Optimal selection of observation times in the linear-quadratic Gaussian control problem

Feedback control is sometimes applied to a dynamic system operating in a stochastic environment under circumstances where only a limited number of observations can be made. The optimal positioning of ana priori fixed number of observations is considered. A direct approach to this problem yields a dynamic programming functional equation, while a second approach involving an ancillary observation cost leads to a rapid and practical numerical solution.     

Asymptotic behavior of the optimal cost functional for a rapidly stabilizing indirect control in the singular case

The optimal control problem for a linear system with fast and slow variables in the form of indirect control with a convex terminal cost functional and a smooth geometric constraint on the control is studied. An asymptotic expansion of the cost functional up to any power of a small parameter is constructed.     

线性脉冲控制系统的能控性与最优控制

首先讨论了一类线性随机脉冲控制系统的精确能控性质,给出了该类控制系统的脉冲精确能控的等价的代数判据.然后提出了一个确定性的二维线性脉冲控制系统的时间-脉冲强度最优控制问题;利用动态规划原理,给出了脉冲最优控制的反馈形式和值函数的显式表达式;说明了值函数在整个平面上是连续的,在左右两个半平面的内部还是连续可微的.     

拟线性椭圆变分不等式的障碍最优控制问题

对拟线性椭圆变分不等式的障碍最优控制问题(即以障碍为控制变量)进行了研究．指标泛函为Lagrange型,其中含有控制变量二阶导数的p次幂,这使得最优性条件的推导颇为不易．对所考虑的问题给出了最优控制的存在性定理以及必要条件．     

Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems

This paper concerns the nonfragile guaranteed cost control problem for a class of nonlinear dynamic systems with multiple time delays and controller gain perturbations. Guaranteed cost control law is designed under two classes of perturbations, namely, additive form and multiplicative form. The problem is to design a memoryless state feedback control law such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Based on the linear matrix inequality (LMI) approach, some delay-dependent conditions for the existence of such controller are derived. A numerical example is given to illustrate the proposed method.     

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.     

An optimal control problem for a prey–predator system with a general functional response

An optimal control problem is studied for a prey–predator system with a general functional response. The control functions represent the rate of mixture of the populations and the cost functional is of Mayer type. The number of switching points of the optimal control is discussed in terms of the sign of a specific constant.     

Robust suboptimal control of nonlinear systems

In this paper, we consider a nonlinear dynamic system with uncertain parameters. Our goal is to choose a control function for this system that balances two competing objectives: (i) the system should operate efficiently; and (ii) the system’s performance should be robust with respect to changes in the uncertain parameters. With this in mind, we introduce an optimal control problem with a cost function penalizing both the system cost (a function of the final state reached by the system) and the system sensitivity (the derivative of the system cost with respect to the uncertain parameters). We then show that the system sensitivity can be computed by solving an auxiliary initial value problem. This result allows one to convert the optimal control problem into a standard Mayer problem, which can be solved directly using conventional techniques. We illustrate this approach by solving two example problems using the software MISER3.     

Min-max optimal control of systems approximated by finite-dimensional models: Nonquadratic cost functional

An open-loop control problem with nonquadratic performance criterion for a dynamical system, described by an abstract linear equation of evolution and approximated by a finite-dimensional model, is solved. A min-max approach is taken: the value of the cost functional in theworst-case output error between the system and the model, under the assumption that the norm of this output error is estimated by the norm of the input, is minimized.The form of the cost functional reproduces itself under maximization, so that the min-max control problem, when only thedistance between the model and the system is given, has the same features and proprieties of the control problem when the system is thorougly known.Existence and uniqueness theorems for the optimal control are proven, using the spectral proprieties of the model transfer function; and, in the case of a time-invariant model, the min-max control computation is reduced to the solution of a constant-coefficients Sturm-Liouville problem followed by the search for the zeros of a very simple numerical function.This work was made within the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche.     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 2: Application

The optimal open-loop control of a beam subject to initial disturbances is studied by means of a maximum principle developed for hyperbolic partial differential equations in one space dimension. The cost functional representing the dynamic response of the beam is taken as quadratic in the displacement and its space and time derivatives. The objective of the control is to minimize a performance index consisting of the cost functional and a penalty term involving the control function. Application of the maximum principle leads to boundary-value problems for hyperbolic partial differential equations subject to initial and terminal conditions. The explicit solution of this system is obtained yielding the expressions for the state and optimal control functions. The behavior of the controlled and uncontrolled beam is studied numerically, and the effectiveness of the proposed control is illustrated.     

Optimal estuary aeration: An application of distributed parameter control theory

The use of artificial induced aeration has been suggested as a means for improving water quality. The task of controlling the aeration input rate to obtain maximum improvement with least cost is addressed as an optimization problem of a distributed parameter control system. A partial differential equation model for the dissolved oxygen balance in streams and estuaries is given, and a criterion functional is proposed in which the control can be found as the solution of an optimization problem in a Hilbert space. An analytic solution for the optimal feedback control of a stream aeration system is found, and a numerical algorithm for the estuary case is applied to an example using historical data from the Delaware estuary. The sensitivity of the control to system and input variations is discussed, and the dollar cost for the example is compared with costs for other suboptimal control schemes.     

Determination of the optimal initial function and the parameters for nonlinear time-delay systems

This paper deals with dynamic systems described by nonlinear differential-difference equations of retarded type. The problem considered is to determine the initial function and certain system parameters which minimize a given cost functional. A computational method is presented and some convergence results are given. Numerical examples of linear and nonlinear systems are also included.     

Dynamic programming for the stochastic burgers equation

We solve a control problem for the stochastic Burgers equation using the dynamic programming approach. The cost functional involves exponentially growing functions and the analog of the kinetic energy; the case of a distributed parameter control is considered. The Hamilton-Jacobi equation is solved by a compactness method and a-priori estimates are obtained thanks to the regularizing properties of the transition semigroup associated to the stochastic Burgers equation; a fixed point argument does not seem to apply here. Entrata in Redazione il 10 dicembre 1998.     

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

Nearly Optimal Impulsive Controls for Reflected Wideband Width Process

Abstract

Near optimal control problem for a wideband noise process with impulsive controls and constrained to a bounded region is considered. The method developed here will show that sequence of physical processes converges (weakly) to a reflected controlled diffusion process with impulses as the “approximating parameter” goes to zero. The cost functional of the wideband system will also converge to the corresponding cost functional of the limit problem. Due to the reflection at the boundary, pseudo path topology will be used in the weak convergence analysis.     

Infinite-horizon periodic minimax control problem

This note analyzes a minimax optimal control problem involving a periodic discrete-time system. The functional to be minimized is the maximum value of the sum over a period of the costs incurred by the system. A dynamic programming algorithm is proposed for the solution.     

Optimal Guaranteed Cost Control of Linear Systems with Mixed Interval Time-Varying Delayed State and Control

This paper deals with the problem of optimal guaranteed cost control for linear systems with interval time-varying delayed state and control. The time delay is assumed to be a continuous function belonging to a given interval, but not necessary to be differentiable. A linear–quadratic cost function is considered as a performance measure for the closed-loop system. By constructing a set of augmented Lyapunov–Krasovskii functional combined with Newton–Leibniz formula, a guaranteed cost controller design is presented and sufficient conditions for the existence of a guaranteed cost state-feedback for the system are given in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.     

On one class of control problems under uncertainty

An approach to solving a control problem for a nonlinear dynamic system containing controlled and uncertain parameters with terminal performance functional is described. The parameters of the controlled process are such that the uncertainty vector dominates over the control vector. A game control in the class of counterstrategies is suggested such that the value of the functional depends on the realized uncertainty. The quality of a counterstrategy is estimated by comparing the values of the functional under the game control and under the optimal control. The results of calculating the control and values of the functional for test parameters of the model and of the class of uncertainties are presented.     

Possibility and necessity constraints and their defuzzification—A multi-item production-inventory scenario via optimal control theory

In this paper, analogous to chance constraints, real-life necessity and possibility constraints in the context of a multi-item dynamic production-inventory control system are defined and defuzzified following fuzzy relations. Hence, a realistic multi-item production-inventory model with shortages and fuzzy constraints has been formulated and solved for optimal production with the objective of having minimum cost. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the present system produces some defective units along with the perfect ones and the rate of produced defective units is constant. Here demand of the good units is time dependent and known and the defective units are of no use. The space required per unit item, available storage space and investment capital are assumed to be imprecise. The space and budget constraints are of necessity and/or possibility types. The model is formulated as an optimal control problem and solved for optimum production function using Pontryagin’s optimal control policy, the Kuhn–Tucker conditions and generalized reduced gradient (GRG) technique. The model is illustrated numerically and values of demand, optimal production function and stock level are presented in both tabular and graphical forms. The sensitivity of the cost functional due to the changes in confidence level of imprecise constraints is also presented.