Том 55, № 4 (2019)

We obtain a global inverse function theorem guaranteeing that if a smooth mapping of finite-dimensional spaces is uniformly nonsingular, then it has a smooth right inverse. Global implicit function theorems are obtained guaranteeing the existence and continuity of a global implicit function under the condition that the mappings in question are uniformly nonsingular. The local Lipschitz property and the smoothness of the global implicit function are studied. The results are generalized to the case of mappings of Hilbert spaces.

An arbitrary diffeomorphism f of class C¹ acting from an open set \(\mathcal{U}\subset \mathbb{R}^{m}\), m ≥ 2, into \(f(\mathcal{U})\subset \mathbb{R}^{m}\) is considered. Sufficient conditions for such a diffeomorphism to admit a hyperbolic mixing attractor are obtained.

We consider the Sturm-Liouville operator generated in the space L²[0, +∞) by the expression l_a,b:= −d²/dx² + x + aδ(x ™ b) and the boundary condition y(0) = 0, where δ is the Dirac delta function and a and b are positive numbers. Regularized trace formulas for this operator are obtained, and some identities for the eigenvalues are found. In particular, we prove that the sum of reciprocal squares of zeros of the Airy function Ai is 4π²/(3¹/³Γ⁴(1/3)), where Γ is the Euler gamma function.

We study the boundary conditions of the Sturm-Liouville problem posed on a star-shaped geometric graph consisting of three edges with a common vertex. We show that the Sturm-Liouville problem has no degenerate boundary conditions in the case of pairwise distinct edge lengths. However, if the edge lengths coincide and all potentials are the same, then the characteristic determinant of the Sturm-Liouville problem cannot be a nonzero constant and the set of Sturm-Liouville problems whose characteristic determinant is identically zero and whose spectrum accordingly coincides with the entire plane is infinite (a continuum). It is shown that, for one special case of the boundary conditions, this set consists of eighteen classes, each having from two to four arbitrary constants, rather than of two problems as in the case of the Sturm-Liouville problem on an interval.

We study two nonclassical boundary value problems for a system of Poisson equations in three-dimensional space whose boundary conditions contain the main first-order differential operations of field theory. The statements of the problems are based on the trace theorem for a linear combination of the vector of normal derivatives, the rotor, and the divergence. We prove two existence and uniqueness theorems for the weak solution of the problems under study.

We consider the target control synthesis problem for a team of single-type plants performing a common motion to a target set under the condition that the team members do not collide with each other. In the process of motion, the team members must stay inside a virtual container forming a standard motion (tube) and avoiding the obstacles known in advance by reconfiguration. The general scheme for solving this problem, which reduces the original problem to a series of subproblems, is given, and Hamiltonian formalism is used for a detailed consideration of the subproblems of terminal control of ellipsoidal tubes, the container motions between moving external obstacles, and the evolution of the team inside the virtual container.

We consider the problem of designing an asymptotic observer for a SISO system of maximal relative order with unknown input. To solve the problem, we propose a cascade of two-dimensional nonlinear feedback observers and present an algorithm for choosing feedback coefficients such that the observation error asymptotically tends to zero.

For abstract integro-differential equations with unbounded operator coefficients in a Hilbert space, we study the well-posed solvability of initial problems and carry out spectral analysis of the operator functions that are symbols of these equations. This allows us to represent the strong solutions of these equations as series in exponentials corresponding to points of the spectrum of operator functions. The equations under study are the abstract form of linear integro-partial differential equations arising in viscoelasticity and several other important applications.

We consider a compact version (the CQGD system) of the quasi-gasdynamic system. All algorithms that have been used to approximate the spatial derivatives in the Navier–Stokes equations can also be applied to the CQGD system. At the same time, the use of the CQGD system permits significantly improving the stability of explicit schemes, which is important for ensuring high-performance parallel computations. As examples of the use of algorithms based on the CQGD system, we present the results of computations of a laminar boundary layer on a plate and of a hypersonic laminar separated flow in a compression angle.