Finding of bounded solutions to linear impulsive systems

The comprehensive system of information security of an enterprise includes both tactical aspects of information and strategic priorities, reflecting the information policy and information strategy of the enterprise. Ensuring a given level of cybersecurity requires the identification of threat actors, their purpose, intentions of attacks on the IT infrastructure, and weak points of the enterprise’s information security. To achieve these goals, enterprises need new information security solutions. In this work, a mathematical model of the process of cyber risk management in the enterprise, which is based on the distribution of piecewise continuous analytical approximating functions of cyber attacks in the Fourier series, is obtained. A constant continuous monitoring and conduction of cyber regulatory control of the enterprise on time makes it possible to effectively ensure the cybersecurity of the enterprise in real time—predicting the emergence of cyber threats to some extent—which, in turn, determines the management of cyber risks arising in the field of information security of the enterprise. Such a Fourier series expansion of the piecewise continuous analytical approximating function of the intensity of cyber attacks on damage to standard software, obtained by approximating empirical–statistical slices of the intensity of cyber attacks on damage to standard software for each time period by analytical functions, opens up new mathematical possibilities of transition to systems of regulatory control of cyber threats of the enterprise from discrete to continuous automated process for such types of control.

The article develops a mathematical model of a piezoelectric gyroscopic sensor based on bulk acoustic shear waves propagating through the thickness of a layer with piezoelectric anisotropy perpendicular to the electrical load. The layer rotates around an axis perpendicular to its lateral faces, and the excited shear waves have the properties of Coriolis dispersion. The obtained conjugate equations of motion are one-dimensional through the layer thickness only in the case of the absence of mechanical forces at the boundaries and homogeneous conditions of the electrical load. An analytical solution to the problem is constructed, and exact formulas for the distributions of stresses and displacements through the layer thickness in traveling shear waves with elliptical polarization of coupled components in the median plane are obtained. In accordance with the exact solution for the thickness component of the electric displacement field, an expression for the conduction current function in the external circuit was obtained, and the amplitude-frequency characteristic of the gyroscopic sensor was constructed on the primary (excited) and secondary (sensor) modes. It was determined that on the first main resonance of the layer, the efficiency of the excited mode is about 24 percent, and the efficiency of the sensor mode is about 1.8 percent at a value of the normalized angular velocity of the order of 0.01.

Two-dimensional periodic Stokes flow of viscous incompressible fluid in a rectangular cavity with constant velocities applied to the top and bottom walls is considered. The study of chaotic advection regimes is reduced to the sequential solving of two problems. To solve the first problem, the analytical method of superposition, that allows to obtain any desired accuracy of the velocity field, is used. To solve the second problem, which is associated with obtaining the trajectories of individual fluid particles, numerical calculations of the Cauchy problem were performed. An analysis of the boundary conditions accuracy was performed based on the control of local integration. The advection of the selected volume of fluid in the rectangular cavity under the periodic motion of walls for a finite period time based on piecewise spline cubic interpolation is modelled. The obtained numerical results agree with well-known experimental data.

In the modern inertial sensors based on the bulk acoustic waves propagation, the change of the polari­za­ti­on vector through a certain finite length of the waveguide, is proposed to determine the angular velocity. It is caused by Coriolis dispersion due to the action of the inertial forces of rotational motion. Among the important issues of calculation of reflected and refracted waves in rotational motion, one can mention joined wave­gui­des, reflec­tion from a non-orthogonal boundary, etc. So, the problem of the incidence and ref­lec­tion of Coriolis- dispersed harmonic waves on the surface of an elastic half-space at an arbitrary angle of pro­pagation plays the key role. Research in this regard besides the study of the effect of Coriolis dis­per­si­on in reflected waves, includes also piezoelectric and thermal material properties of waveguides. In the ma­thematical modelling the authors of [6] consider the general solutions for mechanical and electrical quan­tities in the some way as in the non-rotating classical case of elastodynamics and thermo­electro­elasticity. In our opinion, the general solutions for incident and reflected waves should be built on the basis of fun­da­­mental solutions for quasi-longitudinal and quasi-transverse waves, propagating in the unbounded ro­ta­ting medium. In the statement of the problem, it should be assumed in advance, that the incident and ref­lec­ted waves are formed by two coupled components of displacements, that is, such waves have elliptical polarization.

A two-dimensional periodic Stokes flow of viscous incompressible fluid in a rectangular cavity with constant velocities applied to the top and bottom walls is considered. The study of advection regimes is reduced to the sequential solving of two problems. To solve the first problem, the analytical method of superposition, which allows one to obtain any desired accuracy of the velocity field, is used. To solve the second problem, which is associated with obtaining the trajectories of individual fluid particles, numerical calculations of the Cauchy problem were performed. An analysis of the boundary conditions accuracy was performed based on the control of local integration. The advection of the selected volume of fluid in the rectangular cavity under the periodic motion of walls for a finite period time based on piecewise spline cubic interpolation is modeled. The obtained numerical results agree with well-known experimental data.