Optimal Open-Loop Control of Vacations,Polling and Service Assignment

We consider in this paper the optimal open-loop control of vacations in queueing systems. The controller has to take actions without state information. We first consider the case of a single queue, in which the question is when should vacations be taken so as to minimize, in some general sense, workloads and waiting times. We then consider the case of several queues, in which service of one queue constitutes a vacation for others. This is the optimal polling problem. We solve both problems using new techniques from [2,4] based on multimodularity.     

Optimal cash management under uncertainty

We solve an agent’s optimization problem of meeting demands for cash over time with cash deposited in bank or invested in stock. The stock pays dividends and uncertain capital gains, and a commission is incurred in buying and selling of stock. We use a stochastic maximum principle to obtain explicitly the optimal transaction policy.     

梁振动边界反馈的最优反馈增益的数值解

本用Legendre谱方法估计一端固定,一端加弯矩耗散线性反馈的梁振动的闭环系统使能量最快衰减的最优反馈增益,我们给出了数值产生的图形结果,通过比较发现另一种非耗散的线性反馈在最优反馈增益下比相应的耗散线性反馈有更好的衰减率。     

Optimal control of inequality under uncertainty

We model the optimal control of inequality for an economy experiencing growth in the mean and variance of the income distribution under conditions of uncertainty. Given quadratic losses in the level of inequality and the strength of the policy instrument, we derive a closed form solution for the optimal policy rule in a finite time horizon model. A calibrated, numerical simulation derives the optimal rule required to return the United States to the level of inequality that it experienced in 1979.     

Optimal hedging under output price uncertainty

This paper provides a theoretical and empirical analysis of optimal hedging under output price uncertainty. The theoretical analysis is facilitated by exploiting the duality between production and cost while the empirical implementation uses the envelope theorem and the indirect expected utility function. Empirically estimable equations are derived by approximating the indirect expected utility function by a Taylor series approximation. The model is tested by using live cattle data as output while using prices of corn, soybeans, and the feeder cattle as inputs. The results support the theoretical predictions and the evidence shows that live cattle farmers exhibit decreasing absolute risk aversion.     

Optimal reinsurance under risk and uncertainty

This paper deals with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also included. The insurer and reinsurer degrees of uncertainty do not have to be identical. The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is totally eliminated.Three main contributions seem to be reached. Firstly, necessary and sufficient optimality conditions are given in a very general setting. Secondly, the optimal contract is often a bang–bang solution, i.e., the sensitivity between the retained risk and the total claims saturates the imposed constraints. Thirdly, the optimal reinsurance problem is equivalent to other linear programming problem, despite the fact that risk, uncertainty, and many premium principles are not linear. This may be important because linear problems may be easily solved in practice, since there are very efficient algorithms.     

Optimal staffing of specialized programme trainees under uncertainty

We propose a stochastic model and provide an easy-to-implement optimization tool for admission decisions to specialized training programmes designed for service industries. The model can be applied for staffing of trainees in medical residency programmes, vocational schools, management trainee programmes, and similar. Especially towards graduation, trainees in these programmes substantially contribute to workforce of their affiliated institutions, thus having a targeted number of advanced level students become a potential performance metric for administration. For uncertain attrition rates and study duration, we model and provide an iterative solution algorithm to find the optimal annual admission number for these programmes. Our numeric analysis results show that the solution is robust to changes in attrition and study duration probabilities; hence, our model is robust against specification errors for these parameters, which could be hard to estimate due to data unavailability and fluctuations in educational and economic conditions.     

Optimal PID controller of robots under stochastic uncertainty

In practice often it is not possible to specify exact model parameters. Hence, precomputed controller based on some parameter estimates can produce bad results. In this presentation the aim is to combine classical PID control theory and stochastic optimisation methods in order to obtain robust optimal feedback control. The method works with cost functions being minimized and takes into account stochastic parameter varations. After Taylor expansion to calculate expected cost functions and a few transformations an approximate deterministic substitute PID control problem follows. Here, retaining only linear terms, approximation of expectations and variances of the expected cost functions can be calculated explicitly. By means of splines, numerical approximations of the objective function and the differential equations are obtained then. Using stochastic optimization methods, random parameter variations are incorporated into the optimal control process. Hence, robust optimal feedback controls are obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Optimal Feedback Control for Stationary Linear Systems

We study linear-quadratic optimal control problems for finite dimensional stationary linear systems A X+B U=Z with output Y=C X+D U from the viewpoint of linear feedback solution. We interpret solutions in relation to system robustness with respect to disturbances Z and relate them to nonlinear matrix equations of Riccati type and eigenvalue-eigenvector problems for the corresponding Hamiltonian system. Examples are included along with an indication of extensions to continuous, i.e., infinite dimensional, systems, primarily of elliptic type.     

Advanced Optimal Feedback Control for Tracking of Real-Life Applications

Calculating the open–loop solution of an optimal control problem is just the first step to cope with the practical realization of real life applications. Feedback controllers, like the classical Linear Quadratic Regulator (LQR), are needed to compensate pertubations appearing in reality. Although these controllers have proven to be a powerful tool in many applications and to be robust enough to countervail most differences between simulation and practice, they are not optimal if disturbances in the system data occur. If these controllers are applied in a real process, the possibility of data disturbances force recomputing the feedback control law in real–time to preserve stability and optimality, at least approximately. For this purpose, variations of the classical closed–loop controller with the extention to a trackingtype controller are analysed by means of an industrial application of container cranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Implementation of Optimal Feedback Control for Nonlinear Systems

A method of on-line constructing an optimal feedback control for nonlinear systems with bang-bang optimal control is suggested. The realization of the algorithm of constructing a closed-loop solution is oriented on fast corrections of optimal open-loop control subject to small variations of initial state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

INFINITE DIMENSIONAL TRACKING OPTIMAL CONTROL VIA DYNAMIC OUTPUT FEEDBACK~*

This paper explores implemantation problems of infinite dimensional linear-quadratictracking optimal control.Based on the closed-loop result,a new formula of optimal controlexpressed by past-time state feedback is proved.From this,on the conditions of observa-bility,expressions of optimal control via dynamic output feedback are derived.The mainfeedback operator functions are given by solution of linear integral equations.     

Optimal production in a competitive market under yield uncertainty

Motivated by the U.S. influenza vaccine market, we study the impact of random yield and production capacity on the number of firms, total supply, consumer surplus and social welfare in a market with identical suppliers. We formulate a two-stage game with endogenous entry, where each entering firm aims to maximize its profit under yield uncertainty subject to a production capacity constraint. We show that if firms produce full capacity in the equilibrium, then there are fewer firms in the equilibrium compared to the social optimum even for small levels of yield uncertainty. Furthermore, we prove that if firms do not produce full capacity in the equilibrium, they will not produce full capacity in the social optimum.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

LQG量测反馈最优控制的精细积分

对于线性二次型高斯(LQG)量测反馈最优控制问题,提出了精细积分解法。根据分离性原理,LQG控制问题可以分成为最优状态反馈控制问题以及最优状态估计问题,即:离线计算的两套黎卡提微分方程的求解以及状态向量的时变微分方程的在线积分解。该算法不仅适用于求解二点边值问题及其相应的黎卡提微分方程,也适用于求解状态估计的时变微分方程。精细积分高精度的特点,对控制和估计都是有利的。数值算例表明了算法的高精度及有效性。     

LTV度量反馈控制问题的最优控制器

In this paper,we consider the measurement feedback control problem for discrete linear time-varying systems within the framework of nest algebra consisting of causal and bounded linear operators.Based on the inner-outer factorization of operators,we reduce the control problem to a distance from a certain operator to a special subspace of a nest algebra and show the existence of the optimal LTV controller in two different ways：one via the characteristic of the subspace in question directly,the other via the duality theory.The latter also gives a new formula for computing the optimal cost.     

Regularity of the Optimal Feedback and the Value Function in Convex Problems of Optimal Control

An optimal feedback mapping, leading to necessary and sufficient conditions for optimality in terms of a closed-loop differential inclusion, is derived in the setting of fully convex generalized problems of Bolza. Results are translated to the format of control problems with linear dynamics and convex costs. Properties of the feedback mapping, with focus on single-valuedness and continuity, are analyzed through those of the value function and of the Hamiltonian. Conditions guaranteeing differentiability of the value function are obtained through the analysis of its subdifferential as a maximal monotone operator and of the generalized Hamiltonian dynamics.