Synchronization is crucial for the correct func-tionality of many natural and man-made complex systems. Inthis work we characterize the formation of synchronizationpatterns in networks of Kuramoto oscillators. Specifically, wereveal conditions on the network weights and structure and onthe oscillators’ natural frequencies that allow the phases of agroup of oscillators to evolve cohesively, yet independently fromthe phases of oscillators in different clusters. Our conditionsare applicable to general directed and weighted networks ofheterogeneous oscillators. Surprisingly, although the oscillatorsexhibit nonlinear dynamics, our approach relies entirely ontools from linear algebra and graph theory. Further, we developa control mechanism to determine the smallest (as measuredby the Frobenius norm) network perturbation to ensure theformation of a desired synchronization pattern. Our procedureallows to constrain the set of edges that can be modified, thusenforcing the sparsity structure of the network perturbation.The results are validated through a set of numerical examples.

Mitochondrial Dynamics (MD) has recently emerged as one of the most interesting topics in biology since the intricate connection between energy production and MD regulates cells development and function. On the other hand, the impairment of such mechanism is strictly related to the emergence of various diseases, among which neurodegenerative disorders. In this work, we provide a simple, yet complete, and well-posed mathematical model to describe the MD and the related phenomena through a population-dynamics approach, together with the ATP-energy turnover, which is an important step to unravel the underlying dynamics of the whole cell system and has a key role in its quality control. With the tools of system theory, we highlight the positiveness of the system and the presence of non-zero equilibria and compute bounds for the involved system state quantities. Furthermore, we consider a situation of impairment in the MD and design a control law, based on input-output linearization and state-feedback control able to allow a damaged system to compensate for the defect and behave as a nominal one. In this scenario, we test two different protocols that could be suggestive for treatment strategies.

A broad class of natural and man-made systems exhibits rich patterns of clustersynchronization in healthy and diseased states, where different groups of interconnectedoscillators converge to cohesive yet distinct behaviors. To provide a rigorous characterizationof cluster synchronization, we study networks of heterogeneous Kuramoto oscillators and wequantify how the intrinsic features of the oscillators and their interconnection paramentersaffect the formation and the stability of clustered configurations. Our analysis shows that clustersynchronization depends on a graded combination of strong intra-cluster and weak inter-clusterconnections, similarity of the natural frequencies of the oscillators within each cluster, andheterogeneity of the natural frequencies of coupled oscillators belonging to different groups. Theanalysis leverages linear and nonlinear controltheoretic tools, and it is numerically validated.

In this paper we discuss a particular case of synchronization involving a finite population of nonlinearly coupled oscillators. We employ a discrete time approximation of the Kuramoto model in order to achieve the coordination of the heading directions of N identical vehicles moving at constant speed in a bidimensional environment; this synchronization model acts as a base for a more complex distributed control, the aim of which is to direct the vehicles towards a target, adjusting their trajectories alongside their formation in the process, while avoiding collisions.

Over the past decade, considerable attention has been devoted to the problem of emergence of synchronization patterns in a network of coupled oscillators, which can be observed in a variety of disciplines, both in the biological and in the engineering fields. In this context, the Kuramoto model is a classical model for describing synchronization phenomena that arise in large-scale systems that exploit local information and interactions. In this work, an extension of such a model is presented, that considers also the spatial distances among the oscillator nodes. In particular, coupling strength and spatial conditions are derived, needed to reach phase cohesiveness and frequency synchronization, both in the scenario when a single population of agents is present and when two different populations interact. These theoretical findings are confirmed by extensive numerical Monte Carlo simulations and statistical analysis.