Optimal dichotomy of temporal scales and boundedness/stability of time-varying multidimensional nonlinear systems

Abstract

This paper develops a new approach to the estimation of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficientsan essential task in various engineering and natural science applications. Known approaches to assessing the stability of such systems rest on the utility of Lyapunov functions and Lyapunov first approximation methodologies, typically providing conservative and computationally elaborate criteria for multidimensional systems of this category. Adequate criteria of boundedness of solutions to nonhomogeneous systems of this kind are rare in the contemporary literature. Lately, we develop a new approach to these problems which rests on bounding the evolution of the norms of solutions to initial systems by matching solutions of a scalar auxiliary equation we introduced in [1], [2] and [3]. Still, the technique advanced in [3] rests on the assumption that the average of the linear components of the underlying system is defined by a stable matrix of general position. The current paper substantially amplifies the application domain of this approach. It is merely assumed that the timedependent linear block of the underlying system can be split into slow and fast varying components by application of any smoothing technique. This dichotomy of temporal scales is determined by the optimal criterion reducing the conservatism of our estimates. In turn, we transform the linear subsystem with slow-varying matrix in a diagonally dominant form by successive applications of the Lyapunov transforms. This prompts the development of novel scalar auxiliary equations embracing the estimation of the norms of solutions to our initial systems. Next, we formulate boundedness/ stability criteria and estimate the relevant regions of the underlying systems using analytical and abridged numerical reasoning. Lastly, we authenticate the developed methodology in inclusive simulations.

Effective Estimation of Trapping/Stability Regions and Bilateral Solutions’ Bounds for Some Multidimensional Nonlinear Systems with Time-Varying Coefficients

“…Furthermore, the modified auxiliary equation gains computational efficacy but brings more conservative estimates than [22] in the common application domain. In turn, [25] developed a methodology escaping some key assumptions of [24] at the expense of more elaborate derivations and ascribes the auxiliary equation in the most general form. Nonetheless, the methodologies that are laid out in [24] and [25] bring rather conservative estimates of the trapping/stability regions.…”

“…In turn, [25] developed a methodology escaping some key assumptions of [24] at the expense of more elaborate derivations and ascribes the auxiliary equation in the most general form. Nonetheless, the methodologies that are laid out in [24] and [25] bring rather conservative estimates of the trapping/stability regions.…”

“…where ( ) Next, we present for convenience some conventional definitions of trapping/stability regions, see also [24] and [25] where such definitions were also adopted. Definition 1.…”

“…Furthermore, in the area of control, this problem can be linked to the estimation of the influence of time-varying disturbances on the controlled system. Some local boundedness criteria and fairly conservative estimates of the region of boundedness were given in [24] and [25]. Successive estimations of the boundaries of the trapping regions were outlined in [23] for a limited class of nonlinear systems.…”