A system of nonlinear discrete (finite-difference) of a general form with a bounded delay is considered. Interest in the tasks of qualitative analysis of such systems has increased significantly in recent years. At the same time, the problem of stability with respect to all variables of the zero equilibrium position, which has a great generality, is mainly analyzed in domestic and foreign literature. The main research method is a discrete-functional analogue of the direct Lyapunov method. In this article, it is assumed that the system under consideration admits a “partial” (in some part of the state variables) zero equilibrium position. The problem of stability of a given equilibrium position is posed, and stability is considered not in all, but only in relation to a part of the variables that determine this equilibrium position. Such a problem belongs to the class of problems of partial stability, which are actively studied for systems of various forms of mathematical description. The proposed statement of the problem complements the scope of the indicated studies in relation to the system under consideration. To solve this problem, a discrete version of the Lyapunov– Krasovskii functionals method is used in the space of discrete functions with appropriate specification of the functional requirements. To expand the capabilities of this method, it is proposed to use two types of additional auxiliary (vector, generally speaking) discrete functions in order to: 1) adjustments of the phase space region of the system in which the Lyapunov–Krasovskii functional is constructed; 2) finding the necessary estimates of the functionals and their differences (increment) due to the system under consideration, on the basis of which conclusions about partial stability are made. The expediency of this approach lies in the fact that as a result, the Lyapunov-Krasovskii functional, as well as its difference due to the system under consideration, can be alternating in the domain that is usually considered when analyzing partial stability. Sufficient conditions of partial stability, partial uniform stability, and partial uniform asymptotic stability of the specified type are obtained. The features of the proposed approach are shown on the example of two classes of nonlinear systems of a given structure, for which partial stability is analyzed in parameter space. Attention is drawn to the expediency of using a one-parameter family of functionals.
The paper considers research of the sustainability plans of active moving objects (AMOs) control system application. AMOs are material objects moving in space and carrying out informational, material and energetic interaction with objects, control points and other AMOs.
Use of AMOs is usually regulated by strict requirements, so any time delay or incomplete achievement of target effect are unacceptable. The reason of failing to perform a purpose task can be the development of control using incorrect plans, so that realization of the plan is unstable due to random factors or targeted effects of the environment and other systems. The paper deals with problematic issues of complex research of active moving objects control application stability plans .
Quantitative and qualitative assessment methods of stability plans , ways and means of ensuring required levels of stability are proposed by the authors. In this case planning task is considered dynamic, which provides plan correction by means of organizational, structural and program-algorithmic measures to ensure stability. The most adequate model of plan implementation is appropriate system of differential equations. Notion of stability probability is defined to account stochastic character of disturbance effect . The expression for stability criterion is obtained. It allows to establish the fact of sustainability and unsustainability based on analysis of properties of a purposefully built Lyapunov function and disturbances characteristics.
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