for quasilinear equations with Gerasimov–Caputo fractional derivatives in Banach spaces whose linear part has an analytic resolving family of operators in the sector. The nonlinear operator is assumed to be a locally Lipschitz operator. We consider equations that are solved with respect to the highest derivative and equations containing a degenerate linear operator acting on the highest derivative. The theorem on the unique solvability of the Cauchy problem proved in the paper is used for the study of the unique solvability of the Showalter–Sidorov problem for degenerate equations. Abstract results are applied to the initial-boundary-value problem for partial differential equations that are not solvable with respect to the highest fractional derivative in time.

For a linear nonstationary singularly perturbed system with small coefficients of higher derivatives, we examine the property of uniform observability, which characterizes the possibility of uniquely determining the state of the system at any time $t$ by the values of the output function and its derivatives up to a certain order only at the point $t$, as well as the property of approximative observability, which means the possibility of accurate estimating the current state of the system without differentiating the output function using $\delta$-sequences.

The uniqueness and existence of a solution to the Cauchy problem for an integro-differential equation associated with a linear peridynamic model in the mechanics of a rigid body with nonlinear elastic properties are proved.