Vol 52, No 6 (2016)

We suggest new tests for the stability and uniform asymptotic stability of an equilibrium in systems of neutral type. By using these tests, we prove conditions for optimal stabilization and derive new estimates for perturbations that can be countered by a system closed by an optimal control. We show that, by using nonmonotone sign-indefinite functionals as Lyapunov functionals, one can obtain conditions for uniform asymptotic stability that do not contain the a priori requirement of stability of the difference operator and do not imply the boundedness of the right-hand side of the system. When studying the action of perturbations on the stabilized systems, these conditions permit one to obtain new estimates of perturbations preserving the stabilizing properties of optimal controls. The obtained estimates do not imply any constraint on the value of perturbations in some domains of the phase space that are defined when constructing an optimal stabilizing control. Some examples are considered.

We construct a class of nonconservative systems of differential equations on the tangent bundle of the sphere of any finite dimension. This class has a complete set of first integrals, which can be expressed as finite combinations of elementary functions. Most of these first integrals consist of transcendental functions of their phase variables. Here the property of being transcendental is understood in the sense of the theory of functions of the complex variable in which transcendental functions are functions with essentially singular points.

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.

We prove the existence of dissipative solutions of the Jeffreys–Oldroyd-α model with the use of the topological approximation method for studying hydrodynamic problems.

We obtain closed-form recursion formulas for the classical solutions of a mixed problem for the general inhomogeneous factorized equation of vibrations of a bounded string with second directional derivatives in the boundary conditions, in which the coefficients multi-plying the first of the two directional derivatives are independent of time. We study the case of boundary conditions in which all first directional derivatives are not directed along the characteristics of the equation. We obtain necessary and sufficient conditions on the right-hand side and the initial and boundary data of the problem for its well-posed global solvability in the set of classical solutions.

We consider a mixed problem of plane isotropic elasticity in a half-plane in which the displacement vector and the normal component of the stress tensor are alternately specified on successive intervals of the real axis. We derive a closed-form expression for the solution of this problem, which is similar to the well-known Keldysh–Sedov formula for the half-plane.

For the computation of lower and upper limits of integral means of piecewise continuous functions as the length of the integration interval tends to +∞, we obtain a formula that permits one, without loss of generality, to assume that the endpoints of the integration interval belong to closed intervals into which the half-line is divided by any sequence in which the difference between neighboring terms tends to +∞.