Optimal control of the genital herpes epidemic

A spatial stochastic model to study the optimal control of the epidemic is introduced. The equilibrium states of the epidemic model are found. The stability and instability in linear approximation of this model are investigated. The optimal control of the unstable equilibrium states is studied. The control functions are obtained from the conditions that ensure the optimal stabilization of these states. Graphical and numerical simulation of the obtained results are presented.     

Optimum control of bobsled steering

Minimum-time and smooth-steering control algorithms are developed for bobsled optimal control. Numerical solutions are obtained both for one-curve optimal control and whole-course piecewise optimal control with application to realistic three-dimensional track surface shapes. Specific results are calculated for the Lillehammer Olympic Track.     

Finite element approximation of elliptic control problems with constraints on the gradient

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.     

Maximum principles for control systems described by measure functional differential equations

Three optimal control problems involving measure functional differential equations are considered. The necessary conditions, in the form of the Pontryagin maximum principle, for an optimal control are obtained. This is accomplished by the application of a theorem by Debovistskii-Milyutin. A simple example is also illustrated to show the applicability of the results obtained.     

Numerical solution of optimization problems in vibrational mechanics using the characteristics method

Optimal control problems with a terminal pay-off functional are considered. The dynamics of the control system consists of rapid oscillatory and slow non-linear motions. A numerical method for solving these problems using the characteristics of the Hamilton–Jacobi–Bellman equation is presented. Estimates of the accuracy of the method are obtained. A theorem is proved which enables one to determine the class of functions containing the optimal preset control to be obtained. The results of the numerical solution of a terminal optimization problem for a fast non-linear pendulum are presented.     

Convergence of the control parameterization Ritz method for nonlinear optimal control problems

This paper studies the local convergence properties of the control parameterization Ritz method in which the control variable is approximated over a finite-dimensional subspace. The nonlinear free-endpoint optimal control problem is considered, and error bounds are derived for both the cost functional and state-control convergence. Explicit error bounds are obtained for the particular case of approximations over spline spaces. On specializing the general results to the linear-quadratic regulator problem, global convergence results are obtained. Computational results supporting the theoretically derived error bounds are presented.This research was supported by the University Grants Committee of New Zealand.     

Near-Optimal Controls of a Class of Volterra Integral Systems

In a recent paper by Zhou (Ref. 1), the concept of near-optimal controls was introduced for a class of optimal control problems involving ordinary differential equations. Necessary and sufficient conditions for near-optimal controls were derived. This paper extends the results obtained by Zhou to a class of optimal control problems involving Volterra integral equations. The results are applied to study near-optimal controls obtained by the control parametrization method.     

Optimization of Nonlinear Systems of Stochastic Difference Equations

We present new results concerning the synthesis of optimal control for systems of difference equations that depend on a semi-Markov or Markov stochastic process. We obtain necessary conditions for the optimality of solutions that generalize known conditions for the optimality of deterministic systems of control. These necessary optimality conditions are obtained in the form convenient for the synthesis of optimal control. On the basis of Lyapunov stochastic functions, we obtain matrix difference equations of the Riccati type, the integration of which enables one to synthesize an optimal control. The results obtained generalize results obtained earlier for deterministic systems of difference equations.     

Optimal control of two interconnected turbogenerators

In this paper, the optimal control of a system with two identical interconnected turbogenerators, which are connected to an infinite bus, is considered. Control of the generators is effected through control of field voltages and turbine torques. The alternators are controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a nonlinear, two-point boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, in contrast to the usual feedback control of a linear model. The numerical values used are indicated in the Appendix.     

Linear square optimal control problem for stochastic difference equations with unknown parameters

The problems of stability and optimal control for stochastic difference equations are receiving important attention now (see, for example, [1–3]). In this paper, the optimal control in final form is obtained for optimal control problem of stochastic linear difference equation with unknown parameters and square cost functional. For stochastic functional differential equations, analogous result are obtained in [4].     

Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control

A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.     

Stability of impulsive control systems with variable times

In this paper, cone-valued Lyapunov functions are employed to study the impulsive control system with variable times. The stability criteria on the non-zero solution of the impulsive control system are given by the cone-valued Lyapunov functions and the results of the controllability on the control system are also obtained.     

一类具有多重时滞的分数阶中立型微分系统的相对可控性

本文对一类具有多重时滞的Caputo分数阶中立型微分控制系统的相对可控性和相对U可控性进行了研究.首先利用Laplace变换得到系统解的一个新的表达式,接着由Grammian矩阵得出系统相对可控的充分必要条件.最后给出了一类非线性分数阶中立型微分控制系统相对U可控的充分必要条件.     

Nonlinear feedback control of chaotic pendulum in presence of saturation effect

In present paper, a feedback linearization control is applied to control a chaotic pendulum system. Tracking the desired periodic orbits such as period-one, period-two, and period-four orbits is efficiently achieved. Due to the presence of saturation in real world control signals, the stability of controller is investigated in presence of saturation and sufficient stability conditions are obtained. At first feedback linearization control law is designed, then to avoid the singularity condition, a saturating constraint is applied to the control signal. The stability conditions are obtained analytically. These conditions must be investigated for each specific case numerically. Simulation results show the effectiveness and robustness of proposed controller. A major advantage of this method is its shorter chaotic transient time in compare to other methods such as OGY and Pyragas controllers.     

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.     

Optimal control and synchronization of Lotka–Volterra model

In this paper we discuss the problem of optimal control for the steady state of Lotka–Volterra model. The conditions of the asymptotic stability of the steady state of this model are used to obtain the optimal control functions. In such study, the optimal Lyapunov function is used. The general solution of the equations of the perturbed state is obtained as a function of time. In addition, the optimal control is also applied to achieve the state synchronization of two identical Lotka–Volterra systems. Graphical and numerical simulation studies of the obtained results are presented.     

非线性经济增长模型的最优控制问题

研究了带有中性技术进步生产函数边界条件的非线性经济增长模型的最优控制问题.利用Banach空间不动点原理,得到了系统解的存在唯一性,利用Gronwall不等式得到了系统解关于控制序列的连续依赖性,借助于法锥和共轭系统,得到了控制问题最优解存在的必要条件.     

Existence of solutions and optimal control for reflecting stochastic differential equations with applications to population control theory *

For the d–dimensional reflecting stochastic differential equations (1) with non-smooth boundary and unbounded domain the existence of a strong solution, (weak solution) is obtained under the conditions that the coefficients are less than linear growth and they are non-Lipschitz, (and the diffusion coefficient is non-degenerate, the drift coefficient is bounded and measurable only). Moreover, the Girsanov theorem and the martingale representation theorem with respect to system (1) are also derived. Then by using the Ekeland lemma and the martingale method the existence, necessary and sufficient conditions for an optimal control and an optimal control are obtained. The results are then applied to solve an optimal control problem for a stochastic population model     

Synchronization for fractional‐order multi‐linked complex network with two kinds of topological structure via periodically intermittent control

This paper concentrates on the global synchronization of the fractional‐order multi‐linked complex network (FMCN) via periodically intermittent control. It should be stressed that periodically intermittent control is employed to the FMCN for the first time. Moreover, the network is defined on digraphs with different weights, and two situations on topological structure of the network are discussed, including each digraph being strongly connected, and the biggest one being strongly connected. Based on Lyapunov method and graph theory, some synchronization criteria are obtained under two situations. And, the obtained synchronization criteria have a close relationship with the order of fractional‐order derivative, coupling strength, control gain, control rate, and control period. Besides, for practicability, theoretical results are applied to studying the synchronization of fractional‐order multi‐linked chaotic systems, and some sufficient conditions are provided. For a special case, fractional‐order multi‐linked Lorenz chaotic systems, numerical simulations are given to indicate the feasibility of theoretical results and the effectiveness of control strategy.     

具终端观测且与年龄相关的非线性种群扩散系统的最优控制

讨论了一类具终端观测且与年龄相关的非线性时变种群扩散系统的最优分布控制问题利用偏微控制理论和先验估计,证明了系统最优分布控制的存在性,得到了控制为最优的一阶必要条件,并进而讨论了系统的最优反馈控制问题.