Solution Bounds, Stability, and Estimation of Trapping/Stability Regions of Some Nonlinear Time-Varying Systems

Pinsky, Mark A.; Koblik, Steve

doi:10.1155/2020/5128430

Cited by 6 publications

(42 citation statements)

References 25 publications

(135 reference statements)

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“…Along these lines we formulate novel bondedness/stability criteria and reinforce our theoretical results in representative simulations. Our inferences advance the methodology developed in [43] and uplift some of its major limitations.…”

Section: A Constmentioning

confidence: 82%

“…In [43] under some conditions we develop an approach to estimation of the upper bound of norms of solutions to (1.1)/(1.2) which leads to more general boundedness/ stability criteria and estimation of the trapping/stability regions for these systems. Still, the developed estimates turn out to be rather conservative, especially for nonhomogeneous systems and ones with large nonlinear components.…”

Section: A Constmentioning

confidence: 99%

“…Next, we recap our approach for estimating the norm of solutions to (1.1) which is used in sections 6 and 7, see [43] for additional details. This paper derives a differential inequality for the norms of solutions to (1)…”

Section: Estimation Of Solutions' Normsmentioning

confidence: 99%

“…This section demonstrates application of our approach to the design of successive approximation to the norms of solutions to a Van der-Pollike model with variable coefficients which was also tackled in [43] by our former technique. This system can be written as (1.1) and assumes the following form in dimensionless variables, ( )…”

Section: Motivating Examplementioning

confidence: 99%

“…This section applies the methodology developed in [43] to estimate the error norm of approximate solutions given by (4.1), i.e., the norm of solution to (4.2) using Lipschitz continuity condition. This, in turn, yields successive approximations to bilateral solution bounds and boundedness/stability conditions for solutions of equations ( 1) and (2).…”

Section: Linear Error Equation and Its Applicationsmentioning

confidence: 99%

See 4 more Smart Citations

Successive estimations of bilateral bounds and trapping/stability regions of solution to some nonlinear nonautonomous systems

Pinsky

2021

Nonlinear Dyn

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Estimation of the degree of stability and the bounds of solutions to nonautonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative conditions which are unable to effectively gage these characteristics in time-varying nonlinear systems. This paper develops a novel methodology providing successive approximations to solutions that are stemmed from the trapping/stability regions of these systems and estimate the errors of such approximations. In turn, this leads to successive approximations of both the bilateral bounds of solutions and the boundaries of trapping/stability regions of the underlying systems. Along these lines we formulate enhanced stability/boundedness criteria and contrast our inferences with inclusive simulations which reveal dependence of the trapping/stability regions upon the structure of timedependent components and initial time-moment.

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Section: A Constmentioning

confidence: 82%

Section: A Constmentioning

confidence: 99%

Section: Estimation Of Solutions' Normsmentioning

confidence: 99%

Section: Motivating Examplementioning

confidence: 99%

Section: Linear Error Equation and Its Applicationsmentioning

confidence: 99%

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Successive estimations of bilateral bounds and trapping/stability regions of solution to some nonlinear nonautonomous systems

Pinsky

2021

Nonlinear Dyn

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

Pinsky

2024

J Nonlinear Sci

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Assessment of the stability/boundedness of solutions to nonlinear systems with variable coefficients brings long-standing and challenging problems which emerge in various application domains. These problems are naturally evolved into more arduous and largely open problems concerned with the estimation of the corresponding stability/boundedness regions. This paper develops a novel approach furnishing computationally tractable boundedness/stability criteria which underscores a methodology providing recursive estimation of the boundaries of the trapping/stability regions for a broad class of multidimensional and nonlinear systems with variable nonperiodic coefficients. Furthermore, our approach naturally conveys the bilateral bounds for the norms of solutions to the corresponding systems via the application of successive approximations which are introduced in this paper.The developed techniques are validated in inclusive simulations which endorse their applications to systems with large and complex nonlinear components and bounded in-norm time-varying disturbances.

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

Pinsky

2022

Math. Control Signals Syst.

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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.

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Solution Bounds, Stability, and Estimation of Trapping/Stability Regions of Some Nonlinear Time-Varying Systems

Cited by 6 publications

References 25 publications

Successive estimations of bilateral bounds and trapping/stability regions of solution to some nonlinear nonautonomous systems

Successive estimations of bilateral bounds and trapping/stability regions of solution to some nonlinear nonautonomous systems

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

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

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