A. Yu. Aleksandrov

Stability analysis for a class of switched nonlinear systems

This paper addresses the stability analysis of a class of switched nonlinear systems. The switched systems have uncertain nonlinear functions constrained in a sector set, which are called admissible sector nonlinearities. A sufficient condition in terms of linear inequalities is presented to guarantee the existence of a common Lyapunov function, and thereby to ensure that the switched system is stable for an arbitrary switching signal and any admissible sector nonlinearities. A constructive algorithm based on the modified Gaussian elimination procedure is given to find the solutions of the linear inequalities. The obtained results are applied to a population model with switchings of parameter values and the conditions of ultimate boundedness of its solutions are investigated. Another example of an automatic control system is considered to demonstrate the effectiveness of the proposed approaches.

On the asymptotic stability of switched homogeneous systems

Abstract The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L > 0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.

On the asymptotic stability of solutions of nonlinear systems with delay

Under study are systems of homogeneous differential equations with delay. We assume that in the absence of delay the trivial solutions to the systems under consideration are asymptotically stable. Using the direct Lyapunov method and Razumikhin’s approach, we show that if the order of homogeneity of the right-hand sides is greater than 1 then asymptotic stability persists for all values of delay. We estimate the time of transitions, study the influence of perturbations on the stability of the trivial solution, and prove a theorem on the asymptotic stability of a complex system describing the interaction of two nonlinear subsystems.

Electrodynamic stabilization of earth-orbiting satellites in equatorial orbits

A satellite with electrodynamic stabilization system is considered. Based on the method of Lyapunov functions, sufficient conditions of the asymptotic stability of direct equilibrium position of this satellite in the orbital coordinate system under perturbing action of a gravitational moment are obtained. These conditions allow one to ensure a rational choice of parametric control coefficients depending on parameters of the satellite and its orbit.

Monoaxial electrodynamic stabilization of earth satellite in the orbital coordinate system

Consideration was given to the Earth satellite moving along a circular arbitrarily inclined orbit. The possibility of using the electrodynamic control system for monoaxial stabilization of the satellite in an indirect position in the orbital coordinate system was analyzed. An extension of the concept of electrodynamic control including a solution of the problem of electrodynamic compensation of the perturbing moment was suggested. Conditions were established for solution of the formulated problem with the use of the electromagnetic control in the presence of perturbing action of the gravitational moment. Sufficient conditions for asymptotic stability of the satellite equilibrium were established for the nonlinear statement of problem.

On stability of the solutions of a class of nonlinear delay systems

Consideration was given to a class of systems of nonlinear differential equations with retarded argument. It was assumed that in the absence of delay the zero solutions of the systems under study are asymptotically stable. Using the method of Lyapunov functions in the form of B.S. Razumikhin, it was proved that if the right-hand sides of these equations are free of the linear terms relative to the phase variables, then the asymptotic stability is retained for any delay.

Stability and stabilization of mechanical systems with switching

Hybrid mechanical systems with switched force fields, whose motions are described by differential second-order equations are considered. We propose two approaches to solving problems of analysis of stability and stabilization of an equilibrium position of the named systems. The first approach is based on the decomposition of an original system of differential equations into two systems of the same dimension but of the first order. The second approach is in direct specifying a construction of a general Lyapunov function for a mechanical system with switching.

On absolute stability of one class of nonlinear switched systems

A certain class of nonlinear switched systems is considered. Methods of developing Lyapynov functions for the systems under study are suggested. Conditions are defined, in the fulfillment of which the asymptotic stability of the zero solution will take place for any witching laws and for any admissible nonlinearities entering into right sides of the equations under discussion.

Stability of Complex Systems in Critical Cases

The stability of the solutions of nonlinear multiconnected systems is investigated by a method based on the use of the Lyapunov second method. Sufficient conditions for the asymptotic stability of certain classes of complex systems in nonlinear approximation are formulated.

On stability and dissipativity of some classes of complex systems

A new form of aggregation of complex systems was proposed and used to determine the conditions for asymptotic stability and uniform dissipativity in nonlinear approximation with the view of establishing criteria for absolute stability and absolute dissipativity. The results obtained were used to analyze dynamics of some the mechanical and biological systems.