Operator-Theoretic Characterization of Eventually Monotone Systems

Abstract-In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory. We exploit the framework of the Koopman operator, which provides a linear infinite-dimensional description of nonlinear dynamical systems and spectral operator-theoretic notions such as eigenvalues and eigenfunctions. The sublevel sets of the dominant eigenfunction form a family of nested forwardinvariant sets and the basin of attraction is the largest of these sets. The boundaries of these sets, called isostables, allow studying temporal properties of the system. Our first observation is that the dominant eigenfunction is increasing in every variable in the case of monotone systems. This is a strong geometric property which simplifies the computation of isostables. We also show how variations in basins of attraction can be bounded under parametric uncertainty in the vector field of monotone systems. Finally, we study the properties of the parameter set for which a monotone system is multistable. Our results are illustrated on several systems of two to four dimensions.

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“…Although the full and reduced systems are not monotone with respect to any orthant, numerical results in [37] indicate…”

Section: Toxin-antitoxin Systemmentioning

confidence: 96%

“…0. The proof of Proposition 4 is almost identical to the proof of a similar result in [37], and hence it is omitted. In both cases, the conditions on λ 1 , v 1 and s 1 are only necessary and not sufficient for monotonicity, which is consistent with the linear case and necessary conditions for positivity.…”

Section: A Spectral Properties and Lyapunov Functionsmentioning

confidence: 96%

Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

Sootla

Mauroy

2017

IEEE Trans. Automat. Contr.

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“…In the terminology of linear systems, for instance, such phenomenon is often also called eventual positivity (see [6,19,27] and references therein) in forward (or backward) time, which means that trajectories starting from positive initial values will become positive in forward (or backward) time only after some initial transient. As a matter of fact, this property received rapidly-increasing attention very recently in both finite-dimensional linear systems [19] and infinite-dimensional linear systems [5,6], as well as applications to ordinary differential equations [20,28], partial differential equations [4,7,8], delay differential equations [5,6] and control theory [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, various examples of systems have been found in [27], which cannot be confirmed to be monotone (cooperative), but are eventually monotone (eventually cooperative). Moreover, they are not limited to near monotone (cooperative) systems in the context of perturbation theory.…”

Section: Introductionmentioning

confidence: 99%

“…Hirsch [10] has ever studied the dynamics of eventually monotone (cooperative) systems. Sootla and Mauroy [27] proposed a spectral characterization of eventually monotone systems from Koopman operators point of view. Based on all these results, it is reasonable to expect that eventually cooperative (or competitive) systems might possess many asymptotic properties of cooperative (or competitive) systems.…”

Section: Introductionmentioning

confidence: 99%

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Non-oscillation principle for eventually competitive and cooperative systems

Niu¹,

Wang²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We presented in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both ω-and α-) limit sets is obtained for such systems; and moreover, we established the Poincaré-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.

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Optimal control formulation of pulse-based control using Koopman operator

2018

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a b s t r a c tIn many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

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Operator-Theoretic Characterization of Eventually Monotone Systems

Cited by 22 publications

References 41 publications

Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

Non-oscillation principle for eventually competitive and cooperative systems

Optimal control formulation of pulse-based control using Koopman operator

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