A constructive tool of nonlinear control system design, the method of control Lyapunov functions (CLFs), has found numerous applications in stabilization problems for continuous-time, discrete-time, and hybrid systems. In this paper, we address the fundamental question: Given a CLF, corresponding to a continuous-time controller with some predefined (e.g., exponential) convergence rate, can the same convergence rate be provided by an event-triggered controller? Under certain assumptions, we give an affirmative answer to this question and show that the corresponding event-based controllers provide positive dwell times between consecutive events. Furthermore, we prove the existence of self-triggered and periodic event-triggered controllers, providing stabilization with a known convergence rate.

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Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group’s array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group’s initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group’s implicit initial decision space.

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This paper is concerned with the rate at which a discrete-time, deterministic, and possibly large network of nonlinear systems generates information, and so with the minimum rate of data transfer under which the addressee can maintain the level of awareness about the current state of the network. While being aimed at development of tractable techniques for estimation of this rate, this paper advocates benefits from directly treating the dynamical system as a set of interacting subsystems. To this end, a novel estimation method is elaborated that is alike in flavor to the small gain theorem on input-to-output stability. The utility of this approach is demonstrated by rigorously justifying an experimentally discovered phenomenon. The topological entropy of nonlinear time-delay systems stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.

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In this paper, we consider the mean-field model of noisy bounded confidence opinion dynamics under exogenous influence of static radical opinions. The long-term behavior of the model is analyzed by providing a sufficient condition for exponential convergence of the dynamics to stationary state. The stationary state is also characterized by a global estimate for a sufficiently large noise. Furthermore, we consider the order-disorder transition in the model in order to identify the effect of the (relative) mass of the radicals on the critical noise level at which this transition occurs. A numerical scheme for approximating the critical noise level is provided and validated through numerical simulations of the mean-field model and the corresponding agent-based model for a particular distribution of radical opinions.

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In this paper, systems of nonlinear integro-differential Volterra equations are examined that can be represented as feedback interconnections of linear time-invariant block and periodic nonlinearities. The interest in such systems is motivated by their numerous applications in mechanical, electrical and communication engineering; examples include, but are not limited to, models of phase-locked loops, pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities are usually featured by multistability and have infinite sequences of (locally) stable and unstable equilibria; their trajectories may exhibit nontrivial (e.g. chaotic) behavior. We offer frequency-domain criteria, ensuring convergence of any solution to one of the equilibria points, and this property is referred to as the gradient-like behavior and corresponds to phase locking in synchronization systems. Although it is hard to find explicitly the equilibrium, attracting a given trajectory, we give a constructive estimate for the distance between this limit equilibrium and the initial condition. The relevant estimates are closely related to the analysis of cycle slipping in synchronization systems. In the case where the criterion of gradient-type behavior fails, a natural question arises-which nonconverging solutions may exist in the system and, in particular, how many periodic solutions it has. We show that a relaxation of the frequency-domain convergence criterion ensures the absence of high-frequency periodic orbits. The results obtained in the paper are based on the method of integral quadratic constraints that has arisen in absolute stability theory and stems from Popov's techniques of "a priori integral indices". We illustrate the analytic results by numerical simulations.

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Whereas development of mathematical models describing the endocrine system as a whole remains a challenging problem, visible progress has been demonstrated in modeling its subsystems, or axes. Models of hormonal axes portray only the most essential interactions between the hormones and can be described by low-order systems of differential equations. This paper analyzes the properties of a novel model of a hypothalamic-pituitary axis, portraying the interactions in a chain of a release hormone (secreted by the hypothalamus), a tropic hormone (produced by the pituitary gland) and an effector hormone (secreted by a target gland). This model, unlike previously published ones, captures two prominent features of neurohormonal regulation systems, namely, the pulsatile (episodic) production of the release hormone and a complex non-cyclic feedback mechanism that maintains the involved hormone concentrations within certain biological limits. At the same time, the discussed model is analytically tractable; in particular, the existence of a so-called 1-cycle featured by a single pulse over one period is proven mathematically.

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The 2017 Nobel Prize in Physiology or Medicine awarded for discoveries of molecular mechanisms controlling the circadian rhythm has called attention to the challenging area of nonlinear dynamics that deals with synchronization and entrainment of oscillations. Biological circadian clocks keep time in living organisms, orchestrating hormonal cycles and other periodic rhythms. The periodic oscillations of circadian pacemakers are self-sustained; at the same time, they are entrainable by external periodic signals that adjust characteristics of autonomous oscillations. Whereas modeling of biological oscillators is a well-established research topic, mathematical analysis of entrainment, i.e. the nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. Along with sustained periodic rhythms, periodically forced oscillators can exhibit various “irregular” behaviors, such as quasiperiodic or chaotic trajectories. This paper presents an overview of the mathematical models of circadian rhythm with respect to endocrine regulation, as well as biological background. Dynamics of the human endocrine system, comprising numerous glands and hormones operating under neural control, are highly complex. Therefore, only endocrine subsystems (or axes) supporting certain biological functions are usually studied. Low-order dynamical models that capture the essential characteristics and interactions between a few hormones can than be derived. Goodwin's oscillator often serves as such a model and is widely regarded as a prototypical biological oscillator. A comparative analysis of forced dynamics arising in two versions of Goodwin's oscillator is provided in the present paper: the classical continuous oscillator and a more recent impulsive one, capturing e.g. pulsatile secretion of hormones due to neural regulation. The main finding of this study is that, while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior due to non-smooth dynamics in entrainment.

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In this paper, we provide an overview of recent works on dynamics of social networks and distributed algorithms for their exploration, contributed by Dr. Roberto Tempo (1956–2017) and his colleagues. These works, based on the recent achievements in multi-agent systems theory and distributed randomized algorithms, contribute in bridging the gap between the two classical sciences of Social Network Analysis and Systems and Control. The topics covered by this survey include distributed algorithms for analysis of complex social networks and novel dynamic models, describing opinion formation processes in such networks.

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Control Lyapunov Functions (CLF) method gives a constructive tool for stabilization of nonlinear systems. To find a CLF, many methods have been proposed in the literature, e.g. backstepping for cascaded systems and sum of squares (SOS) programming for polynomial systems. Dealing with continuous-time systems, the CLF-based controller is also continuous-time, whereas practical implementation on a digital platform requires sampled-time control. In this paper, we show that if the continuous-time controller provides exponential stabilization, then an exponentially stabilizing event-triggered control strategy exists with the convergence rate arbitrarily close to the rate of the continuous-time system.

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Systems that can be decomposed as feedback interconnections of stable linear blocks and periodic nonlinearities arise in many physical and engineering applications. The relevant models e.g. describe oscillations of a viscously damped pendulum, synchronization circuits (phase, frequency and delay locked loops) and networks of coupled power generators. A system with periodic nonlinearities usually has multiple equilibria (some of them being locally unstable). Many tools of classical stability and control theories fail to cope with such systems. One of the efficient methods, elaborated to deal with periodic nonlinearities, stems from the celebrated Popov method of 'integral indices', or integral quadratic constraints; this method leads, in particular, to frequency-domain criteria of the solutions' convergence, or, equivalently, global stability of the equilibria set. In this paper, we further develop Popov's method, addressing the problem of robustness of the convergence property against external disturbances that do not oscillate at infinity (allowing the system to have equilibria points). Will the forced solutions also converge to one of the equilibria points of the disturbed system? In this paper, a criterion for this type of robustness is offered.

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To understand the sophisticated control mechanisms of the human's endocrine system is a challenging task that is a crucial step towards precise medical treatment of many dysfunctions and diseases. Although mathematical models describing the endocrine system as a whole are still elusive, recently some substantial progress has been made in analyzing theoretically its subsystems (or axes) that regulate the production of specific hormones. Secretion of many vital hormones, responsible for growth, reproduction and metabolism, is orchestrated by feedback mechanisms that are similar in structure to the model of simple genetic oscillators, proposed first by B.C. Goodwin. Unlike the celebrated Goodwin's model, the endocrine regulation mechanisms are in fact known to have non-cyclic structures and involve multiple feedbacks; a Goodwin-type model thus represents only a part of such a complicated mechanism. In this paper, we examine a non-cyclic feedback system of hormonal regulation, obtained from the classical Goodwin's oscillator by introducing an additional negative feedback. We establish global properties of this model and show, in particular, that the local instability of its unique equilibrium implies that almost all system's solutions oscillate; furthermore, under additional restrictions these solutions converge to periodic or homoclinic orbits.@en

Recent years have witnessed a significant trend towards filling the gap between Social Network Analysis (SNA) and control theory. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the development of algorithms and software for data analysis and the tremendous progress in understanding complex networks and multi-agent systems (MAS) dynamics. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with dynamic models of social networks and processes over them, to the attention of the broad research community. In its first part (Proskurnikov & Tempo, 2017), we have considered the most classical models of social dynamics, which have anticipated and to a great extent inspired the recent extensive studies on MAS and complex networks. This paper is the second part of the tutorial, and it is focused on more recent models of social processes that have been developed concurrently with MAS theory. Future perspectives of control in social and techno-social systems are also discussed.

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In this paper, we consider synchronization of dynamical systems interconnected via nonlinear integral coupling. Integral coupling allows one to achieve synchronization with lower interaction levels (coupling gains) than with linear coupling. Previous results on this topic were obtained for synchronization of several systems with all-to-all interconnections. In this paper, we relax the requirement of all-to-all interconnections and provide two results on exponential synchronization under nonlinear integral coupling for networks with topologies different from all-to-all interconnections. In particular, we provide a high-gain result for an arbitrary interconnection topology and a non-high-gain method for analysis of synchronization for specific topologies. The results are illustrated by simulations of Hindmarsh-Rose neuron oscillators.

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The Goodwin oscillator is a simple yet illustrative model of a biochemical system with a stable limit cycle. Considered as a prototypical biological oscillator, Goodwin's model is broadly used e.g. to describe circadian rhythms, hormonal cycles, self-oscillatory metabolic pathways.These periodic or non-periodic oscillations are self-sustained; at the same time, they are entrainable by external periodic signals, adjusting the characteristics of the autonomous oscillatory behavior. Mathematical analysis of entrainment phenomena, i.e. nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. This paper presents a comparative analysis of forced dynamics arising in two versions of Goodwin's oscillator: the classical continuous oscillator and a more recent impulsive one, e.g. capturing pulsatile secretion of hormones. The main finding of this study is that while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior thus highlighting the role of non-smooth dynamics in entrainment.

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In this paper, we propose an algorithm for path-following control of the nonholonomic mobile robot based on the idea of the guiding vector field (GVF). The desired path may be an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the robot's kinematic model, we design a GVF, whose integral curves converge to the trajectory. A nonlinear motion controller is then proposed, which steers the robot along such an integral curve, bringing it to the desired path. We establish global convergence conditions for our algorithm and demonstrate its applicability and performance by experiments with wheeled robots.

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Investigation of social influence dynamics requires mathematical models that are “simple” enough to admit rigorous analysis, and yet sufficiently “rich” to capture salient features of social groups. Thus, the mechanism of iterative opinion pooling from (DeGroot, 1974), which can explain the generation of consensus, was elaborated in (Friedkin and Johnsen, 1999) to take into account individuals’ ongoing attachments to their initial opinions, or prejudices. The “anchorage” of individuals to their prejudices may disable reaching consensus and cause disagreement in a social influence network. Further elaboration of this model may be achieved by relaxing its restrictive assumption of a time-invariant influence network. During opinion dynamics on an issue, arcs of interpersonal influence may be added or subtracted from the network, and the influence weights assigned by an individual to his/her neighbors may alter. In this paper, we establish new important properties of the (Friedkin and Johnsen, 1999) opinion formation model, and also examine its extension to time-varying social influence networks.

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Whereas obtaining a global model of the human endocrine system remains a challenging problem, visible progress has been demonstrated in modeling its subsystems (axes) that regulate production of specific hormones. The axes are typically described by Goodwin-like cyclic feedback systems. Unlike the classical Goodwin oscillator, obeying a system of ordinary differential equations, the feedback mechanisms of brain-controlled hormonal regulatory circuits appear to be pulsatile, which, in particular, exclude the possibility of equilibrium solutions. The recent studies have also revealed that the regulatory mechanisms of many vital hormones (including e.g. testosterone and cortisol regulation) are more complicated than Goodwin-type oscillators and involve multiple negative feedback loops. Although a few “multi-loop” extensions of the classical Goodwin model have been studied in literature, the analysis of impulsive endocrine regulation models with additional negative feedbacks has remained elusive. In this paper, we address one of such models, obtained from the impulsive Goodwin-type oscillator by introducing an additional linear feedback. Since the levels of hormones’ concentrations oscillate periodically, examination of endocrine regulation circuits is primarily focused on periodic solutions. We prove the existence and uniqueness of periodic solutions of a special type, referred to as 1-cycles and featured by the unique discontinuous point in each period. Procedures for computing such a solution and testing its stability are discussed. The results are confirmed by numerical simulations.

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Many distributed algorithms for multi-agent coordination employ the simple averaging dynamics, referred to as the Laplacian flow. Besides the standard consensus protocols, examples include, but are not limited to, algorithms for aggregation and containment control, target surrounding, distributed optimization and models of opinion formation in social groups. In spite of their similarities, each of these algorithms has been studied using separate mathematical techniques. In this paper, we show that stability and convergence of many coordination algorithms involving the Laplacian flow dynamics follow from the general consensus dichotomy property of a special differential inequality. The consensus dichotomy implies that any solution to the differential inequality is either unbounded or converges to a consensus equilibrium. In this paper, we establish the dichotomy criteria for differential inequalities and illustrate their applications to multi-agent coordination and opinion dynamics modeling.

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Recently the dynamics of signed networks, where the ties among the agents can be both positive (attractive) or negative (repulsive) have attracted substantial attention of the research community. Examples of such networks are models of opinion dynamics over signed graphs. It has been shown that under mild connectivity assumptions these protocols provide the convergence of opinions in absolute value, whereas their signs may differ. This 'modulus consensus' may correspond to the bipartite consensus (the opinions split into two clusters, converging to two opposite values) or the asymptotic stability of the system (the opinions always converge to zero). In this paper, we demonstrate that the phenomenon of modulus consensus in a signed network is a manifestation of a more general, regarding the solutions of special recurrent inequalities, associated to conventional first-order consensus algorithms. Although such a recurrent inequality does not provide the uniqueness of a solution, it can be shown that, under some natural assumptions, each of its bounded solutions has a limit and, moreover, converges to consensus. A similar property has previously been established for special continuous-time differential inequalities in [1]. Besides analysis of signed networks, we link the consensus properties of recurrent inequalities to the convergence properties of distributed optimization algorithms and stability properties of substochastic matrices.

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In recent years, we have observed a significant trend towards filling the gap between social network analysis and control. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the advancement in complex networks theory and multi-agent systems, and the development of modern computational tools for big data analysis. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with applications to social systems, to the attention of the broad research community. This paper is the first part of the tutorial, and it is focused on the most classical models of social dynamics and on their relations to the recent achievements in multi-agent systems.

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