Imprecise parameters for near-optimal control of stochastic SIV epidemic model

The change of parameters may influence the dynamic behaviors of epidemic diseases. Biological system parameters can also be changed due to diverse uncertainties such as lack of data and errors in the statistical approach. The problem of how to define and decide the optimal-control strategies of epidemic diseases with imprecise parameters deserves further researches. The paper presents a stochastic susceptible, infected, and vaccinated (SIV) system that includes imprecise parameters. Firstly, we give the method of parameter estimates of the SIV model. Then, by using Ekeland's principle and Hamiltonian function, we obtain the sufficient conditions and necessary conditions of near-optimal control of the SIV epidemic model with imprecise parameters. At last, numerical examples prove our theoretical results.     

Monotonicity and bounds for convex stochastic control models

We consider a general convex stochastic control model. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the transition kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on convex stochastic orderings of probability measures. We derive several interesting sufficient conditions of these ordering concepts, where we make also use of the Blackwell ordering. The structural results are illustrated by partially observed control models and Bayesian information models.     

Mathematical analysis of crime dynamics in and out of prisons

Recently, there have been many attempts to develop a mathematical model that captures the nature of crime. One of the successful models has been based on diffusion‐type differential equations that describe how criminals spread in a specific area. Here, we propose a dynamic model that focuses on the effect of interactions between distinct types of criminals. The accumulated criminal records show that serious and minor crimes differ in many measures and are related in a complex way. While some of those who have committed minor crime spontaneously evolve into serious criminals, the transition from minor crime to major crime involves many social factors and has not been fully understood yet. In this work, we present a mathematical model to describe how minor criminals turn into major criminals inside and outside of prisons. The model assumes that a population can be divided into a set of compartments, according to the level of crime and whether arrested or not, and individuals have equal probability to change compartment. The model is design to implement two social effects which respectively have been conceptualized in popular terms “broken windows effect” and “prison as a crime school.” Analysis of the system shows how the crime‐related parameters such as the arrest rate, the period of imprisonment, and the in‐prison contact rate affect the criminal distribution at equilibrium. Without proper control of contact between prisoners, the longer imprisonment rather increases occurrence of serious crimes in society. An optimal allocation of the police resources to suppress crimes is also discussed.     

Expectation-Stock Dynamics in Multi-Agent Fisheries

In this paper we consider a game-theoretic dynamic model describing the exploitation of a renewable resource. Our model is based on a Cournot oligopoly game where n profit-maximizing players harvest fish and sell their catch on m markets. We assume that the players do not know the law governing the reproduction of the resource. Instead they use an adaptive updating scheme to forecast the future fish stock. We analyze the resulting dynamical system which describes how the fish population and the forecasts (expectations) of the players evolve over time. We provide results on the existence and local stability of steady states. We consider the set of initial conditions which give non-negative trajectories converging to an equilibrium and illustrate how this set can be characterized. We show how such sets may change as some structural parameters of our model are varied and how these changes can be explained. This paper extends existing results in the literature by showing that they also hold in our two-dimensional framework. Moreover, by using analytical and numerical methods, we provide some new results on global dynamics which show that such sets of initial conditions can have complicated topological structures, a situation which may be particularly troublesome for policymakers.     

Stability of periodically switched discrete-time linear singular systems

In this paper we present some necessary and sufficient conditions for the stability of periodically switched discrete-time linear index-1 singular system, (PSSS). In particular, it is proved that, if at least one subsystem of a PSSS is asymptotically stable, then there is a switching rule, so that the whole system is also uniformly exponentially stable. Furthermore, for a periodically switched control system with no stable subsystems, there exist a switching rule and feedback matrices, such that the obtained PSSS is uniformly exponentially stable.     

Fixed-order robust compensators via min-max principle

An important (some say, the major) reason for using feedback control is the presence of uncertain parameters which are a natural part of any real dynamical model. In this paper, we consider uncertain constant parameters in a time-invariant linear plant and announce some new results concerning robust compensator synthesis. Using the min-max principle, we derive necessary conditions for fixed-order linear robust controllers assuring asymptotic stability or relative stability. These necessary conditions are an extension of the Lagrange multiplier method. This is achieved using a cost function based on the inverse of the so-called critical constraint. We present both matrix and polynomial versions; the latter allows controllers of fixed structure. We suggest a probability-one homotopy algorithm and solve some examples from the literature.The authors wish to thank Professor R. Bental, Faculty of Industrial Engineering at the Technion, for his suggestion to replace the cost function based on the inverse critical polynomial by a logarithmic function.     

Emergent behaviors of a thermodynamic Cucker-Smale flock with a time-delay on a general digraph

We present emergent flocking dynamics of a thermodynamic Cucker-Smale (TCS) flock on a general digraph with spanning trees under the effect of communication time-delays. The TCS model describes a temporal evolution of mechanical and thermodynamic observables such as position, velocity and temperature of CS particles. In this paper, we study how variations in mechanical and thermodynamic variables can decay to zero along a time-independent network with position dependent weights from initial state configuration. For this, we provide a sufficient framework for a mechanical and thermodynamical flocking in terms of initial configuration, network topology, and system parameters. We also present several numerical examples and compare them with analytical results.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Impulsive control for differential systems with delay

This paper deals with the impulsive control for a class of differential systems with delay. Using Lyapunov functions and the comparison principle, we present some sufficient conditions for the asymptotic stability and exponential stability of impulsive control systems with delay. Moreover, we give an estimate of the upper bound of impulse interval. The results in this paper extend and improve the earlier publications. Copyright © 2012 John Wiley & Sons, Ltd.     

Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations

We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the solution is specified at an initial moment of time). We then show how our method applies in fluid mechanics. More specifically, we present a complete solution to the problem of long waves run-up in inclined bays of arbitrary shape with nonzero initial velocity.     

Structural results for partially observed control models

A general partially observed control model with discrete time parameter is investigated. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the observation kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on two multivariate stochastic orderings: theTP ₂ ordering and the Blackwell ordering.Dedicated to Prof. Dr. K. Hinderer on the occassion of his 60^th birthday     

基于观测器的模糊时滞系统指数稳定的一种设计方法

研究了一类模糊时滞系统的指数稳定问题.首先利用T-S模型对非线性不确定性时滞系统进行建模,在此基础上设计了基于观测器的模糊状态反馈控制器,通过巧妙选取Lyapunov函数给出了模糊闭环时滞系统的条件及稳定裕度且模糊反馈增益和模糊观测器增益可通过求解线性矩阵不等式获得.     

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.     

Quality of service statistics over heterogeneous networks: Analysis and applications

Heterogeneous wireless/wired networks and ubiquitous environments are gaining ever more attention by research community. To properly control and manage such puzzles a deep knowledge of quality of service parameters is needed and, therefore, a complete and robust performance assessment is necessary. This paper deals with a performance evaluation and measurement of a number of heterogeneous end-to-end paths taking into account a wide range of statistics. To study the behavior of QoS parameters, an active measurement approach has been introduced for the analysis of properties we called (i) concise statistics (mean, standard deviation, inter quantile range, minimum, maximum, and median) and (ii) detailed statistics (Probability Density Function, Auto-correlation Function, Entropy, Complementary Cumulative Distribution Function, and Bivariate Probability Density Function). We show how, thanks to this view on QoS statistics, a more complete understanding of QoS parameters behavior is possible. In addition, we show how the measured statistics can be fruitfully used in the context of network control and management. More precisely, we present two proof of concepts regarding frameworks for QoS-based anomaly detection and for QoS-based identification of network elements.     

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.     

A parameter estimation scheme for a class of sequential hybrid systems

It may happen that the equations governing the response of dynamical systems have some parameters whose values may not be known a priori and have to be obtained using parameter estimation schemes. In this article, we present a parameter estimation scheme for a class of sequential hybrid systems. By hybrid systems, we refer to those systems whose response is described by different governing equations corresponding to various regimes/modes of operation along with some criteria to switch between the same. In a sequential hybrid system, the different modes are arranged in a specific sequence and the system can switch from a given mode to either the previous mode or the following mode in this sequence. Here, we consider those systems whose governing equations consist of ordinary differential equations and algebraic equations. The conditions for switching between the various modes (referred to as transition conditions) are in the form of linear inequalities involving the system output. We shall first consider the case where the transition conditions are known completely. We present a parameter update scheme along with sufficient conditions that will guarantee bounded parameter estimation errors. Then, we shall consider the case where the transition conditions are not known in the sense that some parameters in these conditions are not known. We present a parameter estimation scheme for this case. We illustrate the performance of the parameter estimation scheme in both cases with some examples.     

Stability analysis for networked control systems with sampling,transmission protocols and input delays

In this paper, the stability problem is investigated for networked control systems. Input delays and multiple communication imperfections containing time-varying transmission intervals and transmission protocols are considered. A unified framework based on the hybrid systems with memory is proposed to model the whole networked control system. Hybrid systems with memory are used to model hybrid systems affected by delays and permit multiple jumps at a jumping instant. The stability analysis depends on the Lyapunov–Krasovskii functional approaches for hybrid systems with memory and the proposed stability theorem does not need strict decrease of the Lyapunov–Krasovskii functional during jumps. Based on the developed stability theorems, stability conditions for networked control systems are established. An explicit formula is given to compute the maximal allowable transmission interval. In the special case that the networked control system contains linear dynamics, an explicit Lyapunov functional is constructed and stability conditions in terms of linear matrix inequalities (LMI) are proposed. Finally, an example of a chemical batch reactor is given to illustrate the effectiveness of the proposed results.     

关于带混合边界条件的半线性反应扩散系统解的整体存在性问题

本文利用Ｇｒｏｎｗａｌ不等式及对偶技巧，研究了一类带混合边界条件的半线性反应扩散系统解的整体存在性问题，给出一些控制条件，突破了以往加在“Ｍｏｒ ｇａｎ”和上的线性控制，得到解的整体存在性，很大程度上改进和推广了以往结果，使该问题的研究前进了一大步     

Traffic flow modelling with junctions

Motivated by the modelling of a roundabout, we are led to study the traffic on a road with points of entry and exit. In this note, we would like to describe the modellisation of a junction and solve the Riemann problem for such a model. More precisely, between each point of discontinuity we use a multi-class extension of the LWR model to describe the evolution of the density of the vehicles, the ‘multi-class’ approach being used in order to distinguish the vehicles after their origin and destination. Then, we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles. In the case of the one-T road we obtain a result of existence and uniqueness. This first step allows us to obtain a similar result for the n-T road. We describe these results and also some properties of the obtained solutions, in order to see how long this model is valid.     

Controllability Properties of Linear Mean-Field Stochastic Systems

We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the control of mean-field systems, we provide a (conditional) viability criterion for approximate null-controllability. In the classical diffusion framework, approximate and approximate null-controllability are equivalent. This is no longer the case for mean-field systems. We provide sufficient (algebraic) invariance conditions implying approximate null-controllability. We also present a general class of systems for which our criterion is equivalent to approximate null-controllability property. We also introduce some rank conditions under which approximate and approximate null-controllability are equivalent. Several examples and counter-examples as well as a partial algorithm are provided.