We prove the existence of a compact finite dimensional global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a thermoelastic Mindlin-Timoshenko plate system with nonlinear viscous damping. We show the upper semi-continuity of the attractor with respect to the parameters related to the coupling terms and the shear modulus of the plate.

Keywords: acoustic model, attractor, upper semi-continuity.

Small motions of a capillary fluid in the tank partitioned by perforated bafflers are considered. A mathematical formulation of the problem with the averaged boundary conditions on perforated sections of the bafflers is formulated. The results of the study of evolutionary and spectral problems are presented.

Keywords: capillary fluid, perforated bafflers, small oscillations, eigenfrequencies of oscillations.

A simple method for calculating the exponential stability of some classes of nonlinear systems containing varying and distributed delays is proposed. An upper bound for the largest Lyapunov exponent is found. Examples, illustrating application of the developed technique, are given.

Keywords: exponential stability, time-delay, Lyapunov exponent.

Representation theorems are important in the theory of subharmonic and $\delta$-subharmonic functions. In the article we sharpen Azarin's variant of representation theorem of $\delta$-subharmonic functions of finite order.

Keywords: : $\delta $-subharmonic function, Gegenbauer's polynomials, Nevanlinna's characteristic function.

Decomposition of the resolvent matrix of Caratheodory problem in terms of Blaschke-Potapov factors is obtained. Each such factor is expressed via generalized Schur parameters. Explicit relations between Blaschke-Potapov factors and Schur parameters are given. Step by step process solution matrix Caratheodory problem in the class ${\cal S}[a,\ b]$ is given.

Keywords: the matrix Caratheodory problem, Blaschke-Potapov product, generalized Schur parameters, step by step Shur process.

For the functional model of a bounded non-dissipative operator that is realized by the operator of multiplication by an independent variable in the special Hilbert space of quadratically integrable functions, the concrete form of the corresponding weight is obtained.

Keywords: functional model, non-dissipative operator.

We obtain the stability theorem for unconditional Schauder decompositions in Hilbert spaces. This result is a generalization of the classical theorem of T. Kato on similarity for sequences of projections in Hilbert spaces to the case of unconditional Schauder decompositions. Also we sharpen one theorem of V.N. Vizitei on the stability of Schauder decompositions in the case of unconditional Schauder decompositions.

Keywords: unconditional Schauder decomposition, projection, isomorphism.

We constructed explicit approximate solutions of the Boltzmann equation for the hard-sphere model, which have the form of bimodal distribution, i.e. linear combination of nonstationary inhomogeneous Maxwellians. They describe the interaction of two gas flows, which are accelerated and packed when moving along a fixed axis. Sufficient conditions for the minimization of the integral error between the sides of Boltzmann equation are found.

Keywords: Boltzmann equation, hard spheres, Maxwellian, approximate explicit solutions, bimodal distribution, "accelerating-packing", the integral error.

We consider new functions generated by central factorials. Graphs of such functions are drawn and some of their properties are proved. We have established their relationship with the hypergeometric functions. It is shown, that constructed functions are solutions of ordinary differential equations, derived in the article.

Keywords: factorial powers, central factorial powers, generalized hypergeometric function, Cauchy problem.

The paper deals with the problem of the global robust feedback synthesis of a bounded control for a system with an unknown bounded perturbations. On the basis of V.I. Korobov's controllability function method we suggest several approaches to delimiting a perturbation. We provide a positional control which is independent of the perturbation and steers an arbitrary initial point to the origin in a finite time; an estimate from above for the time of motion is given.

Keywords: robust feedback synthesis problem, positional bounded control, unknown bounded perturbation.