Pinning dynamic systems of networks with Markovian switching couplings and controller-node set

In this paper, we study pinning control problem of coupled dynamical systems with stochastically switching couplings and stochastically selected controller-node set. Here, the coupling matrices and the controller-node sets change with time, induced by a continuous-time Markovian chain. By constructing Lyapunov functions, we establish tractable sufficient conditions for exponentially stability of the coupled system. Two scenarios are considered here. First, we prove that if each subsystem in the switching system, i.e. with the fixed coupling, can be stabilized by the fixed pinning controller-node set, and in addition, the Markovian switching is sufficiently slow, then the time-varying dynamical system is stabilized. Second, in particular, for the problem of spatial pinning control of network with mobile agents, we conclude that if the system with the average coupling and pinning gains can be stabilized and the switching is sufficiently fast, the time-varying system is stabilized. Two numerical examples are provided to demonstrate the validity of these theoretical results, including a switching dynamical system between several stable sub-systems, and a dynamical system with mobile nodes and spatial pinning control towards the nodes when these nodes are being in a pre-designed region.Comment: 9 pages; 3 figure

Adaptive Control of Dynamically Complex Networks with Saturation Couplings

This paper considers the control problem of dynamically complex networks with saturation couplings. Two novel control schemes in terms of adaptive control are presented to deal with such saturation couplings. Based on the robust idea, the underlying complex network is firstly transformed into a strongly connected network having time-varying uncertainty. However, the upper bound of uncertainty is unknown. Because of such an unavailable bound, a kind of adaptive controller added to each node is proposed such that the closed-loop auxiliary network is uniformly bounded. In particular, the original system states are asymptotically stable. Moreover, in order to avoid adding the desired controller to every node, another different kind of adaptive controller based on the pinning control idea is proposed. Compared with the former method, it is only applied to a part of nodes and has a good operability. Finally, a numerical example is provided to show the effectiveness of our results

Impulsive Control of Dynamical Networks

Dynamical networks (DNs) consist of a large set of interconnected nodes with each node being a fundamental unit with detailed contents. A great number of natural and man-made networks such as social networks, food networks, neural networks, WorldWideWeb, electrical power grid, etc., can be effectively modeled by DNs. The main focus of the present thesis is on delay-dependent impulsive control of DNs. To study the impulsive control problem of DNs, we firstly construct stability results for general nonlinear time-delay systems with delayed impulses by using the method of Lyapunov functionals and Razumikhin technique. Secondly, we study the consensus problem of multi-agent systems with both fixed and switching topologies. A hybrid consensus protocol is proposed to take into consideration of continuous-time communications among agents and delayed instant information exchanges on a sequence of discrete times. Then, a novel hybrid consensus protocol with dynamically changing interaction topologies is designed to take the time-delay into account in both the continuous-time communication among agents and the instant information exchange at discrete moments. We also study the consensus problem of networked multi-agent systems. Distributed delays are considered in both the agent dynamics and the proposed impulsive consensus protocols. Lastly, stabilization and synchronization problems of DNs under pinning impulsive control are studied. A pinning algorithm is incorporated with the impulsive control method. We propose a delay-dependent pinning impulsive controller to investigate the synchronization of linear delay-free DNs on time scales. Then, we apply the pinning impulsive controller proposed for the delay-free networks to stabilize time-delay DNs. Results show that the delay-dependent pinning impulsive controller can successfully stabilize and synchronize DNs with/without time-delay. Moreover, we design a type of pinning impulsive controllers that relies only on the network states at history moments (not on the states at each impulsive instant). Sufficient conditions on stabilization of time-delay networks are obtained, and results show that the proposed pinning impulsive controller can effectively stabilize the network even though only time-delay states are available to the pinning controller at each impulsive instant. We further consider the pinning impulsive controllers with both discrete and distributed time-delay effects to synchronize the drive and response systems modeled by globally Lipschitz time-delay systems. As an extension study of pinning impulsive control approach, we investigate the synchronization problem of systems and networks governed by PDEs

Biological Protein Patterning Systems across the Domains of Life: from Experiments to Modelling

Distinct localisation of macromolecular structures relative to cell shape is a common feature across the domains of life. One mechanism for achieving spatiotemporal intracellular organisation is the Turing reaction-diffusion system (e.g. Min system in the bacterium Escherichia coli controlling in cell division). In this thesis, I explore potential Turing systems in archaea and eukaryotes as well as the effects of subdiffusion. Recently, a MinD homologue, MinD4, in the archaeon Haloferax volcanii was found to form a dynamic spatiotemporal pattern that is distinct from E. coli in its localisation and function. I investigate all four archaeal Min paralogue systems in H. volcanii by identifying four putative MinD activator proteins based on their genomic location and show that they alter motility but do not control MinD4 patterning. Additionally, one of these proteins shows remarkably fast dynamic motion with speeds comparable to eukaryotic molecular motors, while its function appears to be to control motility via interaction with the archaellum. In metazoa, neurons are highly specialised cells whose functions rely on the proper segregation of proteins to the axonal and somatodendritic compartments. These compartments are bounded by a structure called the axon initial segment (AIS) which is precisely positioned in the proximal axonal region during early neuronal development. How neurons control these self-organised localisations is poorly understood. Using a top-down analysis of developing neurons in vitro, I show that the AIS lies at the nodal plane of the first non-homogeneous spatial harmonic of the neuron shape while a key axonal protein, Tau, is distributed with a concentration that matches the same harmonic. These results are consistent with an underlying Turing patterning system which remains to be identified. The complex intracellular environment often gives rise to the subdiffusive dynamics of molecules that may affect patterning. To simulate the subdiffusive transport of biopolymers, I develop a stochastic simulation algorithm based on the continuous time random walk framework, which is then applied to a model of a dimeric molecular motor. This provides insight into the effects of subdiffusion on motor dynamics, where subdiffusion reduces motor speed while increasing the stall force. Overall, this thesis makes progress towards understanding intracellular patterning systems in different organisms, across the domains of life