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A well-known property of linear resistive electrical networks is that the current distribution minimizes the total dissipated power. When the circuit includes resistors with nonlinear monotonic characteristic, the current distribution minimizes in general a different functional. We show that, if the nonlinear characteristic is a threshold-like function and the current generator is concentrated in a single point, as in the case of lightning or dielectric discharge, then the current flow is concentrated along a single path, which is a minimum path to the ground with respect to the threshold. We also propose a dynamic model that explains and qualitatively reproduces the lightning transient behavior: initial generation of several plasma branches and subsequent dismissal of all branches but the one reaching the ground first, which is the optimal one.

We proposed network-decentralized control strategies, in which each actuator can exclusively rely on local information, without knowing the network topology and the external input, ensuring that the flow asymptotically converges to the optimal one with respect to the p -norm. For 1 < p < ∞ , the flow converges to a unique constant optimal up∗. We show that the state converges to the optimal Lagrange multiplier of the optimization problem. Then, we consider networks where the flows are affected by unknown spontaneous dynamics and the buffers need to be driven exactly to a desired set-point. We propose a network-decentralized proportional-integral controller that achieves this goal along with asymptotic flow optimality; now it is the integral variable that converges to the optimal Lagrange multiplier. The extreme cases p=1 and p=∞ are of some interest since the former encourages sparsity of the solution while the latter promotes fairness. Unfortunately, for p=1 or p=∞ these strategies become discontinuous and lead to chattering of the flow, hence no optimality is achieved. We then show how to approximately achieve the goal as the limit for p 1 or p ∞.

Given a class of (bio)Chemical Reaction Networks (CRNs) identified by a stoichiometric matrix $S$ , we define as dual reaction network , CRN $^{*}$ , the class of (bio)Chemical Reaction Networks identified by the transpose stoichiometric matrix $S^\top $ . We consider both the dynamical systems describing the time evolution of the species concentrations and of the reaction rates. First, based on the analysis of the Jacobian matrix, we show that the structural (i.e., parameter-independent) local stability properties are equivalent for a CRN and its dual CRN $^{*}$ . We also assess the structural global stability properties of the two dual networks, analysing both concentration and rate representations. We prove that the existence of a polyhedral (or piecewise-linear) Lyapunov function in concentrations for a CRN is equivalent to the existence of a piecewise-linear in rates Lyapunov function for the dual CRN $^{*}$ ; in fact, if $V$ is a polyhedral Lyapunov function for a CRN, the dual polyhedral function $V^{*}$ is a piecewise-linear in rates Lyapunov function for the dual network. We finally show how duality can be exploited to gain additional insight into biochemical reaction networks.

Fibromyalgia (FM) is an unsolved central pain processing disturbance. We aim to provide a unifying model for FM pathogenesis based on a loop network involving thalamocortical regions, i.e., the ventroposterior lateral thalamus (VPL), the somatosensory cortex (SC), and the thalamic reticular nucleus (TRN). The dynamics of the loop have been described by three differential equations having neuron mean firing rates as variables and containing Hill functions to model mutual interactions among the loop elements. A computational analysis conducted with MATLAB has shown a transition from monostability to bistability of the loop behavior for a weakening of GABAergic transmission between TRN and VPL. This involves the appearance of a high-firing-rate steady state, which becomes dominant and is assumed to represent pathogenic pain processing giving rise to chronic pain. Our model is consistent with a bulk of literature evidence, such as neuroimaging and pharmacological data collected on FM patients, and with correlations between FM and immunoendocrine conditions, such as stress, perimenopause, chronic inflammation, obesity, and chronic dizziness. The model suggests that critical targets for FM treatment are to be found among immunoendocrine pathways leading to GABA/glutamate imbalance having an impact on the thalamocortical system.

We consider the problem of assessing the sensitivity of uncertain biochemical systems in the presence of input perturbations (either constant or periodic) around a stable steady state. In particular, we propose approaches for the robust sensitivity analysis of systems with uncertain parameters assumed to take values in a hyper-rectangle. We highlight vertex results, which allow us to check whether a property is satisfied for all parameter choices in the hyper-rectangle by simply checking whether it is satisfied for all parameter choices at the vertices of the hyper-rectangle. We show that, for a vast class of systems, including (bio)chemical reaction networks with mass-action kinetics, the system Jacobian has a totally multiaffine structure (namely, all minors of the Jacobian matrix are multiaffine functions of the uncertain parameters), which can be exploited to obtain several vertex results. We consider different problems: robust non-singularity; robust stability of the steady-state; robust steady-state sensitivity analysis, in the case of constant perturbations; robust frequency-response sensitivity analysis, in the presence of periodic perturbations; and robust adaptation analysis. The developed theory is then applied to gain insight into some examples of uncertain biochemical systems, including the incoherent feed-forward loop, the coherent feed-forward loop, the Brusselator oscillator and the Goldbeter oscillator.

Despite the progress in medical care, combined population-wide interventions (such as physical distancing, testing and contact tracing) are still crucial to manage the SARS-CoV-2 pandemic, aggravated by the emergence of new highly transmissible variants. We combine the compartmental SIDARTHE model, predicting the course of COVID-19 infections, with a new data-based model that projects new cases onto casualties and healthcare system costs. Based on the Italian case study, we outline several scenarios: mass vaccination campaigns with different paces, different transmission rates due to new variants, and different enforced countermeasures, including the alternation of opening and closure phases. Our results demonstrate that non-pharmaceutical interventions (NPIs) have a higher impact on the epidemic evolution than vaccination, which advocates for the need to keep containment measures in place throughout the vaccination campaign. We also show that, if intermittent open-close strategies are adopted, deaths and healthcare system costs can be drastically reduced, without any aggravation of socioeconomic losses, as long as one has the foresight to start with a closing phase rather than an opening one.

International audience; We consider a linear plant with decentralized input saturation, whose limits are not necessarily symmetric. We propose a nonlinear static state feedback stabilizer that is asymmetric, in such a way that the non-symmetric nature of the saturation is fully exploited in the control design, for larger regions of attraction. We show by example that the proposed technique provides significantly larger regions of attractions as compared to the symmetric solution results, in a case where the positive and negative saturation limits are different.

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete-time linear systems. We mainly focus on the so-called Krasnoselskij–Mann iteration, ${x}$ ( $k + 1$ ) = ( $1-\alpha _{k}$ ) ${x}$ ( ${k}$ ) + $\alpha _{k} A~{x}$ ( ${k}$ ), which is relevant for distributed computation in optimization and game theory, when $A$ is not available in a centralized way. We show that strict pseudocontractiveness of the linear operator $x \mapsto Ax$ is not only sufficient (as known) but also necessary for the convergence to a vector in the kernel of $I-A$ . We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.

In late December 2019, a novel strand of Coronavirus (SARS-CoV-2) causing a severe, potentially fatal respiratory syndrome (COVID-19) was identified in Wuhan, Hubei Province, China and is causing outbreaks in multiple world countries, soon becoming a pandemic. Italy has now become the most hit country outside of Asia: on March 16, 2020, the Italian Civil Protection documented a total of 27980 confirmed cases and 2158 deaths of people tested positive for SARS-CoV-2. In the context of an emerging infectious disease outbreak, it is of paramount importance to predict the trend of the epidemic in order to plan an effective control strategy and to determine its impact. This paper proposes a new epidemic model that discriminates between infected individuals depending on whether they have been diagnosed and on the severity of their symptoms. The distinction between diagnosed and non-diagnosed is important because non-diagnosed individuals are more likely to spread the infection than diagnosed ones, since the latter are typically isolated, and can explain misperceptions of the case fatality rate and of the seriousness of the epidemic phenomenon. Being able to predict the amount of patients that will develop life-threatening symptoms is important since the disease frequently requires hospitalisation (and even Intensive Care Unit admission) and challenges the healthcare system capacity. We show how the basic reproduction number can be redefined in the new framework, thus capturing the potential for epidemic containment. Simulation results are compared with real data on the COVID-19 epidemic in Italy, to show the validity of the model and compare different possible predicted scenarios depending on the adopted countermeasures.

Correction for ‘Stabilization of negative capacitance in ferroelectric capacitors with and without a metal interlayer’ by T. Rollo, et al., Nanoscale, 2020, 12, 6121–6129, DOI: 10.1039/C9NR09470A.

This paper proposes a decentralised explicit (closed-form) iterative formula that solves convex programming problems with linear equality constraints and interval bounds on the decision variables. In particular, we consider a team of decision agents, each setting the value of a subset of the variables, and a team of information agents, in charge of ensuring that the equality constraints are fulfilled. The structure of the constraint matrix imposes a communication pattern between decision and information agents, which can be represented as a bipartite graph. We associate each information agent with an integral variable and each decision agent with a saturated function, which takes the interval bounds into account, and we design a decentralised dynamic mechanism that globally converges to the optimal solution. Under mild conditions, the convergence is shown to be exponential. We also provide a discrete-time algorithm, based on the Euler system, and we give an upper bound for the step parameter to ensure convergence. Although the considered optimisation problem is static, we show that the proposed scheme can be successfully applied to find the optimal solution of network-decentralised dynamic control problems.

A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globally-stable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarily-varying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted.

Genetic oscillators have a fundamental role in the regulation not only of intracellular, but also of intercellular functions: for instance, in the segmentation clock, the synchronised oscillation of neighbouring cells generates spatial travelling waves that induce segmentation of precursors of the vertebral column. To investigate this type of phenomena, we consider the behaviour of two genetic negative feedback oscillators, each operating in a different cell, coupled by an intercellular positive-feedback interconnection with delays. The two coupled systems are nominally identical, but can be different due to noise and cell-to-cell variability. When they can be different in general, we study the effect of positive-feedback and of delay in inducing an oscillatory behaviour. When they are identical, we study how the intercellular feedback delay affects the phase difference between the two oscillators.

We present a revised analysis of Negative Capacitance (NC) in ferroelectric-insulator capacitors, and particularly of the difference between systems with and without a metal interlayer. We develop a model accounting for the three-dimensional electrostatics and report analytical and numerical results based on Landau-Ginzburg equations. Our results explain the lack of NC operation in capacitors having an interlayer metal, compare well with recent experiments and enlighten the role of traps at the ferroelectric-oxide interface.

In biochemical networks, the steady-state input-output influence is the sign of the output steady-state variation due to a persistent positive input perturbation; if the sign does not depend on the value of the strictly positive system parameters, the influence is structural. As recently shown for small perturbations, when the linearized system approximation is valid, steady-state input-output influences can be structurally assessed, for biochemical networks with $m$ unknown parameters, by means of a vertex algorithm with complexity $2^{m}$ . This letter shows that the structural input-output influence of a biochemical network is a global property, which does not require any small-perturbation assumption. It also shows that, using a new algorithm, the complexity can be reduced down to $2^{m-n}$ , where $n$ is the system order, thus drastically reducing the computation time. Finally, when the uncertain parameters belong to known intervals, non-conservative bounds are given for the steady-state ratio between output and input, allowing for sensitivity analysis.

International audience; — To detect faults in a system we can adopt an observer, designed for the healthy system, and monitor the discrepancy between actual and expected behaviour of the residual (difference between the system output and its estimate). To isolate faults, we can compute the invariant sets associated with each fault, and their projection in the residual space (limit set): faults can be isolated if the associated limit sets are separated when a (constant) test input is applied. However, the explicit computation of limit sets can be hard even for low-dimensional systems. As a main contribution, we show that, by adopting an implicit representation of limit sets, very efficient procedures can be used to solve the problem, based on convex quadratic programming or linear programming. Simulations show that the approach is effective in solving even large dimensional problems, which makes it suitable for large-scale networked systems.

Feedback is both a pillar of control theory and a pervasive principle of nature. For this reason, control-theoretic methods are powerful to analyse the dynamic behaviour of biological systems and mathematically explain their properties, as well as to engineer biological systems so that they perform a specific task by design. This paper illustrates the relevance of control-theoretic methods for biological systems. The first part gives an overview of biological control and of the versatile ways in which cells use feedback. By employing control-theoretic methods, the complexity of interlaced feedback loops in the cell can be revealed and explained, and layered feedback loops can be designed in the cell to induce the desired behaviours, such as oscillations, multi-stability and activity regulation. The second part is mainly devoted to modelling uncertainty in biology and understanding the robustness of biological phenomena due to their inherent structure. Important control-theoretic tools used in systems biology are surveyed. The third part is focused on tools for model discrimination in systems biology. A deeper understanding of molecular pathways and feedback loops, as well as qualitative information on biological networks, can be achieved by studying the “dynamic response phenotypes” that appear in temporal responses. Several applications to the analysis of biological systems are showcased.

This paper investigates the principal properties of time varying operation modes for tuneable vibration absorbers mounted on distributed structures to reduce vibrations produced by stationary broad frequency band disturbances. The study considers a practical application where an electro-mechanical tuneable vibration absorber, comprising a seismic coil-magnet linear transducer, is fixed on a thin walled circular duct flexible structure. The absorber is commanded to either periodically switch or continuously vary the stiffness and damping of the elastic suspension holding the moving magnet. As a result, the tuneable vibration absorber fundamental natural frequency and damping ratio are respectively switched or swept to cyclically reduce the resonant response of multiple flexural natural modes of the duct structure. The study analytically shows, and confirms with simulations and experiments, that the vibration control produced by a simple “blind sweep” operation mode of the tuneable vibration absorber does not differ significantly from the (sub)optimal performance attainable via a switching operation mode.

We study diffusion of cooperation in an n-population game in continuous time. At each instant, the game involves n random individuals, one from each population. The game has the structure of a prisoner's dilemma, where each player can choose a continuous decision variable associated with the probability of cooperating or defecting. We turn the game into a positive dynamical system. Then, we propose a novel strategy that is the saturation of a polynomial function. The strategy requires to each player exclusively the knowledge of her/his own current average payoff, along with her/his own payoffs in the cooperative and noncooperative equilibria; no information about other players' payoffs is required. The proposed strategy guarantees local stability of the cooperative equilibrium if the degree p of the polynomial is greater than or equal to 2. Conversely, the noncooperative equilibrium becomes unstable, for p large enough, if and only if a certain Metzler matrix depending on the payoffs has a positive Frobenius eigenvalue. We prove that the n-dimensional box of all payoffs between the noncooperative and the cooperative ones is positively invariant. Finally we show that, for p large, the domain of attraction of the cooperative equilibrium inside this box becomes arbitrarily close to the box itself.

Biological oscillators are present in nearly all self-regulating systems, from individual cells to entire organisms. In any oscillator structure, a negative feedback loop is necessary, but not sufficient to guarantee the emergence of periodic behaviors. The likelihood of oscillations can be improved by careful tuning of the system time constants and by increasing the loop gain, yet it is unclear whether there is any general relationship between optimal time constants and loop gain. This issue is particularly relevant in genetic oscillators resulting from a chain of different subsequent biochemical events, each with distinct (and uncertain) kinetics. Using two families of genetic oscillators as model examples, we show that the loop gain required for oscillations is minimum when all elements in the loop have the same time constant. On the contrary, we show that homeostasis is ensured if a single element is considerably slower than the others.

For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally non-singular, under positivity assumptions for the system partial derivatives, the existence of an equilibrium is guaranteed. For systems subject to positivity constraints, it is shown that, if the system admits a PLF, under structural non-singularity assumptions, global convergence within the positive orthant is structurally ensured, while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise-linear-in-rate Lyapunov function.

International audience; This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming , for computing the test input. The developed methodology offers a competent continuous-time solution to the optimization-based computation of the test input via Hahn–Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems.

This paper investigates the topology-independent stability of homogeneous dynamical networks, composed of interconnected equal systems. Precisely, dynamical systems with identical nominal transfer function $F(s)$ are associated with the nodes of a directed graph, whose arcs account for their dynamic interactions, described by a common nominal transfer function $G(s)$ . It is shown that topology-independent stability is guaranteed for all possible interconnections with interaction degree (defined as the maximum number of arcs leaving a node) equal at most to $N$ if the $\infty$ -norm of the complementary sensitivity function $NF(s)G(s)[1+NF(s)G(s)]^{-1}$ is less than 1. This bound is nonconservative in that there exist graphs with interaction degree $N$ that are unstable for an $\infty$ -norm greater than 1. When nodes and arcs transferences are affected by uncertainties with norm bound $K > 0$ , topology-independent stability is robustly ensured if the $\infty$ -norm is less than $1/(1+2NK)$ . For symmetric systems, stability is guaranteed for all topologies with interaction degree at most $N$ if the Nyquist plot of $NF(s)G(s)$ does not intersect the real axis to the left of $-1/2$ . The proposed results are applied to fluid networks and platoon formation.

We propose a novel approach to the problem of inverse kinematics for possibly redundant planar manipulators. We show that, by considering the joints as point masses in a fictitious gravity field, and by adding proper constraints to take into account the length of the links, the kinematic inversion may be cast as a convex programming problem. Convex constraints in the decision variables (in particular, linear constraints in the workspace) are easily managed with the proposed approach. We also show how to exploit the idea for avoiding obstacles while tracking a reference end-effector trajectory and discuss how to extend the results to some kinds of non-planar manipulators. Simulation results are reported, showing the effectiveness of the approach.

Oscillators are essential to fuel autonomous behaviours in molecular systems. Artificial oscillators built with programmable biological molecules such as DNA and RNA are generally easy to build and tune, and can serve as timers for biological computation and regulation. We describe a new artificial nucleic acid biochemical reaction network, and we demonstrate its capacity to exhibit oscillatory solutions. This network can be built in vitro using nucleic acids and three bacteriophage enzymes, and has the potential to be implemented in cells. Numerical simulations suggest that oscillations occur in a realistic range of reaction rates and concentrations.

Given a static plant described by a differentiable input-output function, which is completely unknown, but whose Jacobian takes values in a known polytope in the matrix space, this paper considers the problem of tuning (i.e., driving to a desired value) the output, by suitably choosing the input. It is shown that, if the polytope is robustly nonsingular (or has full rank, in the nonsquare case), then a suitable tuning scheme drives the output to the desired point. The proof exploits a Lyapunov-like function and applies a well-known game-theoretic result, concerning the existence of a saddle point for a min-max zero-sum game. When the plant output is represented in an implicit form, it is shown that the same result can be obtained, resorting to a different Lyapunov-like function. The case in which proper input or output constraints must be enforced during the transient is considered as well. Some application examples are proposed to show the effectiveness of the approach.

We consider flow-inducing networks, a class of models that are well-suited to describe important biochemical systems, including the MAPK pathway and the interactions at the trans-Golgi network. A flow-inducing network is given by the interconnection of subsystems (modules), each associated with a stochastic state matrix whose entries depend on the state variables of other modules. This results in an overall nonlinear system; when the interactions are modeled as mass action kinetics, the overall system is bilinear. We provide preliminary results concerning the existence of single or multiple equilibria and their positivity. We also show that instability phenomena are possible and that entropy is not a suitable Lyapunov function. The simplest non-trivial module is the duet, a second order system whose variables represent the concentrations of a species in its activated and inhibited state: under mass action kinetics assumptions, we prove that: 1) a negative loop of duets has a unique equilibrium that is unconditionally stable and 2) a positive loop of duets has either a unique stable equilibrium on the boundary or two equilibria, of which one is unstable on the boundary and one is strictly positive and stable; both properties 1) and 2) hold regardless of the number of duets in the loop.

Most of the existing techniques for palmprint recognition rely on metrics, typically based on static functions, which evaluate the distance between a pair of features. In this paper, we propose a new technique for palmprint verification based on a dynamical system approach for principal palm lines matching. The proposed dynamic algorithm is recursive and involves a positive linear dynamical system, whose evolution depends on the matching level between the two input images. In a preprocessing phase, the procedure iteratively erodes both of the images to be compared, by eliminating points in each image that do not have enough close neighboring points both in the image itself and the comparison image. As a result of the iterations, only the points that have enough neighboring points in both the image itself and in the comparison image can survive. Thus, the output of the dynamical system converges either to zero, when a deep mismatch exists between the two images, or to a high value, when a good matching is observed. The results, in terms of verification, are in line with the state-of-the-art results in the current literature. The main advantage of the approach is its robustness when dealing with low-resolution and noisy images. The impact of noise (e.g., salt and pepper noise) is effectively reduced: images corrupted with such noise are easily recognized, while a randomly generated image is rejected even when compared with itself.

We consider the problem of controlling the transmission rate in a communication network where each node adjusts its own transmission rate exclusively based on the physical medium occupation (the band occupancy of the aggregate complementary nodes). We show how to design a decentralized control for maximizing band occupancy and fairness. If the network is fully connected, the problem admits a simple solution. Difficulties arise in the case of partially connected networks, and in the presence of time-varying network topologies and delays. General conditions are given which, by properly tuning control parameters, ensure stability. These conditions are conservative and affect system performance. However, we show that in the case of symmetric connections, stability can be studied based on the system eigenvalues even in the presence of topology switchings. Less conservative bounds can be inferred by exploiting known properties of the eigenvalues of the adjacency matrix of a graph. We finally consider the multichannel case, in which nodes may jump among channels: the previous scheme can be extended to this case and asymptotically ensures uniform channel exploitation. 1 1 Some of the presented results have been previously published in a preliminary form in [1].

We provide necessary and sufficient structural conditions for multistationarity and oscillations in aggregate monotone systems, defined as the interconnection of stable monotone components. Our classification is based on the presence of exclusively positive or exclusively negative cycles in the system aggregate graph, whose nodes are the monotone subsystems. The results presented here scale up the structural classification of oscillatory and multistationary behaviors in sign-definite biochemical networks previously proposed by the authors. Models of biomolecular systems are discussed to demonstrate the applicability of our classification.

Molecular titration is emerging as an important biochemical interaction mechanism within synthetic devices built with nucleic acids and the CRISPR/Cas system. We show that molecular titration in the context of feedback circuits is a suitable mechanism to enhance the emergence of oscillations and bistable behaviors. We consider biomolecular modules that can be inhibited or activated by input monomeric regulators; the regulators compete with constitutive titrating species to determine the activity of their target. By tuning the titration rate and the concentration of titrating species, it is possible to modulate the delay and convergence speed of the transient response, and the steepness and dead zone of the stationary response of the modules. These phenomena favor the occurrence of oscillations when modules are interconnected to create a negative feedback loop; bistability is favored in a positive feedback interconnection. Numerical simulations are supported by mathematical analysis showing that the capacity of the closed loop systems to exhibit oscillations or bistability is structural.

The problem of synthesizing network-decentralized observers arises when several agents, corresponding to the nodes of a network, exchange information about local measurements to asymptotically estimate their own state. The network topology is unknown to the nodes, which can rely on information about their neighboring nodes only. For homogeneous systems, composed of identical agents, we show that a network-decentralized observer can be designed by starting from local observers (typically, optimal filters) and then adapting the gain to ensure overall stability. The smallest eigenvalue of the so-called generalized Laplacian matrix is crucial: stability is guaranteed if the gain is greater than the inverse of this eigenvalue, which is strictly positive if the graph is externally connected. To deal with uncertain topologies, we characterize the worst-case smallest eigenvalue of the generalized Laplacian matrix for externally connected graphs, and we prove that the worst-case graph is a chain. This general result provides a bound for the observer gain that ensures robustness of the network-decentralized observer even under arbitrary, possibly switching, configurations, and in the presence of noise.

Structural properties, independent of specific parameter values, can explain the robustness of biochemical systems. In this paper we consider the framework previously proposed by the authors to assess structural stability of biochemical reaction networks with monotone reaction rates, which considers systems in concentration coordinates, and we show that the results can be applied to systems in reaction coordinates (whose stability was first investigated by Al-Radhawi and Angeli): the same numerical test can be employed to find a polyhedral Lyapunov function and thus certify stability. Under suitable assumptions on the rank of structural matrices, we prove the equivalence between the test performed for the system in concentration coordinates and in reaction coordinates. We finally illustrate the approach by examples.

The paper presents and analyses some nonlinear continuous-time dynamic models of social systems whose members, groups or individuals, may change partners and/or opponents at any time, according to a greedy criterion. The main structural properties of these models, which belong to the class of positive switching systems, are investigated with particular regard to the existence of solutions, their positivity, boundedness and asymptotic behaviour. Simulations show how the cooperative or hostile attitudes of the participants affect their yield.

This paper addresses set invariance properties for linear time-delay systems. More precisely, the first goal of the article is to review known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by linear discrete time-delay difference equations (dDDEs). Secondly, we address the construction of invariant sets in the original state space (also called D -invariant sets) by exploiting the forward mappings. The notion of D -invariance is appealing since it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in some appropriate extended state space model. The present paper contains a sufficient condition for the existence of ellipsoidal D -contractive sets for dDDEs, and a necessary and sufficient condition for the existence of D -invariant sets in relation to linear time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D -invariance of linear dDDEs. In short, it is shown that the convex hull of the union of two or more D -invariant sets is not necessarily D -invariant, while the convex hull of a non-convex D -invariant set is D -invariant.