On the uniqueness of the solution of nonlinear differential-algebraic systems

We consider the Cauchy problem for a system of nonlinear ordinary differential equations unsolved for the derivative of the unknown vector function and identically degenerate in the domain. We prove a theorem on the coincidence of two smooth solutions of the considered problem. We show that, under some additional assumptions, the above-mentioned problem cannot have classical solutions with less smoothness. We obtain conditions under which the problem has a fixed finite number of solutions.     

On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.     

On the determination of the potential function from given orbits

The paper deals with the problem of finding the field of force that generates a given (N ? 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand’s problem. At the end we establish the relation between Bertrand’s problem and the solutions to the Heun differential equation. We illustrate our results by several examples.     

Regularization of the solution of the cauchy problem for the generalized Moisil-Theodoresco system

We consider the problem of analytic continuation of a solution of the generalized Moisil-Theodoresco system in a spatial domain on the basis of its values on part of the boundary of this domain, that is, the Cauchy problem. We construct an approximate solution of this problem on the basis of the Carleman matrix method.     

On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum

We consider the sum of the Sturm-Liouville operator and a convolution operator. We study the inverse problem of reconstructing the convolution operator from the spectrum. This problem is reduced to a nonlinear integral equation with a singularity. We prove the global solvability of this nonlinear equation, which permits one to show that the asymptotics of the spectrum is a necessary and sufficient condition for the solvability of the inverse problem. The proof is constructive.     

Asymptotics of solutions of the linear conjugation problem at the corner points of the curve

We consider the classical linear conjugation problem for analytic functions on a piecewise smooth curve in the entire scale of weighted Hölder spaces. We derive a closed-form power-logarithmic asymptotics of the solution of this problem at the corner points of the curve under the assumption that the right-hand side of the problem admits a similar asymptotics.     

Solving the optimal control problem of the parabolic PDEs in exploitation of oil

In this paper, the optimal control problem is governed by weak coupled parabolic PDEs and involves pointwise state and control constraints. We use measure theory method for solving this problem. In order to use the weak solution of problem, first problem has been transformed into measure form. This problem is reduced to a linear programming problem. Then we obtain an optimal measure which is approximated by a finite combination of atomic measures. We find piecewise-constant optimal control functions which are an approximate control for the original optimal control problem.     

Numerical solution of the three-dimensional Dirichlet problem for inhomogeneous media by the method of integral equations

We consider the three-dimensional Dirichlet problem for equations of elliptic type in inhomogeneous media. The problem can be reduced to a system of loaded Fredholm integral equations of the second kind over the volume. We prove the uniqueness of a classical solution of the problem. We suggest a numerical solution algorithm of iterative type. An example of the numerical solution of the problem is considered, and the convergence of the iterative procedure is demonstrated numerically.     

Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem

We study the homogenization problem for the Poisson equation in a periodically perforated domain with a nonlinear boundary condition for the flux on the cavity boundaries. We show that, under certain relations on the problem scale, the homogenized equations may have different character of the nonlinearity. In each case considered, we obtain estimates for the convergence of solutions of the original problem to the solution of the homogenized problem in the corresponding Sobolev spaces.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

Optimal design of the damping set for the stabilization of the wave equation

We consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in RN, N=1,2. In a first theoretical part, we reformulate the problem into an equivalent non-convex vector variational one using a characterization of divergence-free vector fields. Then, by means of gradient Young measures, we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi-convexification. This implies that the new relaxed problem is well-posed in the sense that there exists a minimizer and, in addition, the infimum of the original problem coincides with the minimum of the relaxed one. In a second numerical part, we address the resolution of the relaxed problem using a first-order gradient descent method. We present some numerical experiments which highlight the influence of the over-damping phenomena and show that for large values of the damping potential the original problem has no minimizer. We then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one.     

热传导方程小波解的点态收敛

本文主要考虑热传导方程uxx=ut,0≤x<1,t≥0;u(1,t)=g(t),其中边界条件g(t)为已知函数．此定解问题为一不适定问题,也就是说当边界条件有微小扰动时,将会引起解大的扰动．本文将利用多分辨率分析构造一小波解,且证明此解是适定的,并给出所定义小波解与定解问题的真正解在点态意义下的误差估计．     

Solution of the inverse quasiperiodic problem for the Dirac system

We present a complete solution of the inverse problem of spectral analysis for the Dirac operator with quasiperiodic boundary conditions. We prove a uniqueness theorem for the solution of the inverse problem and obtain necessary and sufficient conditions for a sequence of real numbers to be the spectrum of a quasiperiodic Dirac problem.     

Optimal control of solutions of the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation

An optimal control problem for a second-order Sobolev type equation with a relatively polynomially bounded operator pencil is considered. We prove the existence and uniqueness of a strong solution of the Showalter-Sidorov problem for this equation. Necessary and sufficient conditions for the existence and uniqueness of an optimal control of such solutions are obtained. We study the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation.     

On the Existence of a Generalized Solution of the Saturated-Unsaturated Filtration Problem

We study the existence of a generalized solution of an initial–boundary value problem describing the process of unsteady filtration of a liquid in a bounded region of an n-dimensional space. We consider the case in which the Kirchhoff transformation used to determine the generalized solution takes the real axis into a semiaxis bounded below. An auxiliary problem is constructed. It is proved that any solution of the auxiliary problem is a solution of the problem under study. The solvability of the auxiliary problem is established by using the method of semidiscretization in time and the Galerkin method.     

On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.     

Control of the spectrum of irregular oscillations of linear systems with coinciding ranks of the matrix multiplying the control and the extended matrix

We consider a linear periodic control system such that the ranks of the matrix multiplying the control and the extended matrix consisting of the averaged coefficient matrix and the matrix multiplying the control are the same. We assume that the control has the form of feedback linear in the state variables and is periodic with the same period as the system itself. We pose the problem of control of the frequency spectrum of strongly irregular periodic oscillations with an objective set, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies. We obtain necessary and sufficient conditions for the solvability of this problem.     

On the computational complexity of continuous routing

We study the problem of routing and broadcasting messages in a network, in which messages are generated at processors at arbitrary times and each message must reach its destination by a specific deadline. We present distributed and global routing algorithms for some restricted continuous routing problems on arrays of processors. We show that distributed algorithms are unlikely to exist in more general situations by giving an NP-hardness proof for their corresponding feasibility problem; i.e., the problem of determining whether all messages can be routed without violating the constraints of the network. We also present a distributed algorithm for the continuous broadcasting problem.     

Application of the decomposition method to identify the distributed parameters of an elliptical equation

We use the Adomian's technique for solving an elliptical boundary problem with an auxiliary condition. We also solve the inverse problem for determining the distributed parameters.     

Oblique derivative problem with variable inclination angle: Location of the spectrum and the absence of the basis property of the system of root functions

We consider the eigenvalue problem for the Laplace equation in the disk with the oblique derivative with variable inclination angle. We show that the root functions of the problem do not form a basis in any Lebesgue space with exponent exceeding unity.