Numerical Schemes for Discontinuous Value Functions of Optimal Control

In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key-point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem.     

A Globally Convergent and Efficient Method for Unconstrained Discrete-Time Optimal Control

Shift schemes are commonly used in non-convex situations when solving unconstrained discrete-time optimal control problems by the differential dynamic programming (DDP) method. However, the existing shift schemes are inefficient when the shift becomes too large. In this paper, a new method of combining the DDP method with a shift scheme and the steepest descent method is proposed to cope with non-convex situations. Under certain assumptions, the proposed method is globally convergent and has q-quadratic local conve rgence. Extensive numerical experiments on many test problems in the literature are reported. These numerical results illustrate the robustness and efficiency of the proposed method.     

Optimal Control of Constrained Piecewise Affine Discrete-Time Systems

This paper makes two contributions; firstly, it provides a characterization of the solution of the optimal control problem for piecewise affine discrete-time systems with a quadratic cost function (the generally preferred option) and, secondly, provides a simple method (reverse transformation) for solving this and the previously solved &ell problem. The characterization is useful for on-line implementation.     

Optimal Control Problem for the Lyapunov Exponents of Random Matrix Products

This paper deals with the optimal control problem for the Lyapunov exponents of stochastic matrix products when these matrices depend on a controlled Markov process with values in a finite or countable set. Under some hypotheses, the reduced process satisfies the Doeblin condition and the existence of an optimal control is proved. Furthermore, with this optimal control, the spectrum of the system consists of only one element.     

Optimal Control of a Stochastic Hybrid System with Discounted Cost

We address the optimal control problem of a very general stochastic hybrid system with both autonomous and impulsive jumps. The planning horizon is infinite and we use the discounted-cost criterion for performance evaluation. Under certain assumptions, we show the existence of an optimal control. We then derive the quasivariational inequalities satisfied by the value function and establish well-posedness. Finally, we prove the usual verification theorem of dynamic programming.     

The Lyapunov spectrum of families of time-varying matrices

For -families of time varying matrices centered at an unperturbed matrix, the Lyapunov spectrum contains the Floquet spectrum obtained by considering periodically varying piecewise constant matrices. On the other hand, it is contained in the Morse spectrum of an associated flow on a vector bundle. A closer analysis of the Floquet spectrum based on geometric control theory in projective space and, in particular, on control sets, is performed. Introducing a real parameter , which indicates the size of the -perturbation, we study when the Floquet spectrum, the Morse spectrum, and hence the Lyapunov spectrum all coincide. This holds, if an inner pair condition is satisfied, for all up to at most countably many -values.

     

Optimal Guaranteed Cost Control of Stochastic Discrete-Time Systems with States and Input Dependent Noise Under Markovian Switching

A problem of robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered. The control is simultaneously applied to both the random and the deterministic components of the system. The noise (the random) term depends on both the states and the control input. The jump Markovian switching is modeled by a discrete-time Markov chain and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Using linear matrix inequalities (LMIs) approach, the robust quadratic stochastic stability is obtained. The proposed control law for this quadratic stochastic stabilization result depended on the mode of the system. This control law is developed such that the closed-loop system with a cost function has an upper bound under all admissible parameter uncertainties. The upper bound for the cost function is obtained as a minimization problem. Two numerical examples are given to demonstrate the potential of the proposed techniques and obtained results.     

一类线性奇异摄动最优控制问题的渐近解

研究了一类线性奇异摄动最优控制问题的空间对照结构,讨论了初始点固定,终端自由的情形.首先根据变分法得到了一阶最优性条件,其次运用退化最优控制问题的解证明了异宿轨道的存在性,从而结合奇异摄动理论证明了原问题空间对照结构解的存在性.进一步根据解的结构,利用边界层函数法构造了奇异摄动最优控制问题一致有效的形式渐近解.最后,通...     

Primal-Dual Strategy for State-Constrained Optimal Control Problems

State constrained optimal control problems represent severe analytical and numerical challenges. A numerical algorithm based on an active set strategy involving primal as well as dual variables, suggested by a generalized Moreau-Yosida regularization of the state constraint is proposed and analyzed. Numerical examples are included.     

Optimal Production Planning in a Stochastic Manufacturing System with Long-Run Average Cost

This paper is concerned with the optimal production planning in a dynamic stochastic manufacturing system consisting of a single machine that is failure prone and facing a constant demand. The objective is to choose the rate of production over time in order to minimize the long-run average cost of production and surplus. The analysis proceeds with a study of the corresponding problem with a discounted cost. It is shown using the vanishing discount approach that the Hamilton–Jacobi–Bellman equation for the average cost problem has a solution giving rise to the minimal average cost and the so-called potential function. The result helps in establishing a verification theorem. Finally, the optimal control policy is specified in terms of the potential function.     

On Smoothness Properties and Approximability of Optimal Control Functions

The theory of discretization methods to control problems and their convergence under strong stable optimality conditions in recent years has been thoroughly investigated by several authors. A particularly interesting question is to ask for a natural smoothness category for the optimal controls as functions of time.In several papers, Hager and Dontchev considered Riemann integrable controls. This smoothness class is characterized by global, averaged criteria. In contrast, we consider strictly local properties of the solution function. As a first step, we introduce tools for the analysis of L elements at a point. Using afterwards Robinson's strong regularity theory, under appropriate first and second order optimality conditions we obtain structural as well as certain pseudo-Lipschitz properties with respect to the time variable for the control.Consequences for the behavior of discrete solution approximations are discussed in the concluding section with respect to L as well as L ₂ topologies.     

Runge–Kutta Based Procedure for the Optimal Control of Differential-Algebraic Equations

A new approach for optimization of control problems defined by fully implicit differential-algebraic equations is described in the paper. The main feature of the approach is that system equations are substituted by discrete-time implicit equations resulting from the integration of the system equations by an implicit Runge–Kutta method. The optimization variables are parameters of piecewise constant approximations to control functions; thus, the control problem is reduced to the control space only. The method copes efficiently with problems defined by large-scale differential-algebraic equations.     

最优值函数的方向导数

本文详细介绍了参数非线性规划问题最优值函数的方向导数的存在性、表示形式和有关的发展历史,本文不仅讨论了这一问题的经典结论,还叙述了最近二十年来的新结果。本文最后指出进一步研究的方向。     

The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.     

Asymptotic Behavior of the Attainable Sets of Linear Periodic Control Systems

This paper deals with the attainable sets of linear periodic control systems. The asymptotic behavior of the attainable sets over a long time interval is investigated in terms of shapes of the sets. The shape of a set stands for the totality of all its images under nonsingular linear transformations. It is shown that there exist limits of the shape of attainable sets corresponding to time instants with the same residue modulo the period of the system and that the limit shapes are different if the system includes a stable subsystem.     

Immunization and Max–Min Optimal Control

A model of a term structure of interest rates is conceived in which disturbances are unknown and bounded. Arbitrage opportunities are ruled out by imposing suitable constraints to the disturbances. This sets the stage for casting a well-known immunization problem as a max-min optimal control problem. Dynamic programming is then used to obtain the analytical solution to such a problem. In this manner, a well-known immunization policy is proved to be optimal in a dynamic setting.     

Optimization Approach to the Estimation and Control of Lyapunov Exponents

In this paper, we describe an algorithm for estimating the Lyapunov exponents from the chaotic dynamics of control systems. Attention is focused on optimization methods for estimating tangent maps from experimental time series data. Our numerical tests show that the algorithm is robust and quite effective, and that its performance is comparable with that of other algorithms. The properties of the algorithm are demonstrated by application to a range of data sets. We consider numerical and experimental data and discuss the computational aspects of the proposed algorithm. New feedback rules for use with optimization techniques in the stimulation of the epileptic brain are proposed. This work was supported by NIH, NSF, and CRDF grants.     

Relaxed LMI Condition for Output Feedback Guaranteed Cost Control of Uncertain Discrete-Time Systems

This paper studies the output feedback guaranteed cost control for a class of uncertain discrete-time systems. The uncertainty is of the linear fractional form. A new relaxed LMI condition is given based on a recently developed stability condition. An extra variable is introduced to obtain a less conservative result. The effectiveness of the method is illustrated with an example to compare with previous results. This work was supported in part by the National Science Foundation of China, Grant 10472001.     

Stability Analysis of Optimal Control Systems via Liapunov Method

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On the Existence of Optimal Solutions to an Optimal Control Problem

In this paper, some results concerning the existence of optimal solutions to an optimal control problem are derived. The problem involves a quasilinear hyperbolic differential equation with boundary condition and a nonlinear integral functional of action. The assumption of convexity, under which the main theorem is proved, is not connected directly with the convexity of the functional of action. In the proof, the implicit function theorem for multimappings is used.Communicated by L. D. Berkovitz