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

Abstract We consider the question as to how precise the localization of invariant compact sets of a time-invariant system by the functional localization method can be in principle. We show that the union of all invariant compact sets can be dramatically different from the intersection of all localizing sets obtained by the functional method. A class of systems for which these two coincide is singled

Abstract In a two-dimensional bounded domain \(\Omega \), we consider the system of Navier–Stokes equations describing the flow of a viscous compressible gas with no allowance for heat exchange with the ambient medium under small variations of density in the equations of motion. We prove the existence of a solution to the problem of exact local controllability of this system by an external force supported

Abstract Many contemporary automatic control problems are characterized by large dimensions, the presence of uncertainty in the description of the system, the presence of uncontrolled exogenous disturbances, the need to analyze large amounts of information online, decentralization/simplification of control systems in multi-agent systems, and a number of other factors that complicate the application

Abstract We consider the trajectory tracking problem for a dynamical system described by nonlinear fractional differential equations with an unknown input disturbance. Based on regularization methods and constructions from guaranteed positional control theory, we propose an information noise- and computational error-robust algorithm solving the problem for the case in which only part of the system

Abstract For two classes of two-dimensional systems and for a fourth-order system describing the development of pancreatic cancer, we use the method of localization of compact invariant sets to establish estimates for their compact invariant sets and indicate conditions for the existence of attractors. A condition for the degeneration of dynamics is found for the four-dimensional system. Examples and

Abstract We show that there exist whole classes of various boundary value problems having the same characteristic determinant, with the respective problems allowed to have differing orders of the differential equations and to be defined both on intervals and on geometric graphs.

Abstract We study how the pattern of perturbations superimposed on a plane-parallel time-periodic flow of a Newtonian viscous fluid evolves in a layer in which one of the boundaries performs longitudinal harmonic vibrations along itself, with the zero-friction slip of material allowed on the other boundary. We pose a generalized Orr–Sommerfeld problem as a linearized problem of hydrodynamic stability

Abstract We study the solvability of the problem for the elliptic second-order differential-operator equation \(\lambda ^2 u(x)-u^{\prime {}\prime }(x)+Au(x)=f(x) \), \(x\in (0,1)\) , in a separable Hilbert space \(H\) with the boundary conditions \( u^{\prime }(1)+\lambda Bu(0)=f_1\) and \(u^{\prime }(0)=f_2\) , where \(\lambda \) is a complex parameter, \(A\) and \(B\) are given linear operators

Abstract For an arbitrary expanding endomorphism of the class \(C^1 \) acting from \(\mathbb {T}^{\infty } \) to \(\mathbb {T}^{\infty } \), where \(\mathbb {T}^{\infty } \) is an infinite-dimensional torus (the quotient space of some Banach space by an integer lattice), we establish the following standard assertions from the hyperbolic theory: the topological conjugacy of this endomorphism with the

Abstract We consider a boundary value problem for an ordinary singularly perturbed second-order differential equation whose right-hand side is a nonlinear function with a discontinuity along some curve that is independent of the small parameter. For this problem, we study the existence of a smooth solution with steep gradient in a neighborhood of some point lying on this curve. The point itself and

Abstract We study the Cauchy problem in the space of continuous functions for a nonlinear Sobolev type differential equation generalizing the equation of torsional vibrations of an infinite rod in a viscoelastic medium. Conditions for the existence of a global solution and for the blow-up of the solution of the Cauchy problem in finite time are considered.

Abstract We prove the existence of continuous weak solutions of the nonlinear equations describing steady-state flows of second-grade fluids in a bounded three-dimensional domain under the no-slip boundary condition. A weak solution is found using the Galerkin method with special basis functions constructed with the help of a perturbed Stokes operator. An energy inequality for the resulting solution

Abstract We study the properties of finite-order entire solutions of a class of algebraic differential equations. It is shown that, under certain conditions, all such solutions are solutions of linear differential equations whose coefficients are polynomials in the independent complex variable.

Abstract We introduce the concepts of Poisson boundedness and partial Poisson boundedness for a solution of a differential system. These properties mean that the solution or, respectively, its projection onto a given subspace is contained in some ball for the values of an independent variable belonging to countably many intervals converging to infinity. Based on the method of Lyapunov vector functions

Abstract Using a modification of the approach previously developed by the authors, we show that finding solutions of one class of systems of linear Fredholm integral equations of the third kind on the real line with finitely many multipoint singularities is equivalent to finding solutions of a system of linear Fredholm integral equations of the second kind on the real line with additional conditions

Abstract We consider model boundary value problems for elliptic pseudodifferential equations in multidimensional cones. A result on the unique solvability and representation of solutions of some boundary value problems in suitable Sobolev–Slobodetskii spaces is obtained. A priori estimates of solutions are given.

Abstract We study the inverse spectral problem of reconstructing nonsplitting boundary conditions for the Sturm–Liouville operator from a minimal amount of spectral data. Necessary and sufficient conditions are derived for the existence of solutions of the problem under consideration and for these solutions to be isolated or nonisolated. We propose a simple analytical algorithm for reconstructing boundary

Abstract For a Sturm–Liouville operator on a piecewise smooth curve, we study the effect that the spectrum of a nonintegrable singularity of the potential on the segment joining the endpoints of this curve has on the operator asymptotics. It is shown that in the case where the singular point does not give rise to the branching of solutions in its neighborhood (the case of trivial monodromy), the spectral

Abstract We study a problem arising in open-closed \(p \)-adic string theory for a convolution type integral equation with a cubic nonlinearity on the real line \(\mathbb {R} \). We establish conditions on the function and the real parameter occurring in the equation under which the problem has an odd continuous solution on \(\mathbb {R}\setminus \{0\}\). The proof of the existence of such a solution

Abstract An application of the integral dispersion equation method to the solution of two nonlinear eigenvalue problems is considered. One of these problems arises when studying the propagation of TE-waves in a dielectric shielded nonlinear layer with Kerr nonlinearity, while the other, a more complex one, generalizes the first and contains, in particular, a nonlinearity multiplying the highest derivative

Abstract For the boundary value problem of diffraction of an external disturbance by a local penetrable body with smooth surface, we construct an integral representation of the solution based on a linear combination of single and double layer potentials with densities distributed over a common auxiliary interior surface. A complete mathematical justification of this representation is given.

Abstract We consider a nonlinear Sobolev equation of the third order describing the electric field potential in a conductor. Six classes of exact solutions are constructed, and their qualitative behavior is analyzed.

Abstract We consider the three-dimensional problem of diffraction of a monochromatic electromagnetic wave by a system of objects of various physical nature, including dielectric bodies (domains), perfectly conducting bodies, and perfectly conducting screens. The perfectly conducting bodies may be placed in an exterior medium or immersed into the dielectric domains. In addition, the perfectly conducting

Abstract We consider a generalized Richards equation with power-law nonlinearities modeling filtration in porous media. Conditions are derived under which the problem can be reduced to the linear heat equation or to nonlinear equations with known solutions. The families of explicit exact solutions that can be expressed via elementary functions or Lambert’s \(W \)-function are found. Some examples illustrating

Abstract We study singular modes of volume singular integral equations describing problems of scattering of electromagnetic wave in anisotropic dielectric structures. An explicit form of these modes is obtained for a certain class of anisotropic media. Example of real media in which singular modes can exist are considered.

Abstract We consider the problem of diffraction of a waveguide wave by an impedance rod in a rectangular waveguide with perfectly conducting walls and the problem of diffraction of a plane two-dimensional electromagnetic wave and the field of a point source by an evenly spaced array formed by infinite cylinders of arbitrary cross-section with perfectly and well conducting walls. Both problems are reduced

Abstract We consider the problem on the analytic continuation of the solution of the system of vibration equations in elasticity theory in a spatial domain based on the values of the solution and the stresses on part of the boundary of this domain, i.e., a Cauchy problem. The problem is ill posed. If the part of the domain on which the Cauchy data are given is real analytic, then the problem has a

Abstract We study inverse problems of magnetostatics that arise when designing two-dimensional multilayered shielding and cloaking devices. It is assumed that the device to be designed has the form of a circular shell consisting of finitely many layers, each filled with a homogeneous isotropic medium. Using an optimization method, the inverse problems under consideration are reduced to finite-dimensional

Abstract A quadrature formula for the direct value of the normal derivative of the single layer potential with smooth density defined on a closed or nonclosed surface is obtained. Single layer potentials for the Laplace and Helmholtz equations are considered. Numerical tests confirm that our quadrature formula gives considerably higher accuracy than the standard quadrature formula. It can be used when

Abstract We state and study the first and second boundary value problems and the transmission problem for complex potentials of plane filtration flows in porous media. In the general case, the flow sources are arbitrary discrete sources that can be located both on the boundaries and outside the medium boundaries. If the boundaries are canonical (modeled by a straight line or a circle), then the solutions

Abstract We consider a system of nonlinear integral equations arising when studying the inverse coefficient problem for a system of partial differential equations. The inverse problem is to determine two coefficients of the system based on additional information on one of the solution components. We prove the local existence and uniqueness of the solution of the nonlinear system of integral equations

Abstract We study a linear integro-differential equation of the third kind with a coefficient having power-order zeros. To solve this equation approximately in the space of generalized functions, we propose and justify a generalized version of the subdomain method based on special Kantorovich polynomials.

Abstract We study abstract integro-differential equations that are operator models of problems in viscoelasticity. We present results based on an approach related to the study of one-parameter semigroups for linear evolution equations. The presented approach can also be used to study other integro-differential equations containing integral terms of the Volterra convolution form. A method is given for

Abstract For a generalized Cauchy–Riemann equation on the plane with a coefficient that has finitely many singular points and with a continuous free term, we find an integral representation of the solution in the class of functions bounded at infinity.

Abstract We consider linear completely integrable Pfaffian systems of differential equations with matrices whose entries are analytic functions. The property of complete controllability of the system in a neighborhood of a regular point is studied. A sufficient condition for a Pfaffian system to have this property is derived. Whether the system has the complete controllability property is determined

Abstract We study abstract integro-differential equations serving as operator models for problems arising in viscoelasticity. The results are based on an approach related to the study of one-parameter semigroups for linear evolution equations. The approach proposed can be used when investigating other integro-differential equations containing integral terms of the form of Volterra convolution.

Abstract For the solutions of the linear differential inequality \(\mathscr {L}(u)\geq 0 \), where \(\mathscr {L} \) is a linear differential operator of order \(l \) defined on functions of one variable, we establish estimates of the form \(\|u; W^l (J^\delta )\|\leq C(\delta )\|u; L(J)\| \), where \(J=[a,b]\subset \mathbb {R} \), \(0<3\delta

Abstract We consider the leaky wave problem for an open inhomogeneous metal–dielectric circular-section waveguide. The problem is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the fact that the spectral parameter occurs in the matching conditions necessitate defining the solution of

Abstract We consider a linear homogeneous \(n\)th-order differential equation, \(n\in \mathbb {N}\), with constant bounded operator coefficients in a Banach space. Under some conditions on the (real and complex) roots of the corresponding characteristic equation, we obtain a formula expressing the general solution of the differential equation via the operator functions given by the exponential, sine

Abstract We study the basis properties of root functions of a spectral problem describing the bending vibrations of a homogeneous rod with a longitudinal force acting in its cross sections. Both rod ends are elastically fixed and either there is a lumped mass or a follower force acts on each of the ends. We establish a sufficient condition for the basis property of the system of root functions of this

Abstract We consider the Cauchy problem for general linear systems of complex partial differential equations in scales of Banach spaces of vector functions of the exponential type with an integral metric. Necessary and sufficient conditions for the well-posed solvability of this problem are obtained. Thus, we describe the structure of linear systems of complex partial differential equations for which

Abstract We consider the well-known Bazykin–Svirezhev model describing the predator–prey interaction. This model is a system of two nonlinear ordinary differential equations with a small parameter multiplying one of the derivatives. The existence and stability of a so-called relaxation cycle in such a system are studied. A peculiar feature of such a cycle is that as the small parameter tends to zero

Abstract For the Gellerstedt equation with a singular coefficient, we prove theorems on the uniqueness and existence of a solution of the problem with local and nonlocal conditions on parts of the boundary characteristic and with discontinuous transmission conditions on the degeneration line.

Abstract We study the existence of oscillatory modes in a controlled nonlinear system closed by a feedback in the form of a bistable hysteretic switch. The system is a simplified mathematical model of a thermal energy harvester. Conditions on the controller and on the system parameters are established that guarantee the existence of oscillatory modes in this system.

Abstract We study the problem of electromagnetic wave diffraction on a junction of two rectangular waveguides. The boundary value problem for the system of Maxwell equations is reduced to a vector pseudodifferential equation in special Sobolev spaces. Sufficient conditions are obtained for the existence of a unique solution of the boundary value problem and the integro-differential equation.

Abstract We consider the analytic properties of solutions to equations of arbitrary order in the generalized hierarchy of the second Painlevé equation. The local properties of solutions, the Bäcklund transformations, and rational solutions and their representations via generalized Yablonskii–Vorob’ev polynomials are studied.

Abstract For the solutions of a nonlinear second-order differential-operator equation and the corresponding three-level operator-difference scheme, we establish sufficient boundedness conditions as well as sufficient conditions for the stability with respect to perturbations of the operators. These results are applied to studying the coefficient stability of a mixed problem for a hyperbolic equation

Abstract We propose methods for solving an exterior boundary value problem for the Laplace equation based on the main integral Green formula. The main technique is the one of setting an artificial integral boundary condition with iterative improvement. It is shown that iterative methods converge at the rate of a geometric progression. The applicability of the methods for solving exterior problems is

Abstract It has recently been established that all fundamental solutions of a multidimensional singular elliptic equation can be expressed via the well-known multivariate Lauricella hypergeometric function. In the present paper, we prove that the generalized Holmgren problem for an elliptic equation with several singular coefficients has a unique solution and find this solution in closed form. When

Abstract This work belongs in the field of mathematical modeling and numerical studies of equilibrium configurations of plasma, magnetic field, and electric current in Galathea traps with current-carrying conductors immersed in a plasma volume. Models of permissible configurations in the annular neighborhood of a straight conductor that are not in contact with its surface are constructed and investigated

Abstract We consider an adaptive interpolation algorithm for numerical integration of systems of ordinary differential equations (ODEs) with interval parameters and initial conditions. At each time moment, a piecewise polynomial function of a prescribed degree is constructed in the course of algorithm operation that interpolates the dependence of the solution on particular values of interval uncertainties

Abstract For a system of partial differential equations, we consider the inverse problem of determining one of the coefficients based on additional information on one of the solution components. An iterative method is proposed for calculating the unknown coefficient based on the reduction of the inverse problem to a nonlinear operator equation. The convergence of the iterative method and the existence

Abstract We consider a quasi-hydrodynamic regularized model of the phase field type that describes the dynamics of an isothermal compressible two-phase viscous mixture with allowance for interphase effects. The dissipativity of the model, i.e., the lack of growth in the complete energy of the closed system as it tends to an equilibrium state, is discussed. A spatially discrete approximation to the