The mean-field computation in a supermarket model with server multiple vacations

While vacation processes are considered to be ordinary behavior for servers, the study of queueing networks with server vacations is limited, interesting, and challenging. In this paper, we provide a unified and effective method of functional analysis for the study of a supermarket model with server multiple vacations. Firstly, we analyze a supermarket model of N identical servers with server multiple vacations, and set up an infinite-dimensional system of differential (or mean-field) equations, which is satisfied by the expected fraction vector, in terms of a technique of tailed equations. Secondly, as N→?∞ we use the operator semigroup to provide a mean-field limit for the sequence of Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Thirdly, we provide an effective algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model, including the mean of stationary queue length in any server and the expected sojourn time that any arriving customer spends in this system. Finally, we use some numerical examples to analyze how the performance measures depend on some crucial factors of this supermarket model. Note that the method of this paper will be useful and effective for performance analysis of complicated supermarket models with respect to resource management in practical areas such as computer networks, manufacturing systems and transportation networks.     

Block-structured supermarket models

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging.This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinite-dimensional system of differential vector equations satisfied by the expected fraction vector, where the invariance of environment factors is given as an important result. (2) Introducing the phase-type structure to the operator semigroup and to the mean-field limit, and a Lipschitz condition can be established by means of a unified matrix-differential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times. Thus the results of this paper give new highlight on understanding influence of non-Poisson inputs and of non-exponential service times on performance measures of more general supermarket models.     

Fast Jackson Networks with Dynamic Routing

A new class of models of queueing networks with load-balanced dynamic routing is considered. We propose a sufficient condition for positive recurrence of the arising Markov process and a limiting mean-field approximation where the process becomes deterministic and is described by a system of nonlinear ordinary differential equations.     

On application of the implemented semigroup to a problem arising in optimal control

In this paper we show that the concept of an implemented semigroup provides a natural mathematical framework for analysis of the infinite-dimensional differential Lyapunov equation. Lyapunov equations of this form arise in various system-theoretic and control problems with a finite time horizon, infinite-dimensional state space and unbounded operators in the mathematical model of the system. The implemented semigroup approach allows us to derive a necessary and sufficient condition for the differential Lyapunov equation with an unbounded forcing term to admit a bounded solution in a suitable space. Whilst our focus is on the differential Lyapunov equation, we show that the same framework is also appropriate for the algebraic version of this equation. As an application we show that the approach can be used to solve a simple decoupling problem arising in optimal control. The problem of infinite time admissibility of the control operator and an infinite-dimensional version of the Lyapunov theorem serve as additional illustrations.     

Sufficient conditions for existence of a fixed point in stochasticreward net-based iterative models

Stochastic Petri net models of large systems that are solved by generating the underlying Markov chain pose the problem of largeness of the state-space of the Markov chain. Hierarchical and iterative models of systems have been used extensively to solve this problem. A problem with models which use fixed-point iteration is the theoretical proof of the existence, uniqueness and convergence of the fixed-point equations, which still remains an “art”. In this paper, we establish conditions, in terms of the net structure and the characteristics of the iterated variables, under which existence of a solution is guaranteed when fixed-point iteration is used in stochastic Petri nets. We use these conditions to establish the existence of a fixed point for a model of a priority scheduling system, at which tasks may arrive according to a Poisson process or due to spawning or conditional branching of other tasks in the system     

Solving analysis and synthesis problems for a spatially two-dimensional distributed object represented with an infinite system of differential equations

We prove theorems that define an algorithm for passing from differential equations with partial derivatives with respect to two spatial variables and time to an infinite-dimensional system of ordinary differential equations in Cauchy form. We study the convergence of resulting solutions and show that it is possible to pass from an infinite system in Cauchy form to a finite one, which opens up the possibilities to use state space methods for controller design in distributed systems. Based on the quadratic quality criterion, we design a controller for the case when controlling influences are applied at the boundaries of the control object. We obtain the solution of this system analysis problem in the form of Fourier series with respect to spatial variables based on orthogonal systems of trigonometric functions and Bessel functions.     

Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT

In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.     

Application of fixed point-collocation method for solving an optimal control problem of a parabolic–hyperbolic free boundary problem modeling the growth of tumor with drug application

In this paper, employing a fixed point-collocation method, we solve an optimal control problem for a model of tumor growth with drug application. This model is a free boundary problem and consists of five time-dependent partial differential equations including three different first-order hyperbolic equations describing the evolution of cells and two second-order parabolic equations describing the diffusion of nutrient and drug concentration. In the mentioned optimal control problem, the concentration of nutrient and drug is controlled using some control variables in order to destroy the tumor cells. In this study, applying the fixed point method, we construct a sequence converging to the solution of the optimal control problem. In each step of the fixed point iteration, the problem changes to a linear one and the parabolic equations are solved using the collocation method. The stability of the method is also proved. Some examples are considered to illustrate the efficiency of method.     

Existence results for the three-point impulsive boundary value problem involving fractional differential equations

In this paper, we consider the existence of solutions for a class of three-point boundary value problems involving nonlinear impulsive fractional differential equations. By use of Banach’s fixed point theorem and Schauder’s fixed point theorem, some existence results are obtained.     

Periodic value problems of differential systemson infinite-dimensional spaces and applications to differential geometry

Periodicity questions of differential systems on infinite-dimensional Hilbert spaces arestudied via a new methodology which is based on Fan-Knaster-Kuratowski-Mazurkiewicz theorem. We obtain in this way an alternative, to the classical fixed point theory, approach to the study of such type of problems with various applications on issues of mathematical analysis and differential geometry. Two examples of such applications are included.     

Desynchronization and inhibition of Kuramoto oscillators by scalar mean-field feedback

Motivated by neuroscience applications, and in particular by the deep brain stimulation treatment for Parkinson’s disease, we have recently derived a simplified model of an interconnected neuronal population under the effect of its mean-field proportional feedback. In this paper, we rely on that model to propose conditions under which proportional mean-field feedback achieves either oscillation inhibition or desynchronization. More precisely, we show that for small natural frequencies, this scalar control signal induces an inhibition of the collective oscillation. For the closed-loop system, this situation corresponds to a fixed point which is shown to be almost globally asymptotically stable in the fictitious case of zero natural frequencies and all-to-all coupling and feedback. In the case of an odd number of oscillators, this property is shown to be robust to small natural frequencies and heterogencities in both the coupling and feedback topology. On the contrary, for large natural frequencies, we show that scalar proportional mean-field feedback is able to induce desynchronization. After having recalled a formal definition for desynchronization, we show how it can be induced in a network of originally synchronized oscillators.     

A paradigm of ill-posedness with respect to time delays

The authors present a simple general model of an infinite-dimensional control system which is stabilized by a velocity feedback, but then destabilized by “small” or “large” time delays in the feedback. This model is in some sense generic for boundary-stabilized conservative hyperbolic linear partial differential equations     

A note on boundary value problems for a coupled system of fractional differential equations

In this work, we establish sufficient conditions for the existence of solutions to a general class of multi-point boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann–Liouville sense. By means of a fixed point theorem, existence results for the solutions are established. We include an example to show the applicability of our results.     

Internal model based tracking and disturbance rejection for stable well-posed systems

In this paper we solve the tracking and disturbance rejection problem for infinite-dimensional linear systems, with reference and disturbance signals that are finite superpositions of sinusoids. We explore two approaches, both based on the internal model principle. In the first approach, we use a low gain controller, and here our results are a partial extension of results by Hämäläinen and Pohjolainen. In their papers, the plant is required to have an exponentially stable transfer function in the Callier-Desoer algebra, while in this paper we only require the plant to be well-posed and exponentially stable. These conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable. Our second approach concerns the case when the second component of the plant transfer function (from control input to tracking error) is positive. In this case, we identify a very simple stabilizing controller which is again an internal model, but which does not require low gain. We apply our results to two problems involving systems modeled by partial differential equations: the problem of rejecting external noise in a model for structure/acoustics interactions, and a similar problem for two coupled beams.     

Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions

In the this paper, we establish sufficient conditions for the existence and nonexistence of positive solutions to a general class of integral boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. Our analysis rely on Banach fixed point theorem, nonlinear differentiation of Leray-Schauder type and the fixed point theorems of cone expansion and compression of norm type. As applications, some examples are also provided to illustrate our main results.     

Input-to-state stability of infinite-dimensional control systems

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.     

Sampled-data H8 state-feedback control of systems with state delays

Sliding mode based feedback control has long been recognized as a powerful, yet easy-to-implement, control method to counteract non-vanishing external disturbances and unmodelled dynamics. Recently, research attention has focused on the development of sliding mode feedback control methods for various classes of infinite-dimensional systems. However, the existing methods are based on the assumption that distributed sensing and actuation is available, which significantly restricts their applicability to distributed process control applications. In this work, a sliding mode output feedback control method is developed for a class of linear infinite-dimensional systems with finite-dimensional unstable part using finite-dimensional sensing and actuation. Modal decomposition is initially used to decompose the original infinite-dimensional system into an interconnection of a finite-dimensional (possibly unstable) system and an infinite-dimensional stable system. Then, a sliding mode-based stabilizing state feedback controller is constructed on the basis of the finite-dimensional system. Subsequently, an infinite-dimensional Luenberger state observer, which utilizes a finite number of measurements, is constructed to provide estimates of the state of the infinite-dimensional system. Finally, an output feedback controller design is completed by coupling the infinite-dimensional Luenberger state observer and the sliding mode-based state feedback controller. Implementation, performance and robustness issues of the sliding-mode output feedback controller are illustrated in a simulation study of a distributed parameter system governed by the linearization around the spatially-uniform steady-state solution of the Kuramoto–Sivashinsky partial differential equation with periodic boundary conditions.     

Boundary control for a constrained two-link rigid–flexible manipulator with prescribed performance

In this paper, we consider a boundary control problem for a constrained two-link rigid–flexible manipulator. The nonlinear system is described by hybrid ordinary differential equation–partial differential equation (ODE–PDE) dynamic model. Based on the coupled ODE–PDE model, boundary control is proposed to regulate the joint positions and eliminate the elastic vibration simultaneously. With the help of prescribed performance functions, the tracking error can converge to an arbitrarily small residual set and the convergence rate is no less than a certain pre-specified value. Asymptotic stability of the closed-loop system is rigorously proved by the LaSalle's Invariance Principle extended to infinite-dimensional system. Numerical simulations are provided to demonstrate the effectiveness of the proposed controller.     

Iterative solution of algebraic matrix Riccati equations in open loop Nash games

In this note we study a fixed point iteration approach to solve algebraic Riccati equations as they appear in general two player Nash differential games on an infinite time horizon, where the information structure is of open loop type. We obtain conditions for existence and uniqueness of non‐negative solutions. The performance of the numerical algorithm is shown in an example. Copyright © 2005 John Wiley & Sons, Ltd.     

-output feedback of infinite-dimensional systems via approximation

As in the finite-dimensional case, a state-space based controller for the infinite-dimensional disturbance-attenuation problem may be calculated by solving two Riccati equations. These operator Riccati equations can rarely be solved exactly. We approximate the original infinite-dimensional system by a sequence of finite-dimensional systems. The solutions to the corresponding finite-dimensional Riccati equations are shown to converge to the solution of the infinite-dimensional Riccati equations. Furthermore, the corresponding finite-dimensional controllers yield performance arbitrarily close to that obtained with the infinite-dimensional controller.