Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

Self-organized criticality and partial synchronization in an evolving network

We describe an evolving network type model of stock market, and present numerical results. Recent works on Self-organized criticality in pulse coupled relaxation oscillators have shown SOC to be related to frustrated attempts of the system to synchronise. Other works have shown the emergence of a self-organized control parameter which feeds back onto an order parameter, causing avalanches which separate periods of stasis. We define an all-to-all network of connected spins which also have an attached fitness. We find a self-organized fitness threshold given by the mean fitness, which separates a solid type state from a gaseous type state. We also find a self-organized control parameter given by the fitness deviation. Approach of this parameter to zero causes `avalanches' of partial synchronizations which occur on all sizes, separated by periods of stasis, and re-order the system.     

Analytic form of optimal boundary controls of transverse vibrations of a rod consisting of several parts with different densities and elasticities but with the same impedances

We consider the problem of optimal boundary displacement control (control minimizing the boundary energy functional) of an elastic rod consisting of several parts with different densities and elasticities but with the same impedances. We obtain a closed-form expression for the optimal control bringing the rod from an arbitrarily given initial state to an arbitrarily given terminal state in a given time T.     

A Markov decision problem in a risk model with interest rate and Markovian environment

We consider the compound binomial model in a Markovian environment presented by Cossette et al.(2004). We modify the model via assuming that the company receives interest on the surplus and a positive real-valued premium per unit time, and introducing a control strategy of periodic dividend payments. A Markov decision problem arises and the control objective is to maximize the cumulative expected discounted dividends paid to the shareholders until ruin minus a discounted penalty for ruin. We show that under the absence of a ceiling of dividend rates the optimal strategy is a conditional band strategy given the current state of the environment process. Under the presence of a ceiling for dividend rates, the character of the optimal control strategy is given. In addition, we offer an algorithm for the optimal strategy and the optimal value function.Numerical results are provided to illustrate the algorithm and the impact of the penalty.     

Impulse Control of One-Dimensional Ito Diffusions with an Expected and a Pathwise Ergodic Criterion

We consider the problem of controlling a general one-dimensional Ito diffusion by means of an impulse control process. The objective is to minimise a long-term expected criterion as well as a long-term pathwise criterion that penalise both deviations of the state process from a given nominal point and the use of impulsive control effort. In particular, each time the controller deploys an impulse to reposition the system's state, a fixed cost and a cost proportional to the impulse's size are incurred. We solve the resulting optimisation problems and we provide an explicit characterisation of an optimal control strategy under general assumptions. The control of a foreign exchange rate or an inflation rate presents a potential application of the model that we study.     

Critical-time boundary displacement control at two ends of an inhomogeneous rod in the case of coincidence of the wave propagation times through each of two homogeneous parts

We study a boundary displacement control at two ends of an inhomogeneous rod that has two parts of distinct densities and elasticities in the case of coinciding wave propagation times over these parts. The control acts on a time interval of critical length. We obtain a closed analytical form of the boundary displacement control bringing the rod in critical time from the initial state of rest into a given terminal state specified by given terminal displacement and terminal velocity.     

A new approach to optimum state estimation in optimum system control

We consider optimum system control, in which the controller has imperfect information on the availability of the state of the plant to control. The system is controlled via an estimation of the state. We propose a general estimation criterion, which should permit determination of the optimal estimator for a given dynamics and a given performance cost. This approach includes the separation theorem as a special case.     

Solution of the control problem for a linear descriptor system with rectangular coefficient matrix

We pose a control problem for linear stationary algebro-differential system whose coefficients are rectangular matrices. We obtain a controllability criterion for the partially given output function and the boundary values of the state function. We construct a control function and a state function under minimal smoothness requirements on the output function.     

Increasing and Lipschitz continuous minimizers in one-dimensional linear-convex systems without constraints: The continuous and the discrete case

We consider a stochastic control model with linear transition law and arbitrary convex cost functions, a far-reaching generalization of the familiar linear quadratic model. Firstly conditions are given under which the continuous state version has minimizersf _n at each stagen which are increasing and in addition either right continuous or continuous or Lipschitz continuous with explicitly given Lipschitz constant. For the computationally important discrete version we verify some analogous properties under stronger assumptions.     

Mathematical models of restoration and control of a single species with Allee effect

According to the initial density of a single species with Allee effect and corresponding management strategy, three kinds of mathematical models are presented to describe the evolutionary process of the species by impulsive differential equations. When the initial density of the species is larger than economic injury level (EIL) (or economical threshold, ET), impulsive harvest control is considered in a finite time to decrease the population of the species. The feasibility of the impulsive harvest control in a finite time is given by the existence of solution of the model with initial and boundary value problem. When the initial density of the species is less than EIL (or ET), the model with state feedback control is formulated according to the state of the species. The existence and stability of periodic solution of the model with state feedback control are discussed. When the initial density of the species is less than the Allee threshold and the species tends to extinction, the model with impulsive release at fixed moments is presented to study the restoration of the species. The conditions for the feasibility of periodic restoration of the species are given. Finally, some discussions are given.     

Variance reduction for Markov chain processes using state space evaluation for control variates

We develop a method for reducing variance in Monte Carlo simulation of Markov chain processes based on extracting accurate control variates from state space evaluation functions. An example is given in the form of a simple combat model, where the net variance reduction (adjusted for additional calculation) is larger than a factor of 80. We also indicate how our algorithm may be applied to discrete event simulations and system dynamic models with discrete random events.     

Optimal control of Markovian jump processes with partial information and applications to a parallel queueing model

We consider a stochastic control problem over an infinite horizon where the state process is influenced by an unobservable environment process. In particular, the Hidden-Markov-model and the Bayesian model are included. This model under partial information is transformed into an equivalent one with complete information by using the well-known filter technique. In particular, the optimal controls and the value functions of the original and the transformed problem are the same. An explicit representation of the filter process which is a piecewise-deterministic process, is also given. Then we propose two solution techniques for the transformed model. First, a generalized verification technique (with a generalized Hamilton–Jacobi–Bellman equation) is formulated where the strict differentiability of the value function is weaken to local Lipschitz continuity. Second, we present a discrete-time Markovian decision model by which we are able to compute an optimal control of our given problem. In this context we are also able to state a general existence result for optimal controls. The power of both solution techniques is finally demonstrated for a parallel queueing model with unknown service rates. In particular, the filter process is discussed in detail, the value function is explicitly computed and the optimal control is completely characterized in the symmetric case.     

A method for solving boundary value problems for nonlinear control systems in the class of discrete controls

We suggest an algorithm for constructing discrete control functions. This algorithm is sufficiently convenient for numerical implementation and, for a wide class of nonlinear systems of ordinary differential equations, provides the passage from the initial state into an arbitrary given or arbitrarily small neighborhood of a given terminal state. We obtain a constructive criterion for the choice of terminal states and discretization steps for which the passage is possible with regard of the constraints imposed on the control and the phase coordinates. We consider an interorbital flight problem, for which we carry out numerical simulation.     

Optimization of the boundary displacement control of vibrations of a rod consisting of two dissimilar parts

We optimize the boundary displacement control that is applied at one end of a rod consisting of two dissimilar parts and brings the rod vibrations from a given initial state to a given terminal state for the case in which the other end of the rod is fixed.     

Boundary control by an elastic force at one end of an inhomogeneous rod with the other end free for the case in which the wave travels either of the homogeneous parts in the same time

We study the boundary control by an elastic force at one end of an inhomogeneous rod that has two parts of different densities and elasticities and whose other end is free. The case in which the wave travels either of the homogeneous parts in the same time is considered. We present a closed-form analytical expression for the boundary control by an elastic force that brings the rod from the initial quiescent state to a given terminal state specified by given terminal displacement and terminal velocity.     

Boundary control in critical time by elastic forces applied at two ends of an inhomogeneous rod for the case in which the wave propagation time over each of the inhomogeneity parts is the same

We study boundary control in critical time by elastic forces at two ends of an inhomogeneous rod consisting of two parts of distinct densities and elasticities for the case in which the wave propagation time over each of these parts is the same. We present a closed-form expression for the boundary control by elastic forces bringing the originally quiescent stick into a given terminal state specified by a given terminal displacement and a given terminal velocity in a given critical time.     

A partial Hamiltonian approach for current value Hamiltonian systems

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.     

On the Existence of a Lipschitz Feedback Control in a Control Problem with State Constraints

We consider a nonlinear control system with state constraints given as a solution set for a finite system of nonlinear inequalities. The problem of constructing a feedback control that ensures the viability of trajectories of the closed system in a small neighborhood of the boundary of the state constraints is studied. Under some assumptions, the existence of a feedback control in the form of a Lipschitz function of the state of the system is proved.     

Optimal State Estimation in Networked Systems with Asynchronous Communication Channels and Switched Sensors

We study a linear discrete-time partially observed system perturbed by a white noise. The observations are transmitted to the estimator via communication channels with irregular transmission times. Various measurement signals may incur independent delays, arrive at the estimator out of order, and be lost or corrupted. The estimator is given a dynamic control over the sensors by taking part in producing the signals to be sent via the channels. The minimum variance state estimate and the optimal sensor control strategy are obtained. Ideas of model predictive control are applied to derive a nonoptimal but implementable in real time method for sensor control This work was supported by the Australian Research Council.     

Low-cost control problems on perforated and non-perforated domains

We study the homogenization of a class of optimal control problems whose state equations are given by second order elliptic boundary value problems with oscillating coefficients posed on perforated and non-perforated domains. We attempt to describe the limit problem when the cost of the control is also of the same order as that describing the oscillations of the coefficients. We study the situations where the control and the state are both defined over the entire domain or when both are defined on the boundary.