A real autonomous differential system of the fifth degree with a degenerate singular point without characteristic directions is obtained. The necessary and sufficient condition for the center at a given point is determined by a function that is not analytic at the boundary point of the set of system parameters for which the singular point of the system is monodromic. An asymptotic representation of this function is calculated at the point where its analyticity is violated.
We study the asymptotic behavior of the spectrum of an integral operator similar to an integral operator with a logarithmic kernel depending on the sum of arguments. By a simple change of variables, the corresponding equation is reduced to an integral equation of convolution type defined on a finite interval (as is well known, such equations in the general case cannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced to a conjugation problem for analytic functions and then to an infinite system of linear algebraic equations, the isolation of the main terms in which allows deriving a relation that determines the spectrum of the original problem.
We consider a Dirac type operator with matrix coefficients. Estimates for the root vector functions are established, and criteria for the Bessel property and the unconditional basis property of the root vector function systems of this operator in the space (L_{2}^{2m}(G) ), where (G=(a,b)subset mathbb {R} ) is a finite interval, are obtained.
We consider the problem of stabilization of a switched interval linear system with slow switchings that are inaccessible to observation. It is proposed to look for a solution in the class of variable structure controllers. To ensure the functionality of such a controller, it is necessary to construct an observer of the switching signal. This paper is devoted to some theoretical issues related to the period of quantization of the neural observer’s operating time.
We consider the differential inclusion (F(t,x,dot {x})ni 0 ) with the constraint (dot {x}(t)in B(t) ), (tin [a, b]), on the derivative of the unknown function, where (F) and (B ) are set-valued mappings, (F:[a,b]times mathbb {R}^ntimes mathbb {R}^ntimes mathbb {R }^mrightrightarrows mathbb {R}^k ) is superpositionally measurable, and ( B:[a,b]rightrightarrows mathbb {R}^n) is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form (f(t,x,dot {x},u)=0), (dot {x}(t)in B(t) ), (u(t)in U(t,x,dot {x}) ), (tin [a,b]).
The inverse problem for a nonlinear mathematical model of sorption dynamics with an unknown variable kinetic coefficient is considered. A theorem on the existence of two solutions of the inverse problem is proved, and an iterative method for solving it is justified. An example of the application of the proposed method to the numerical solution of the inverse problem is given.
We study the nonstationary system of Navier–Stokes equations for an incompressible fluid. Based on a regularized problem that takes into account the relaxation of the velocity field into a solenoidal field, the existence of a pressure function almost everywhere in the domain under consideration for solutions in the Hopf class is substantiated. Using the proposed regularization, we prove the existence of more regular weak solutions of the original problem without smallness restrictions on the original data. A uniqueness theorem is proven in the two-dimensional case.
The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.
We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions are obtained in closed form. It is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. The unique solvability of the problem is also proved; the solution is written in the form of a series if the spectral parameter is not equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability conditions are obtained under which the family of solutions is found in the form of a series. A condition for the solvability of the problem depending on the eigenvalues is obtained. The constructed analytical solutions can be used efficiently in numerical modeling of transonic gas dynamics problems.
We consider a continuous approximation to the Sturm–Liouville problem with a nonlinearity discontinuous in the phase variable. The approximating problem is obtained from the original one by small perturbations of the spectral parameter and by approximating the nonlinearity by Carathéodory functions. The variational method is used to prove the theorem on the proximity of solutions of the approximating and original problems. The resulting theorem is applied to the one-dimensional Gol’dshtik and Lavrent’ev models of separated flows.
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