On the protective layer boundary determining error in the thermal conductivity inverse problem

The article studies the problem of determining the error introduced by the inaccuracy of determining the thickness of a protective heat-resistant coating for composite materials. The mathematical problem is the heat conduction equation on an inhomogeneous half-line. The temperature on the outer side of the half-line (x = 0) is considered unknown on an infinite time interval. To find it, the temperature is measured at the media section at the point x = x0. An analytical study of the direct problem is carried out in the work. It made it possible to formulate the inverse problem mathematically rigorously and to define functional spaces in which it is convenient to solve the inverse problem. The main difficulty to be solved in the article is to obtain an estimate of the approximate solution error. The projection regularization method is used to estimate the modulus of conditional correctness, with the help of which order-exact estimates are obtained.

On the Hurst exponents, Markov processes, and fractional Brownian motion

There is much confusion in the literature over Hurst exponent (H). The purpose of this paper is to illustrate the difference between fractional Brownian motion (fBm) on the one hand and Gaussian Markov processes where H is different to 1/2 on the other. The difference lies in the increments, which are stationary and correlated in one case and nonstationary and uncorrelated in the other. The two- and one-point densities of fBm are constructed explicitly. The two-point density does not scale. The one-point density for a semi-infinite time interval is identical to that for a scaling Gaussian Markov process with H different to 1/2 over a finite time interval. We conclude that both Hurst exponents and one-point densities are inadequate for deducing the underlying dynamics from empirical data. We apply these conclusions in the end to make a focused statement about nonlinear diffusion.

PROBLEMS OF STABILITY OF LINEAR NON-STATIONARY SYSTEMS ON A FINITE TIME INTERVAL

When studying various processes taking place in real life, we have to deal with one of the most important concepts - the concept of stability of movement. The foundations of the theory of stability of motion were developed at the end of the last century by the great Russian scientist A. M. Lyapunov. As is known, Lyapunov stability is considered on an infinite time interval, which is a serious obstacle for many applications, since most of the objects of research function for a finite period of time. The concept of stability, introduced for an unlimited period of time, cannot be used to evaluate the properties of motion within a finite period of time. The study of motion stability by analyzing solutions of the corresponding equations is permissible and makes sense only if the mathematical model of physical reality is fully adequate. The purpose of this work is to study the stability and stabilization of the motion of linear non-stationary systems.

Problems of Optimal Control for a Class of Linear and Nonlinear Systems of the Economic Model of a Cluster

For the mathematical model of a three-sector economic cluster, the problem of optimal control with fixed ends of trajectories is considered. An algorithm for solving the optimal control problem for a system with a quadratic functional is proposed. Control is defined on the basis of the principle of feedback. The problem is solved using the Lagrange multipliers of a special form, which makes it possible to find a synthesizing control. The problem of optimal stabilization for a class of nonlinear systems with coefficients that depend on the state of the control object is considered. The results obtained for nonlinear systems are used in the construction of control parameters for a three-sector economic cluster on an infinite time interval.

Generalized Memory: Fractional Calculus Approach

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

Investigation of the behavior of solutions of differential systems with argument deviation

In this paper systems of differential equations with deviation of an argument with nonlinearity of general form in each equation are considered. The asymptotic properties of solutions of systems with a pair and odd number of equations on an infinite time interval are studied

Regional Boundary Strategic Sensors Characterizations

This paper, deals with the linear infinite dimensional distributed parameter systems in a Hilbert space where the dynamics of system is governed by strongly continuous semi-groups. More precisely, for parabolic distributed systems the characterizations of regional boundary strategic sensors have been discussed and analyzed in different cases of regional boundary observability in infinite time interval. Furthermore, the results so obtained are applied in two-dimensional systems and sensors studied under which conditions guarantee regional boundary observability in a sub-region of the system domain boundary. Also, the authors show that, the existent of a sensor for the diffusion system is not strategic in the usual sense, but it may be regional boundary strategic of this system.