22 research outputs found

    A nonlinear structured population model: Global existence and structural stability of measure-valued solutions

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    This paper is devoted to the study of the global existence and structural stability of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on positive real numbers. In distinction to previous studies, where the L^1 norm was used, we apply the flat metric, similar to the Wasserstein W^1 distance. We argue that stability using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Structural stability and the uniqueness of the weak solutions are shown under the assumption about the Lipschitz continuity of the kinetic functions. The stability result is based on the duality formula and the Gronwall-type argument. Using a framework of mutational equations, existence of solutions to the equations of the model is also shown under weaker assumptions, i.e., without assuming Lipschitz continuity of the kinetic functions

    Coordination of multi-agent systems: stability via nonlinear Perron-Frobenius theory and consensus for desynchronization and dynamic estimation.

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    This thesis addresses a variety of problems that arise in the study of complex networks composed by multiple interacting agents, usually called multi-agent systems (MASs). Each agent is modeled as a dynamical system whose dynamics is fully described by a state-space representation. In the first part the focus is on the application to MASs of recent results that deal with the extensions of Perron-Frobenius theory to nonlinear maps. In the shift from the linear to the nonlinear framework, Perron-Frobenius theory considers maps being order-preserving instead of matrices being nonnegative. The main contribution is threefold. First of all, a convergence analysis of the iterative behavior of two novel classes of order-preserving nonlinear maps is carried out, thus establishing sufficient conditions which guarantee convergence toward a fixed point of the map: nonnegative row-stochastic matrices turns out to be a special case. Secondly, these results are applied to MASs, both in discrete and continuous-time: local properties of the agents' dynamics have been identified so that the global interconnected system falls into one of the above mentioned classes, thus guaranteeing its global stability. Lastly, a sufficient condition on the connectivity of the communication network is provided to restrict the set of equilibrium points of the system to the consensus points, thus ensuring the agents to achieve consensus. These results do not rely on standard tools (e.g., Lyapunov theory) and thus they constitute a novel approach to the analysis and control of multi-agent dynamical systems. In the second part the focus is on the design of dynamic estimation algorithms in large networks which enable to solve specific problems. The first problem consists in breaking synchronization in networks of diffusively coupled harmonic oscillators. The design of a local state feedback that achieves desynchronization in connected networks with arbitrary undirected interactions is provided. The proposed control law is obtained via a novel protocol for the distributed estimation of the Fiedler vector of the Laplacian matrix. The second problem consists in the estimation of the number of active agents in networks wherein agents are allowed to join or leave. The adopted strategy consists in the distributed and dynamic estimation of the maximum among numbers locally generated by the active agents and the subsequent inference of the number of the agents that took part in the experiment. Two protocols are proposed and characterized to solve the consensus problem on the time-varying max value. The third problem consists in the average state estimation of a large network of agents where only a few agents' states are accessible to a centralized observer. The proposed strategy projects the dynamics of the original system into a lower dimensional state space, which is useful when dealing with large-scale systems. Necessary and sufficient conditions for the existence of a linear and a sliding mode observers are derived, along with a characterization of their design and convergence properties

    Optimization Theory and Dynamical Systems: Invariant Sets and Invariance Preserving Discretization Methods

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    Invariant set is an important concept in the theory of dynamical systems and it has a wide range of applications in control and engineering. This thesis has four parts, each of which studies a fundamental problem arising in this field. In the first part, we propose a novel, simple, and unified approach to derive sufficient and necessary conditions, which are referred to as invariance conditions for simplicity, under which four classic families of convex sets, namely, polyhedral, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for linear discrete or continuous dynamical systems. This novel method establishes a solid connection between optimization theory and dynamical systems. In the second part, we propose novel methods to compute valid or largest uniform steplength thresholds for invariance preserving of three classic types of discretization methods, i.e., forward Euler method, Taylor type approximation, and rational function type discretization methods. These methods enable us to find a pre-specified steplength threshold which preserves invariance of a set. The identification of such steplength threshold has a significant impact in practice. In the third part, we present a novel approach to ensure positive local and uniform steplength threshold for invariance preserving on a set when a discretization method is applied to a linear or nonlinear dynamical system. Our methodology not only applies to classic sets, discretization methods, and dynamical systems, but also extends to more general sets, discretization methods, and dynamical systems. In the fourth part, we derive invariance conditions for some classic sets for nonlinear dynamical systems. This part can be considered as an extension of the first part to a more general case

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    Model Order Reduction

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    An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

    Cortical resting state circuits: connectivity and oscillations

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    Ongoing spontaneous brain activity patterns raise ever-growing interest in the neuroscience community. Complex spatiotemporal patterns that emerge from a structural core and interactions of functional dynamics have been found to be far from arbitrary in empirical studies. They are thought to compose the network structure underlying human cognitive architecture. In this thesis, we use a biophysically realistic computer model to study key factors in producing complex spatiotemporal activation patterns. For the first time, we present a model of decreased physiological signal complexity in aging and demonstrate that delays shape functional connectivity in an oscillatory spiking-neuron network model for MEG resting-state data. Our results show that the inclusion of realistic delays maximizes model performance. Furthermore, we propose embracing a datadriven, comparative stance on decomposing the system into subnetworks.Últimamente, el interés de la comunidad científica sobre los patrones de la continua actividad espontanea del cerebro ha ido en aumento. Complejos patrones espacio-temporales emergen a partir de interacciones de un núcleo estructural con dinámicas funcionales. Se ha encontrado que estos patrones no son aleatorios y que componen la red estructural en la que la arquitectura cognitiva humana se basa. En esta tesis usamos un modelo computacional detallado para estudiar los factores clave en producir los patrones emergentes. Por primera vez, presentamos un modelo simplificado de la actividad cerebral en envejecimiento. También demostramos que la inclusión del desfase de transmisión en un modelo para grabaciones magnetoencefalográficas del estado en reposo maximiza el rendimiento del modelo. Para ello, aplicamos un modelo con una red de neuronas pulsantes (’spiking-neurons’) y con dinámicas oscilatorias. Además, proponemos adoptar una posición comparativa basada en los datos para descomponer el sistema en subredes
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