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The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.
In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.
We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.
We focus on the convergence rate of the alternating direction method of multipliers (ADMM) in a complex domain. First, the complex form of variational inequality (VI) is established by using the Wirtinger calculus technique. Second, the $O(1/K)$ convergence rate of the ADMM in a complex domain is provided. Third, the ADMM in a complex domain is applied to the least absolute shrinkage and selectionator operator (LASSO). Finally, numerical simulations are provided to show that ADMM in a complex domain has the $O(1/K)$ convergence rate and that it has certain advantages compared with the ADMM in a real domain.
In this work, we examine the mathematical analysis and numerical simulation of pattern formation in a subdiffusive multicomponents fractional-reaction-diffusion system that models the spatial interrelationship between two preys and predator species. The major result is centered on the analysis of the system for linear stability. Analysis of the main model reflects that the dynamical system is locally and globally asymptotically stable. We propose some useful theorems based on the existence and permanence of the species to validate our theoretical findings. Reliable and efficient methods in space and time are formulated to handle any space fractional reaction-diffusion system. We numerically present the complexity of the dynamics that are theoretically discussed. The simulation results in one, two and three dimensions show some amazing scenarios.
We investigate the state feedback pinning synchronization of fractional-order complex networks. Based on the stability theory of fractional-order differential systems and state feedback control by a single controller, synchronization conditions for fractional-order complex networks are given. We assume that the coupling matrix is irreducible, and provide a numerical example to illustrate the validity of the proposed conclusions.
The aim of this work is to prove the existence of a positive almost periodic solution to a multifinite time delayed nonlinear differential equation that describes the so-called hematopoiesis model. The approach uses the Hilbert projective metric in a cone. With some additional assumptions, we construct a fixed point theorem to prove the desired existence and uniqueness of the solution.
We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier-Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less than what is suggested by the analytical study; and is in fact comparable to the number of numerically determining Fourier modes, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest.
This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lipγ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.
We introduce a class of discrete-time stochastic processes, called disjunctive processes, which are important for reliable simulations in random iteration algorithms. Their definition requires that all possible patterns of states appear with probability 1. Sufficient conditions for nonhomogeneous chains to be disjunctive are provided. Suitable examples show that strongly mixing Markov chains and pairwise independent sequences, often employed in applications, may not be disjunctive. As a particular step towards a general theory we shall examine the problem arising when disjunctiveness is inherited under passing to a subsequence. An application to the verification problem for switched control systems is also included.
This paper presents an integrated guidance and control (IGC) design method for an unmanned aerial vehicle with static stability which is described by a nonlinear six-degree-of-freedom (6-DOF) model. The model is linearized by using small disturbance linearization. The dynamic characteristics of pitching mode, rolling mode and Dutch rolling mode are obtained by analysing the linearized model. Furthermore, an IGC design procedure is also proposed in conjunction with a proportional–integral–derivative (PID) control method and fuzzy control method. A PID controller is applied in the control loop of the elevator and aileron, and the attitude angle and attitude angular velocity are used as compensation feedback, giving a simple and low-order control law. A fuzzy control method is applied to perform the cross-coupling control of rolling and yawing. Finally, the 6-DOF simulation shows the effectiveness of the developed method.
We present an application of optimal control theory to a simple SIR disease model of avian influenza transmission dynamics in birds. Basic properties of the model, including the epidemic threshold, are obtained. Optimal control theory is adopted to minimize the density of infected birds subject to an appropriate system of ordinary differential equations. We conclude that an optimally controlled seasonal vaccination strategy saves more birds than when there is a low uniform vaccination rate as in resource-limited places.
The inadequacy of the traditional sliding mode variable structure (SMVS) control method for cruise missiles is addressed. An improved SMVS control method is developed, in which the reaching mode segment of the SMVS control is decomposed into an acceleration accessing segment, a speed keeping segment, and a deceleration buffer segment. A time-fuel optimal control problem is formulated as an optimal control problem involving a switched system with unknown switching times and subject to a continuous state inequality constraint. The new design method is developed based on a control parametrization, a time scaling transform and the constraint transcription method. A sequence of approximate optimal parameter selection problems is obtained with fixed switching time points and a canonical state inequality constraint. Each approximate optimal parameter selection problem can be solved effectively by using existing gradient-based optimization techniques. The convergence of these approximate optimal solutions to the true optimal solution is assured. Simulation results show that the proposed method is highly effective. The response speed of the missile under the control law obtained by the proposed method is improved significantly, while the elevator of the missile is constrained to operate within its permitted range.
This paper studies the problem of delay-dependent robust H∞ control for singular systems with multiple delays. Based on a Lyapunov–Krasovskii functional approach, an improved delay-dependent bounded real lemma (BRL) for singular time-delay systems is established without using any of the model transformations and bounding techniques on the cross product terms. Then, by applying the obtained BRL, a delay-dependent condition for the existence of a robust state feedback controller, which guarantees that the closed-loop system is regular, impulse free, robustly stable and satisfies a prescribed H∞ performance index, is proposed in terms of a nonlinear matrix inequality. The explicit expression for the H∞ controller is designed by using linear matrix inequalities and the cone complementarity iterative linearization algorithm. Numerical examples are also given to illustrate the effectiveness of the proposed method.
A queueing network with arbitrary topology, state dependent routing and flow control is considered. Customers may enter the network at any queue and they are routed through it until they reach certain queues from which they may leave the system. The routing is based on local state information. The service rate of a server is controlled based on local state information as well. A distributed policy for routing and service rate control is identified that achieves maximum throughput. The policy can be implemented without knowledge of the arrival and service rates. The importance of flow control is demonstrated by showing that, in certain networks, if the servers cannot be forced to idle, then no maximum throughput policy exists when the arrival rates are not known. Also a model for exchange of state information among neighboring nodes is presented and the network is studied when the routing is based on delayed state information. A distributed policy is shown to achieve maximum throughput in the case of delayed state information. Finally, some implications for deterministic flow networks are discussed.
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