TIME OPTIMAL CONTROL SYSTEMS.
TIME OPTIMAL CONTROL SYSTEMS.
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DOI：
10.1073/pnas.45.4.573
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发表时间：
1959-04
影响因子：
11.1
通讯作者：
J. P. Lasalle
中科院分区：
文献类型：
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作者：
J. P. Lasalle
1. 介绍。一段时间以来，人们一直有一个直观的假设，即如果控制系统由有限的电源运行，那么系统可以在最短的时间内从一个状态移动到另一个状态。每时每刻都要适当地利用一切可用的能量。这个假设被称为“砰砰原理”。Bushaw接受了这一假设，并在1952年证明了对于一些具有一个自由度的简单系统，在所有的砰砰系统（即在任何时候都使用最大功率的系统）中存在一个最优系统。“1953年，我观察到，所有砰砰系统中最好的，如果存在的话，是所有系统中最好的，使用相同的电源。最近，Bellman、Glicksberg和gross以及后来（但似乎是独立的）Krasovskii4和Gamkrelidze获得了相当普遍的结果。在1958年爱丁堡的国际数学家大会上，LS Pontryagin宣布了一个“极大原理”，这是一个更普遍理论的开端。-我们将自己局限于控制系统的时间最优问题，它是线性的，因为被控制的元素是线性的，并且作为时间的函数，控制是线性的。这种系统的微分方程是x (t)= A (t) x (t)+ B (t) u (t)+ f (t),(1)其中x和f是n维向量函数（x (t)是系统在时刻t的状态），A是一个（n x n）矩阵函数，B是一个（n x r）矩阵函数。因此(1)表示微分方程组n r xii (t) E aij (t) x,(t)+ E bik (t) uk (t)+ f (t), i= 1，。， n j= 1 k= 1
1. Introduction.-It has been an intuitive assumption for some time that if a control system is beingoperated from a limited source of power then the system can be moved from one state to another in the shortest. time by at all times utilizing properly all available power. This hypothesis is called the" bang-bang principle." Bushaw accepted this hypothesis and in 1952 showed for some simple systems with one degree of freedom that of all bang-bang systems (that is, systems which at all times utilize maximum power) there is one that is optimal.'In 1953 I made the observation that the best of all bang-bang systems, if it exists, is then the best of all systems operating from the same power source. 2More recently fairly general results have been obtained by Bellman, Glicksberg, and Gross3 and later (but seemingly independently) by Krasovskii4 and Gamkrelidze. 6 At the 1958 International Congress of Mathematicians in Edinburgh, LS Pontryagin announced a" maximum principle" whichis the beginning of an even moregeneral theory.-We confine ourselves here to the time optimal problem for control systems which are linear in the sense that the elements being controlledare linear and as a function of time the control enters linearly. The differential equation for such systems is x (t)= A (t) x (t)+ B (t) u (t)+ f (t),(1) where x and f are n-dimensional vector functions (x (t) is the state of the system at time t), A is an (n X n) matrix function, and B is an (n X r) matrix function. Thus (1) represents the system of differential equations n r xii (t) E aij (t) x,(t)+ E bik (t) uk (t)+ f (t), i= 1,., n. j= 1 k= 1