前面我在第二章最优控制理论 二、哈密尔顿函数法 给出了Hamilton函数法一些重要推导过程和一些常用公式。最近翻看，觉得写得太多了，于是把一部分不重要的贴到下面，另成一篇。
沿最优轨线
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\frac{\mathrm{d} H}{\mathrm{d} t}=\frac{\partial H}{\partial t}
d t d H = ∂ t ∂ H
证明 ：对Hamiltonian按照链式求导法则全导数：
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\frac{\mathrm{d} H}{\mathrm{d} t}=\frac{\partial H^{\mathrm{T}}}{\partial x} \dot{x}+\frac{\partial H^{\mathrm{T}}}{\partial \lambda} \dot{\lambda}+\frac{\partial H^{\mathrm{T}}}{\partial u} \dot{U}+\frac{\partial H}{\partial t}
d t d H = ∂ x ∂ H T x ˙ + ∂ λ ∂ H T λ ˙ + ∂ u ∂ H T U ˙ + ∂ t ∂ H
考虑到最优轨线附近满足
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\begin{aligned} -\frac{\partial H}{\partial x}&=\dot\lambda\\ \frac{\partial H}{\partial u}&=0\\ \frac{\partial H}{\partial \lambda}&=f=\dot x \end{aligned}
− ∂ x ∂ H ∂ u ∂ H ∂ λ ∂ H = λ ˙ = 0 = f = x ˙
代入则公式
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\frac{\mathrm{d} H}{\mathrm{d} t}=\frac{\partial H}{\partial t}=0
d t d H = ∂ t ∂ H = 0
于是可得
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Const
H(x^*(t),u^*(t),\lambda^*(t),t)=\text{Const}
H ( x ∗ ( t ) , u ∗ ( t ) , λ ∗ ( t ) , t ) = Const
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H(*,t)=H(0)
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H(*,t_f)=0
H ( ∗ , t f ) = 0
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H(*,t)=0
H ( ∗ , t ) = 0
第3 部分我们只考虑了终端等式约束，但是实际的动力学和控制问题里常有其他类型的等式约束，如
积分方程约束
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\int_0^{t_f}N(x(t),u(t),t)\text d t=\beta
∫ 0 t f N ( x ( t ) , u ( t ) , t ) d t = β
控制输入的约束
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N(u(t),t)=0
N ( u ( t ) , t ) = 0
控制输入和状态变量的等式约束
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N(x(t),u(t),t)=0
N ( x ( t ) , u ( t ) , t ) = 0
下面可以证明，以上这些等式约束都可以用Hamilton函数法解决。
对
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\int_0^{t_f}N(x(t),u(t),t)\text d t=\beta\in\Reals^q\tag{integral Eq constraints}
∫ 0 t f N ( x ( t ) , u ( t ) , t ) d t = β ∈ R q ( i n t e g r a l E q c o n s t r a i n t s )
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y(t)\in\Reals^q
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(terminal Eq constraints)
\dot y=N(x,u,t)\\ y(0)=0,y(t_f)=\int_0^{t_f}N(x(t),u(t),t)\text d t\beta\tag{terminal Eq constraints}
y ˙ = N ( x , u , t ) y ( 0 ) = 0 , y ( t f ) = ∫ 0 t f N ( x ( t ) , u ( t ) , t ) d t β ( t e r m i n a l E q c o n s t r a i n t s )
则把上面这个积分方程约束化为终端状态约束，两个方程等价。仍可套用终端状态约束的框架，需要注意由于状态扩充为
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设状态变量和等式约束具有以下形式：
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(state+control Eq constraints)
N(x(t),u(t),t)=0,t_0\lt t\lt t_f\tag{state+control Eq constraints}
N ( x ( t ) , u ( t ) , t ) = 0 , t 0 < t < t f ( s t a t e + c o n t r o l E q c o n s t r a i n t s )
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\mu(t)\in\Reals^q
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H\triangleq L+\lambda^{\mathrm T}f+\mu^{\mathrm T}N
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\dot\lambda=-\frac{\partial H}{\partial x}=-L_x-\lambda^{\mathrm T}f_x-\mu^{\mathrm T}N_x \\ \frac{\partial H}{\partial u}=L_u+\lambda^{\mathrm T}f_u+\mu^{\mathrm T}N_u =0
λ ˙ = − ∂ x ∂ H = − L x − λ T f x − μ T N x ∂ u ∂ H = L u + λ T f u + μ T N u = 0
纯控制变量约束
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前面的假设是控制
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u ( t ) 控制
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\bar H(x,\dot x,\lambda,t)=L(x,\dot x,t)-\lambda^{\mathrm T}(f(x,\dot x,t)-\dot x)%
H ˉ ( x , x ˙ , λ , t ) = L ( x , x ˙ , t ) − λ T ( f ( x , x ˙ , t ) − x ˙ )
由角点处的Weierstrass-Erdmann条件，有
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\begin{aligned} \left.\bar{H}_{\dot{X}}\right|_{t_i-} &=\left.\bar{H}_{\dot{X}}\right|_{t_i+} \\ \bar{H}-\left.\dot{X}^{\mathrm{T}} \bar{H}_{\dot{X}}\right|_{t_i-} &=\bar{H}-\left.\dot{X}^{\mathrm{T}} \bar{H}_{\dot{X}}\right|_{t_i+} \end{aligned}
H ˉ X ˙ ∣ ∣ t i − H ˉ − X ˙ T H ˉ X ˙ ∣ ∣ ∣ t i − = H ˉ X ˙ ∣ ∣ t i + = H ˉ − X ˙ T H ˉ X ˙ ∣ ∣ ∣ t i +
对标准形式的Hamilton函数
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H\triangleq L+\lambda^{\mathrm T}f
H ≜ L + λ T f
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\begin{aligned} \lambda({t_i-})&=\lambda({t_i+})\\ \left.H\right|_{{t_i-}} &=\left.{H}\right|_{t_i-+} \\ \end{aligned}
λ ( t i − ) H ∣ t i − = λ ( t i + ) = H ∣ t i − +
即Hamilton函数连续，且协态变量连续。
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x ( t ) 若状态变量
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\mathbf x(t_i-)=\mathbf x(t_i+)
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\psi^{(i)}(\mathbf x(t_i),t_i)=0,\quad\psi^{(i)}\in\Reals^{q_i}, t_0\lt t_i\lt t_f,i=1,2,\cdots,N
ψ ( i ) ( x ( t i ) , t i ) = 0 , ψ ( i ) ∈ R q i , t 0 < t i < t f , i = 1 , 2 , ⋯ , N
此种情况下，在每一个角点处条件是：
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\lambda(t_i-)=\lambda(t_i+)+\mathbf\mu^{\mathrm T}\frac{\partial \psi^{(i)}}{\partial\mathbf x}\\ H(t_i-)=H(t_i+)+\mathbf\mu^{\mathrm T}\frac{\partial\psi^{(i)}}{\partial t}\tag {10}
λ ( t i − ) = λ ( t i + ) + μ T ∂ x ∂ ψ ( i ) H ( t i − ) = H ( t i + ) + μ T ∂ t ∂ ψ ( i ) ( 1 0 )
其中
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μ ∈ R q i
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\begin{aligned}\dot\mathbf x&=\left\{\begin{matrix}f^{(1)}(\mathbf x(t),u(t),t)&t\in[t_0,t_1]\\ \cdots\\ f^{(q)}(\mathbf x(t),u(t),t)&t\in(t_{q-1},t_q]\\ \end{matrix}\right.\\ &\mathbf x(t_i-)=\mathbf x(t_i+),i=1,2,\cdots,N \end{aligned}
x ˙ = ⎩ ⎨ ⎧ f ( 1 ) ( x ( t ) , u ( t ) , t ) ⋯ f ( q ) ( x ( t ) , u ( t ) , t ) t ∈ [ t 0 , t 1 ] t ∈ ( t q − 1 , t q ] x ( t i − ) = x ( t i + ) , i = 1 , 2 , ⋯ , N
简单来说就是，Hamiltonian和协态变量
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\mathbf x(t_i-)\neq\mathbf x(t_i+)
x ( t i − ) = x ( t i + )
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(\ddag)
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H^{(i)}\triangleq L^{(i)}+\lambda^{\mathrm T}f^{(i)}\\ \Phi\triangleq \phi+\sum_{j=0}^{N}[\nu^{(i)}]^{\mathrm T}\psi^{(i)}
H ( i ) ≜ L ( i ) + λ T f ( i ) Φ ≜ ϕ + j = 0 ∑ N [ ν ( i ) ] T ψ ( i )
间断点处的状态变量不连续，但遵循约束条件；协态变量有；
控制变量由
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\lambda(t)
λ ( t )
[1] 邢继祥. 最优控制应用基础[M]. 科学出版社, 2003.
