So the first variation becomes
tf
J (u * (t ), u(t )) [ H ( x * (t ), u * (t ) u (t ), * (t ), t ) H ( x * (t ), u * (t ), * (t ), t )]dt
t0

The necessary condition for the optimal control u(t) to minimize J is that the first variation
So the first variation becomes
tf
J (u * (t ), u (t ))
t0

(
H )' u (t )dt u
This means that by definition
H ( x * (t ), u * (t ), * (t ), t ) ( )' u (t ) u H ( x * (t ), u * (t ) u (t ), * (t ), t ) H ( x * (t ), u * (t ), * (t ), t )
x(t 0 ) x0 ; x(t f ) is free and t f is free
H H ( x* (t ), u * (t ), * (t),t ) V ( x* (t ), u * (t ), t ) *' (t) f ( x* (t ), u * (t ), t )

L
x
)' | * tf
x f 0
L L(x * (t ), x (t ), u * (t ), (t), t )
* * H H ( x ( t ), u (t ), * (t),t ) S S * * * V ( x (t ), u (t ), t ) ( )' x (t ) ( ) * V ( x* (t ), u* (t ), t ) *' (t) f ( x* (t ), u* (t ), t ) x t
t0
tf

S S ' [ - (t)]*t f x f [ H ]*t t f x t f
In the above,
1) If the optimal state equations are satisfied, it results in the state relation
tf
t0

[ H ( x * (t ), u * (t ) u (t ), * (t ), t ) H ( x * (t ), u * (t ), * (t ), t )]dt 0
For all admissible u(t) less than a small value,thereis
求 u * (t ) ，使性能指标 J [u (t )] 最小，以实现最优控制，则有： 1）正则方程组 状态方程： x (t )
*
H * [ x (t ), u * (t ), (t )] f [ x * (t ), u * (t )]
协态方程: (t )
2）极值条件：
最小值原理的证明
由第二章变分法得到的一阶变分为：
L d L J [( )* ( )* ]'x(t )dt x dt t0 x
tf

tf
L L ( )'* u (t )dt L |t f t f [( )'* ]x(t )] |t f t f u t0 x
H ( x * (t ), u * (t ) u(t ), * (t ), t ) H ( x * (t ), u * (t ), * (t ), t )
H ( x * (t ), u(t ), * (t ), t ) H ( x * (t ), u * (t ), * (t ), t )

2)极值条件
H [ x * (t ), u * (t ), (t ), t ] min H [ x * (t ), u (t ), (t ), t ]
u (t )U
3）端点约束
[ x* (t *0 ), t *0 ] 0 [ x* (t * f ), t * f ] 0
所谓最小值原理，是指当控制作用u(t)的大小限
制在一定范围内时，由最优控制规律所确定的最优轨
线在整个作用范围内一定取一个最小值。
Q2. 连续系统最小值原理?
设受控系统的状态方程为
x f [ x(t ), u (t ), t ]

m u ( t ) U R [t 0 , t f ] 容许控制u(t)：
So the control, state and costate equations in terms of the Hamiltonian
H ( )* 0 u
H ( ) * (t ) x
*
*
H ( ) * x (t )
The general boundary condition for free-end point system 自由终端系统的通用横截条件
4）极小值原理只给出了最优控制的必要条件，并 非充分条件。极小值原理也没有涉及到解的存在性 和唯一性问题。如果由实际问题的物理意义已经能
够判定所论问题的解是存在的，而由极小值原理所
求的控制又只有一个。则这一控制就是最优控制。
四、最小值原理的几种具体形式 1.时不变情况 设受控系统的状态方程为： x f [ x(t ), u (t )] 容许控制u(t)： u (t ) U
4）横截条件
T * * (t 0 ) [ x * (t 0 ), t 0 ] x(t 0 )
T S * * * * * * (t * ) [ x ( t ), t ] [ x ( t ), t f f f f f] x(t f ) x(t f )
2) Thrust of a rocket engine used in a space shuttle launch control system;
3) Speed of an electric motor used in a typical speed control system;
2. Concept
S S * ' [ H ( )]t f t f [( )* (t )] |t f x f 0 t x
*
Q1. 最小值原理概念?
1. considering safety, cost, and other inherent limitations, there have some constraints on the inputs, internal variables and (or) outputs. 1) In a D. C. motor used in a typical positional control system;
|u (t )|U
min {H ( x * (t ), u (t ), * (t ), t )} H ( x * (t ), u * (t ), * (t ), t )
三、极小值原理的意义：
1）容许控制条件的放宽 极小值原理中的极值条件对于通常的控制约束都是适用的。 2） u * (t ) 使H取全局最小值，H取强极小值，而变分法中 u * (t ) 使H取弱极小值。 3）极小值原理并没有H对u的可微性要求。

R
m
[t 0 , t f ]
始端和终端约束条件为：
x(t 0 ) x0
t f 固定，x(t f )自由
目标泛函为：
tf
J [u (t )] S [ x(t f )] V [ x(t ), u (t )]dt
t0

定义Hamilton函数：
H[ x(t ), u(t ), (t )] V [ x(t ), u(t )] T (t ) f [ x(t ), u(t )]
始端和终端约束条件为：
[ x(t 0 ), t 0 ] 0 [ x(t f ), t f ] 0
目标泛函为：
tf
J [u (t )] S[ x(t f ), t f ] V [ x(t ), u (t ), t ]dt
t0

定义Hamilton函数：
H[ x(t ), u(t ), (t ), t ] V [ x(t ), u(t ), t ] T (t ) f [ x(t ), u(t ), t ]

H * [ x (t ), u * (t ), (t )] x
H [ x * (t ), u * (t ), (t )] min H [ x * (t ), u (t ), (t )]
u (t )U
3）端点约束：
x * (t 0 ) x 0
4）横截条件：
极小值原理
Consider the plant The performance index is
x (t ) f [ x(t ), u(t ), t ]
tf t0

J (u (t )) S ( x(t f ), t f ) V ( x(t ), u (t ), t ) dt
Boundary conditions
H ( ) * x (t ) *
2)
If the constate* (t ) is slectedso that thecoefficient of the dependentvariation x(t) in theintegrandis identicall y zero, it resultsin thecostateconditate.