Solvability Of Problem Research Articles

Safe control of logical control networks with random impulses

Under the framework of a hybrid-index model, this paper investigates safe control problems of state-dependent random impulsive logical control networks (RILCNs) on both finite and infinite horizons, respectively. By using the ξ-domain method and the constructed transition probability matrix, the necessary and sufficient conditions for the solvability of safe control problems have been established. Further, based on the technique of state-space partition, two algorithms are proposed to design feedback controllers such that RILCNs can achieve the goal of safe control. Finally, two examples are shared to demonstrate the main results.

On the Weak Solvability of a Nonlocal Boundary Value Problem for the Laplace Equation in an Unbounded Domain

On the Weak Solvability of a Nonlocal Boundary Value Problem for the Laplace Equation in an Unbounded Domain

Spectral properties of local and nonlocal problems for the diffusion-wave equation of fractional order

The paper investigates the issues of solvability and spectral properties of local and nonlocal problems for the fractional order diffusion-wave equation. The regular and strong solvability to problems stated in the domains, both with characteristic and non-characteristic boundaries are proved. Unambiguous solvability is established and theorems on the existence of eigenvalues or the Volterra property of the problems under consideration are proved.

Evasion problem in linear parametric discrete games

We consider the evasion problem in the formulation by Pontryagin and Mishchenko for linear discrete games depending on a parameter. We obtain sufficient conditions and domains of parameter values that ensure the solvability of the evasion problem. The obtained results are applied to the solution of the evasion problem for the well-known problem in the theory of differential games “Isotropic Rockets”-“Boy and Crocodile” in the discrete version.

Solvability of a Boundary Value Problem Involving Fractional Difference Equations

In this current work, we apply the topological degree and fixed point theorems to investigate the existence, uniqueness, and multiplicity of solutions for a boundary value problem associated with a fractional-order difference equation. Moreover, we provide some appropriate examples to verify our main conclusions.

Задачи граничного управления для нелинейной модели реакции-диффузии-конвекции

The solvability of the boundary control problem for a nonlinear model of mass transfer is proven in the case, when the reaction coefficient depends nonlinearly on concentration of substance and depends on spatial variables. The role of the control is played by the concentration value specified on the entire boundary of the domain.

On the solvability of fourth-order boundary value problems with accretive operators

On the solvability of fourth-order boundary value problems with accretive operators

Weak solvability of the initial-boundary value problem for inhomogeneous incompressible Kelvin–Voigt fluid motion model of arbitrary finite order

Weak solvability of the initial-boundary value problem for inhomogeneous incompressible Kelvin–Voigt fluid motion model of arbitrary finite order

Analysis of Control Problems for Stationary Magnetohydrodynamics Equations under the Mixed Boundary Conditions for a Magnetic Field

The optimal control problems for stationary magnetohydrodynamic equations under the inhomogeneous mixed boundary conditions for a magnetic field and the Dirichlet condition for velocity are considered. The role of controls in the control problems under study is played by normal and tangential components of the magnetic field given on different parts of the boundary and by the exterior current density. Quadratic tracking-type functionals for velocity, magnetic field or pressure are taken as cost functionals. The global solvability of the control problems under consideration is proved, an optimality system is derived and, based on its analysis, a mathematical apparatus for studying the local uniqueness and stability of the optimal solutions is developed. On the basis of the developed apparatus, the local uniqueness of solutions of control problems for specific cost functionals is proved, and stability estimates of optimal solutions are established.

Inverse problem for a time‐fractional differential equation with a time‐ and space‐integral conditions

We study the inverse problem for a differential equation of order with the time Caputo fractional derivative and a source with values in Schwartz‐type distributions. The generalized solution of the Cauchy problem for such an equation with initial values and a time‐dependent reaction coefficient are unknown. We find sufficient conditions for the unique solvability of the inverse problem under a time‐ and space‐integral overdetermination conditions.

Global wellposedness of the 3D compressible Navier–Stokes equations with free surface in the maximal regularity class

This paper concerns the global well posedness issue of the compressible Navier–Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal L p -L q estimate and the L p -L q decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal L p -L q regularity class for some p > 2 and q > 3. Namely, the solution is bounded as L p in time and L q in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal L p -L q regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.

On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition

Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.

On the numerical solvability of the initial problem with weight for ordinary linear differential systems with singularities

Abstract The effective sufficient conditions are established for the numerical approximation of solutions of the initial problem with weight for linear systems of ordinary differential equations with singularities.

Two-time capture of coordinated evaders in a simple pursuit problem

In a finite-dimensional Euclidean space, the problem of pursuit of two evaders by a group of pursuers described by a system of the form $$ \dot z_{ij} = u_i - v,\quad u_i,v \in V, $$ is considered. It is assumed that all evaders use the same control. The pursuers use counterstrategies based on information about the initial positions and control history of the evaders. The set of admissible controls $V$ is a unit ball centered at zero, target sets are the origin of coordinates. The goal of the pursuers' group is to capture at least one evader by two pursuers. In terms of initial positions and game parameters a sufficient condition for the capture is obtained. In the study, the method of resolving functions is used as a basic one, which allows obtaining sufficient conditions for the solvability of the approach problem in some guaranteed time.

An inverse problem of determining the fractional order in the TFDE using the measurement at one space-time point

In this paper we are concerned with an inverse problem for determining the order of time fractional derivative in a time fractional diffusion equation (TFDE in short), where the available measurement is given at a single space-time point. The inverse problem is transformed to a nonlinear algebraic equation of the fractional order based on the solution to the forward problem. We prove that the algebraic equation possesses a unique solution by the strict monotonicity of the Mittag-Leffler function, and the inverse problem is of uniqueness. Numerical examples are presented to show the unique solvability of the inverse problem.

On Discrete Boundary-Value Problems and Their Approximation Properties

Discrete analogs of pseudo-differential operators and equations in discrete Sobolev–Slobodetsky spaces are considered. Using suitable discrete boundary conditions, we prove the unique solvability of the discrete boundary-value problem.

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved.

Periodic solutions to history-dependent differential hemivariational inequalities with applications

In this article, we investigate a hybrid model combined by a parabolic differential equation and a parabolic hemivariational inequality (so-called differential hemivariational inequality of parabolic–parabolic type) in general infinite dimensional spaces which includes the history-dependent operator. The solvability of initial value problems as well as the periodic problems of the hemivariational inequality and the differential hemivariational inequality have been proved. In application, we study a contact problem with normal compliance driven by a history-dependent dynamical system.

A global in time existence and uniqueness result of a multidimensional inverse problem

ABSTRACT This study is the multidimensional inverse problem for a viscoelasticity equation. This inverse problem aims to reconstruct the displacement and convolutional kernel of the integral term simultaneously, from the measurement described as the overdetermination condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions continuously in the time variable t and analytic in the space variable. The methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions are applied.

Optimal Control Problem for a Differential Inclusion with Rapidly Oscillating Coefficients on the Semiaxis

We study the optimal control problem for a system of differential inclusions with rapidly oscillating coefficients on the semiaxis. For this purpose, we use the averaging method. The solvability of the original exact problem and the corresponding averaged problem is substantiated. It is proved that the optimal controls and optimal trajectories of solutions of the exact problem converge to the optimal control and optimal trajectory of solutions of the averaged problem. It is also shown that the optimal control of the averaged problem is ”almost” optimal for the original exact problem.