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

Pinsky, Mark A.

doi:10.1007/s11071-020-06033-3

Cited by 6 publications

(15 citation statements)

References 39 publications

(41 reference statements)

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“…Let us recall now that, due to (14), ‖x(t, x 0 )‖ ≤ z(t, z 0 ), z 0 � ‖V − 1 x 0 ‖, ∀t ≥ t 0 , where x(t, x 0 ) and z(t, z 0 ) are solutions to planar versions of either equation ( 1) or (2) and equation (25) or (26), respectively.…”

Section: Motivation Examplementioning

confidence: 99%

“…Yet, the utility of these estimates for precisely defined systems provides a numerical affirmation of our boundedness and stability criteria. Nonetheless, combining the techniques outlined in this paper with the pertinent successive approximations should considerably refine the accuracy of the corresponding estimates (see [14], where such a strategy was developed and implemented for suitable systems).…”

Section: Coupled System Of Duffing-likementioning

confidence: 99%

“…and N is a set of positive integers. Note that additional and more complex examples of this kind are provided in Section 5 (see also [13,14]).…”

Section: Introduction and Motivation Examplementioning

confidence: 99%

“…Nonetheless, such estimates may become more conservative if f * (t, x) is defined explicitly. Consequently, we refined this methodology in [14], where (13) was used to estimate the error of successive approximations of solutions to (1) or (2) stemming from the trapping/stability regions of these systems.…”

Section: Introduction and Motivation Examplementioning

confidence: 99%

“…Nonetheless, the methodologies developed in [13,14] work under the condition that c(t) < ∞, ∀t ≥ t 0 , which considerably limits the scope of its applications. Nonetheless, it follows from (11) that frequently lim t⟶∞ c(t) � ∞ even if A is a stable and time-invariant matrix.…”

Section: Introduction and Motivation Examplementioning

confidence: 99%

See 4 more Smart Citations

Stability and Boundedness of Solutions to Some Multidimensional Time-Varying Nonlinear Systems

Pinsky

2022

Mathematical Problems in Engineering

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Assessment of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent and nonperiodic coefficients is an important problem in various applied areas which has no adequate resolution yet. Most of the known techniques provide computationally intensive and conservative stability criteria in this field and frequently fail to estimate the regions of boundedness/stability of solutions to the corresponding systems. Recently, we outlined a new approach to this problem which is based on the analysis of solutions to a scalar auxiliary equation bounded from the above time histories of the norms of solutions to the original system. This paper develops a novel technique casting the auxiliary equation in a modified form which extends the application domain and reduces the computational burden of our prior approach. Consequently, we developed more general boundedness/stability criteria and estimated trapping/stability regions for some multidimensional nonlinear systems with nonperiodic time-dependent coefficients that are common in various application domains. This lets us to assess in target simulations the extent of boundedness/stability of some multidimensional, nonlinear, and time-varying systems which were intractable with our prior technique.

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Section: Motivation Examplementioning

confidence: 99%

Section: Coupled System Of Duffing-likementioning

confidence: 99%

“…and N is a set of positive integers. Note that additional and more complex examples of this kind are provided in Section 5 (see also [13,14]).…”

Section: Introduction and Motivation Examplementioning

confidence: 99%

Section: Introduction and Motivation Examplementioning

confidence: 99%

Section: Introduction and Motivation Examplementioning

confidence: 99%

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Stability and Boundedness of Solutions to Some Multidimensional Time-Varying Nonlinear Systems

Pinsky

2022

Mathematical Problems in Engineering

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

Cited by 6 publications

References 39 publications

Stability and Boundedness of Solutions to Some Multidimensional Time-Varying Nonlinear Systems

Stability and Boundedness of Solutions to Some Multidimensional Time-Varying Nonlinear 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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