Abstract The connections between stochastic differential equations in which continuous and discontinuous random processes serve as sources of randomness and deterministic equations for the probabilistic characteristics of solutions of these stochastic equations are studied. In the study, we use various approaches based on the stochastic change of variables formula (Itô’s formula), on the analysis of

Abstract Initial value problems with various boundary conditions at the ends are studied for the beam vibration equation taking into account the beam rotational motion under bending. An energy inequality is established, which implies the uniqueness of the solution of the four initial–boundary value problems posed. In the case of hinged ends, theorems on the existence and stability of the solution of

Abstract We study a mixed problem with nonlinear dissipative boundary conditions for systems of one-dimensional semilinear wave equations with a focusing nonlinear source that has a variable growth exponent. Theorems on the blow-up of solutions in finite time are proved.

Abstract Using interval Taylor models (TM), we construct algorithms for the computer-assisted proof of the existence of periodic trajectories in systems of ordinary differential equations (ODEs). Although TMs allow one to construct guaranteed estimates for families of solutions of systems of ODEs when integrating ODEs over large time intervals, the interval residual included in the TMs begins to grow

Abstract We prove the existence and uniqueness of the solution of the Cauchy problem for an integro-differential equation related to the peridynamic model in mechanics of solids.

Abstract We prove that a necessary and sufficient condition for the existence of a classical solution of the inhomogeneous biharmonic equation in a bounded plane domain is the requirement of stronger continuity of the function on the right-hand side in the equation.

Abstract We prove the existence and uniqueness of a solution of the Darboux problem for the fourth-order Bianchi equation. The Riemann–Hadamard function is determined for the Darboux problem. The solution of the problem is constructed in terms of this function.

Abstract We extend an analog of the Chetaev–Malkin–Massera theorem on stability by the first approximation for differential systems to discrete-time systems in which the linear first approximation system is subjected to linear and nonlinear perturbations. The result is stated in terms of the characteristic exponents and the Lyapunov irregularity coefficient of the linear first approximation system

Abstract For an integro-differential equation of the convolution type with a power nonlinearity and a variable coefficient defined on the half-line \([0,\infty )\), we use the method of weight metrics in the cone of the space \(C^1(0,\infty ) \) formed by functions positive on \((0,\infty ) \) and vanishing at the origin to prove a global theorem on the existence and uniqueness of a solution belonging

Abstract We study the Tricomi problem for an advance–delay mixed type equation. Theorems on the uniqueness and existence of a twice continuously differentiable solution are proved.

Abstract We establish estimates (interior and up to the boundary) of solutions of the inequality \(\mathscr {L}(u)\geq 0\), where \(\mathscr {L} \) is a linear second-order differential operator with impulse singularities. The results in particular apply to the Neumann boundary value problem. The relationship between this inequality and the class of concave functions of one variable is studied.

Abstract We use the Lyapunov vector function method and the guiding function method to obtain sufficient conditions for the existence of Poisson bounded and partially Poisson bounded solutions of systems of differential equations.

Abstract We consider the polynomial Liénard system \(\dot {x}=-y \), \(\dot {y}=x+A(x)-B(x)y \) under the assumption that the real polynomials \(A(x) \) and \(B(x) \) and the derivative \(A^{\prime }(x) \) satisfy the conditions \(A(0)=B(0)=A^{\prime }(0)=0 \). We prove that this system has an isochronous center at the singular point \(O(0,0)\) if and only if the polynomials \( A(x)\) and \(B(x) \)

Abstract We consider nonautonomous linear systems of differential equations admitting a time-dependent first integral that is a quadratic form. The duality between mutually adjoint linear systems with quadratic integrals is established. Conditions for the spectrum of such linear systems to be symmetric about zero are indicated. We prove that a linear system is stable if and only if it admits a first

Abstract We study the asymptotics of the spectrum of the Dirac operator on the real line with a potential in \(L_2 \). It is shown that the spectrum of such an operator lies in a domain of the complex plane symmetric about the real axis and bounded by the graph of some continuous real-valued square integrable function. To prove this, we use the \(L_1 \)-functional calculus for self-adjoint operators

Abstract We consider a system of \(n\)th-order ordinary differential equations whose right-hand side is the sum of a linear function of the solution with a constant matrix, an essential nonlinearity of the relay type with hysteresis, and a perturbing continuous periodic function. The matrix of the linear function has only real simple nonzero eigenvalues, of which at least one is positive. We study

Abstract We study how a boundary value problem with a nonlocal integral condition of the first kind for a mixed type equation with a singular coefficient in a rectangular domain depends on a numerical parameter occurring in the equation. A uniqueness criterion is established, and theorems on the existence and stability of a solution of the problem are proved. The solution is constructed in closed form

Abstract For a linear periodic system of differential equations, we indicate a method permitting one to find a matrix similar to the matrix of the Poincaré map in closed form.

Abstract We prove that the \(p\)th moments, \(p\ge 1 \), of strong solutions of a mixed-type stochastic differential equation driven by a standard Brownian motion and a fractional Brownian motion with Hurst index greater than \(1/2\) are finite provided that the coefficients of the equation, together with all of their partial derivatives of order \(\le 2 \), are continuous and bounded and the \(p \)th

Abstract We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla \) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated

Abstract We study the existence of periodic solutions for a class of systems of first-order nonlinear ordinary differential equations on the plane. Using an a priori estimate of periodic solutions, we establish the homotopy invariance of the solvability property of the periodic problem, give a description of the homotopy classes, and prove a theorem about a necessary and sufficient condition for the

Abstract We establish the property of completeness in the space \(L_2(D) \), where \(D \) is a domain in \(\mathbb {R}^n \), \(n\geq 3\), of the products of all possible regular solutions of the equation \(\Delta u-\kappa ^2 u=0 \), \(\kappa \in \mathbb {R}_{+}\cup i\mathbb {R}_{-} \), by the fundamental solutions of this equation with singularities on a straight line that does not meet \(\overline

Abstract We prove theorems on the Lyapunov stability, asymptotic stability, and instability of the equilibrium position of time-varying semilinear differential-algebraic equations (DAEs) as well as theorems about the asymptotic (complete) stability of these equations. As an example of applying the theorems on the global solvability of time-varying semilinear DAEs proved in the first part of this paper

Abstract We present methods for proving new sufficient stability and asymptotic stability tests for the zero solution of a time-varying differential equation using nonmonotone sign-indefinite Lyapunov functions. As an example of application of our statements, we establish new stability tests for a gradient system in which the equilibrium is not isolated and the function defining the right-hand side

Abstract For linear neutral type systems, we construct a finite time observer in the form of the output of a delayed type system with commensurable concentrated delays and finite spectrum. The observer permits one to obtain an exact estimate of the solution of the original system in finite time. The results are illustrated by an example.

Abstract We consider a linear second-order differential equation with coefficients depending on a spectral parameter and present conditions on the coefficients under which there exist no degenerate two-point boundary conditions for the equation.

Abstract We consider initial value problems for a number of hyperbolic equations with a power-law degeneracy and with operator coefficients in a Banach space and establish sufficient conditions for the unique solvability of these problems in terms of the coefficients of the equation, the degeneracy order, and the initial elements.

Abstract We study the spectral properties of the Cauchy problem for the differential operator \(-u^{{\prime \prime }}(x)+\alpha u^{{\prime \prime }}(-x) \) with an involution for \(\alpha \) satisfying the inequalities \(0<|\alpha |<1 \). Based on the analysis of the spectrum and the Green’s function constructed here, it is shown that if the parameter \(\varkappa =\sqrt {(1-\alpha )/(1+\alpha )}\)

Abstract We construct a regularized asymptotics of the solution of the first boundary value problem for a singularly perturbed two-dimensional differential equation of the parabolic type for the case in which the limit equation has a regular singularity. There arise power-law and corner boundary layers along with parabolic ones in such problems.

Abstract We consider the problem of reconstructing the solutions of a generalized Cauchy–Riemann system in a multidimensional bounded spatial domain from their values on part of the boundary of the domain; i.e., we construct an approximate solution of this problem based on the Carleman–Yarmukhamedov matrix method.

Abstract For time-varying semilinear differential-algebraic equations, we prove theorems on the existence and uniqueness of global solutions and theorems on Lagrange stability (boundedness of global solutions), dissipativity (ultimate boundedness of solutions), and Lagrange instability (absence of global solutions).

Abstract We consider an optimal feedback control problem for an initial–boundary value problem describing the motion of a nonlinearly viscous fluid. We prove the existence of an optimal solution minimizing a given performance functional. To prove the existence of an optimal solution, we use a topological approximation method for studying hydrodynamic problems.

Abstract We prove a theorem on the equivalence of a scalar Riemann boundary value problem with a shift-reflection on the real axis and a matrix Riemann boundary value problem without the shift. A relationship between the boundary value problem in question and a boundary value problem with orientation-preserving shift-rotation on the unit circle is established. The operator identities constructed by

Abstract We consider a controlled linear system of differential-algebraic equations with infinitely differentiable coefficients that is allowed to have an arbitrarily high unsolvability index. It is assumed that the matrix multiplying the derivative of the desired vector function has a constant rank. We prove a theorem on the existence of a solution in the class of Sobolev–Schwartz type generalized

Abstract We consider the spectral problem for a second-order differential operator with periodic boundary conditions and an integral perturbation. For this operator, we obtain the eigenvalue asymptotics as well as estimates for the deviations of spectral projections.

Abstract A Hilbert type problem is solved for a generalized Cauchy–Riemann system whose lower-order coefficients admit a strong singularity on a circle and a weak singularity at a point.

Abstract We consider an initial–boundary value problem for a system of third-order partial differential equations in a rectangular domain. By introducing a new unknown function, we reduce the problem to an equivalent one consisting of a nonlocal problem for a system of second-order hyperbolic equations with parameters and integral relations. An iterative algorithm is proposed for approximately solving

Abstract We use separation of variables to find the eigenvalues and the corresponding eigenfunctions of the Gellerstedt–Frankl boundary value problem (with deviation from the characteristics) for a mixed-type Lavrent’ev–Bitsadze equation in a special domain.

Abstract For the differential equation \(y^{\prime \prime \prime }(x)=\lambda y(x) \) on the interval \([0,1] \), we consider the three-point eigenvalue problem \(a_i y^{(i-1)}(0)+y(c)+b_i y^{(i-1)}(1)=0\), \(i=1,2,3 \), where \(\lambda \) is the spectral parameter, the point \(c\in (0,1) \) is fixed, and \(a_i \) and \(b_i \), \(i=1,2,3\), are some complex numbers. Necessary and sufficient conditions

Abstract For a first-order hyperbolic system of integro-differential equations with a convolution-type integral term, we study the inverse problem of determining the convolution kernel. The direct problem is an initial–boundary value problem for this system on a finite interval \([0, H] \). Under some data consistency conditions, the inverse problem is reduced to a system of Volterra type integral

Abstract Using asymptotic methods, we derive homogenization error estimates of the order \(O(\varepsilon ) \) for the periodic problem for a generalized Beltrami equation with rapidly oscillating \(\varepsilon \)-periodic coefficients.

Abstract For the Kipriyanov operator \(\Delta _B\) written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator \(\Delta _B\) on the sphere and the corresponding spherical functions (\(B \)-harmonics) are introduced. The main properties of the

Abstract We consider the problem of transforming two linear control systems to a form with relative degree by one and the same linear time-invariant change of outputs. A constructive algorithm is proposed that permits one to verify if this problem has a solution, and if it does, find the desired change. Moreover, it is shown that transforming three control systems by one and the same linear time-invariant

Abstract A nonlocal generalized implicit function theorem is obtained for mappings acting in Hilbert spaces. Its proof is based on the theory of ordinary differential equations. Under natural assumptions, we use this theorem to derive a global generalized implicit function theorem as well as a global implicit function theorem and a global inverse function theorem as particular cases of the former.

Abstract We consider the problem of designing a tracking system for nonlinear affine SISO plants under unmatched exogenous and parametric disturbances and incomplete measurements. In the framework of the block approach, we develop a discontinuous dynamic feedback control law ensuring a tracking with a given accuracy for the output variable of the command signal under the assumption of exogenous actions

Abstract For any parameters \(m>1\), \(\lambda _1\le \lambda _2<0 \), and \(\varepsilon >0 \) and for two sequences \(\{S_{in}\} \) of uniformly bounded arbitrary Suslin sets \(S_{1n}\subset [\lambda _1+\varepsilon ,b_1]\) and \(S_{2n}\subset [\max \{\lambda _2+\varepsilon ,b_1\},b_2]\), we prove the existence of a two-dimensional nonlinear differential system with a linear approximation that has characteristic

Abstract For a linear differential equation in a Banach space with a degenerate operator multiplying the fractional Gerasimov–Caputo derivative, we prove a theorem on the existence of a unique solution of the inverse problem with an unknown time-dependent coefficient. It is assumed that the relative boundedness condition is satisfied for the pair of operators occurring in the equation and an overdetermation

Abstract We study two problems on planar steady-state weakly supersonic potential flows of an ideal perfect gas with detached shock wave. In the first problem, we consider the flow past a finite wedge in an unbounded stream; the second problem deals with the flow past an infinite wedge in a steady jet. The free-stream velocity is close to the speed of sound, and therefore, the entropy increments \(\Delta

Abstract For a linear differential equation with the Hukuhara derivative and constant coefficient matrix, we obtain a necessary and sufficient condition under which every solution that is a polyhedron at the initial time remains a polyhedron (not necessarily with the same number of vertices) at all subsequent times.

Abstract We consider an inhomogeneous boundary value problem for a system of semilinear elliptic equations modeling radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index. The unique solvability of the boundary value problem is proved without requiring the initial data to be small.

Abstract We prove the existence and study the stability of time-periodic boundary layer solutions for a two-dimensional reaction–diffusion problem with a small parameter multiplying the parabolic operator for the case of singularly perturbed boundary conditions of the third kind. An asymptotic approximation to such solutions with respect to the small parameter is constructed. Conditions under which

Abstract We study the inverse problem of determining the right-hand side of a subdiffusion equation with Riemann–Liouville fractional derivative whose elliptic part has the most general form and is defined in an arbitrary multidimensional domain (with sufficiently smooth boundary). The Fourier method is used to prove theorems on the existence and uniqueness of the classical solution of the initial–boundary

Abstract We introduce a new class of first-order partial differential equations (coined as covariantly equipped systems of equations) invariant with respect to (pseudo)orthogonal changes of Cartesian coordinates in a (pseudo)Euclidean space. We propose a procedure for reducing the Cauchy problem for a system of equations to a Cauchy problem for a covariantly equipped system of equations (the covariant

Abstract We study local decompositions of nonlinear dynamical control systems. Orbital equivalences are used to transform systems into decomposable forms. Similar to the case of a classical decomposition, each orbital decomposition is defined by an invariant distribution. We consider an invariant multivector field composed of fields lying in this distribution. For two orbital decompositions of a system

Abstract We consider the problem of transforming a hyperoutput linear control system to a form with relative degree using a nonsingular change of outputs. For square systems this problem is solved by introducing generalizations of relative degree. Likewise, the present paper extends various generalizations of relative degree to the case of hyperoutput systems and uses these generalizations to introduce

Abstract We consider a target control problem for a system of differential equations of special form containing nonlinear terms that depend on the state variables. The solution is based on passing from the original control problem to an auxiliary problem for a piecewise linear system with a disturbance, applying the comparison principle, and using a continuous piecewise quadratic value function. We

Abstract We solve the problem of constructing a digital controller superstabilizing a continuous-time switched system whose operation modes are interval linear systems. The proposed approach to stabilization includes constructing a continuous/discrete-time closed-loop system with a digital controller, passing to its discrete-time model, and constructing a controller that simultaneously superstabilizes

Abstract We study boundary value and control problems using methods based on results on operator equations in partially ordered spaces. Sufficient conditions are obtained for the existence of a coincidence point for two mappings acting from a partially ordered space into an arbitrary set, an estimate for such a point is found, and corollaries about a fixed point for a mapping that acts in a partially

Abstract We consider the Cauchy problem for a differential inclusion with a Caputo fractional derivative of order \( \alpha \in (0, 1)\). It is assumed that the set-valued mapping defining the right-hand side of the inclusion has nonempty, convex, and compact values, is upper semicontinuous, and satisfies the sublinear growth condition. We prove that the solution set of the Cauchy problem is nonempty

Abstract We consider the problem of reconstructing distributed inputs (disturbances) in linear parabolic equations. An algorithm for solving this problem is given. An upper bound for the convergence rate is established for the case in which the input is a function of bounded variation. The algorithm combines the optimal preset and positional control methods and permits reconstruction based on inaccurate