Stability estimates in nonlinear differential equations of a special kind

Quite a lot of works have been devoted to problems of stability theory and, in particular, to the use of the second Lyapunov method for this. The main ones are the following [1-7]. The main attention in these works is paid to obtaining stability conditions. At the same time, when solving practical problems, it is important to obtain quantitative characteristics of the convergence of solutions to an equilibrium position. In this paper, we consider nonlinear scalar differential equations with nonlinearity of a special form (weakly nonlinear equations). Differential equations of this type are encountered in the study of processes in neurodynamics [8,9]. In this paper, we obtain stability conditions for a stationary solution of scalar equations of this type. And also the characteristics of the convergence of the process are calculated. It is shown that the solution of stability problems is closely related to optimization problems [10-12]. Pages of the article in the issue: 67 - 71 Language of the article: Ukrainia

Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay

Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced

A problem of modal control in a linear neutral system

A problem of modal control is considered for a class of linear multidimensional differential delay systems of neutral type. The control vector is sought in the form that results in a given in advance characteristic equation of the closed system. The problem is completely solved for systems of a special form, the so-called canonical systems. A two-dimensional example is considered in full detail

Control of Oscillating Systems with a Single Delay

Systems are considered related to the control of processes described by oscillating second-order systems of differential equations with a single delay. An explicit representation of solutions with the aid of special matrix functions called a delayed matrix sine and a delayed matrix cosine is used to develop the conditions of relative controllability and to construct a specific control function solving the relative controllability problem of transferring an initial function to a prescribed point in the phase space