The Calculus of Variations and Optimal Control
The Calculus of Variations and Optimal Control
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变分计算和最优控制
DOI：
10.1115/1.3139697
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发表时间：
1982
影响因子：
1.7
通讯作者：
D. Hrovat
中科院分区：
文献类型：
--
作者：
G. Leitmann;D. Hrovat
正如其标题所暗示的，这本书分为两个主要部分：变分法（第一部分）和最优控制（第二部分）。这种按时间顺序的划分看起来很自然，所以第二部分很好地建立在前73页中提供的更经典的材料上。第一部分只处理变分法的简单问题，其中函数极值化仅限于具有固定初始和最终时间的积分泛函。此外，关联的被积函数只包含标量相关变量lex（t）及其导数x（t）。这些限制以及其他放松在书的第二部分，其中还包括处理状态相关控制约束和非自治系统。第二部分是基于作者的原始贡献领域的最优控制通过几何方法在他的工作与教授Blaquiere。2主要的理论发展通过增广的（n +1）维状态空间来说明，其中（n+1）-st轴与成本泛函密切相关。应该指出的是，只有确定性的最优控制问题的书中讨论，因此，例如，著名的LQG（线性二次高斯）理论是缺乏其”G部分”和相关的对偶之间的最优调节器和卡尔曼布西滤波器。现将每章的详细情况介绍如下。前三章发展了著名的欧拉积分泛函弱局部极小的必要条件。包括在这些章节是一个详细的处理欧拉-拉格朗日方程及其相关的特殊形式。第四章讨论了一个有趣的反问题：给定一个二阶常微分方程（Euler-Lagrange方程的候选方程），求出相应的平稳性原理。强局部极小的必要条件（Weierstrass条件）和光滑极值假设下的必要条件（Jacobi条件）分别在第5章和第6章中介绍。第一部分结束与Erdmann-Weierstrass角条件，必须满足分段光滑极值。
As its title alludes, the book is divided into two major parts: calculus of variations (Part I), and optimal control (Part II). This chronological division appears quite natural, so that Part II builds nicely on the more classical material presented in the first 73 pages. The first part treats only the simple problem of the calculus of variations, where the functional extremization is limited to integral functionals with fixed initial and final times. Furthermore, the associated integrands contain only the scalar dependent variablex (t) and its derivative x (t). These constraints as well as others are relaxed in the second part of the book which also includes the treatment of statedependent control constraints and nonautonomous systems. The second part is based on the author's original contribution to the field of optimal control viewed through the geometrical approach developed in his work with Professor Blaquiere. 2 Major theoretical developments are illustrated via augmented,(«+ l)-dimensional state-space where the (n+ l)-st axis is closely associated with the cost functional. It should be pointed out that only the deterministic optimal control problem is discussed in the book, so that, for example, the well-known LQG (Linear-Quadratic-Gaussian) theory is devoid of its" G-part" and the associated duality between the optimal regulator and Kalman-Bucy filter. A more detailed account of each chapter is now given below. The first three chapters develop the celebrated Euler's necessary conditions for weak local minima of integral functionals. Included in these chapters is a detailed treatment of the Euler-Lagrange equation and its associated special forms. Chapter 4 addresses an interesting inverse problem: given a second-order ordinary differential equation (candidate for the Euler-Lagrange equation) find the corresponding stationality principle. The necessary conditions for strong local minima (Weierstrass conditions) and necessary conditions under the assumption of smooth extremals (Jacobi's conditions) are introduced in Chapters 5 and 6, respectively. The first part ends with the Erdmann-Weierstrass corner conditions which must be met by piecewise smooth extremals.