Abstract:
Consider the following optimal control problem:
$$
\dot{q}=u_1 f_1(q)+u_2 f_2(q), \qquad q \in \mathbb{R}^n, \quad u \in
\mathbb{R}^2,
$$
where $q = (x_1,x_2,\dots,x_n)^T, \ f_1=(1,0,-x_2,-x_3,\dots,-x_{n-1})^T,f_2=(0,1,0,0,\ldots,0)^T,$ boundary conditions:
$$
q(0)=q_0, \ q(t_1)=q_1,
$$
quality functional:
$$
l=\int_{0}^{t_1} \sqrt{u^2_1+u^2_2} \ dt \rightarrow min,
$$
where the point $q \in \mathbb{R}^n$ determines the state of the system,
$u=(u_1,u_2)$ is a control, $t_1$ being fixed.
$ $ Notice that $f_1, f_2$ can be chosen just as they are up to any diffeomorphism. This is how the commutators of $f_1$ and $f_2$ look like:
\begin{align*}
f_3&= \frac{\partial{f_2}}{\partial{q}}f_1 -
\frac{\partial{f_1}}{\partial{q}}f_2 =[f_1,f_2] =(0,0,1,0,\dots,0)^T,
\\
f_4&= \frac{\partial{f_3}}{\partial{q}}f_1 -
\frac{\partial{f_1}}{\partial{q}}f_3 =[f_1,f_3]=(0,0,0,1,\dots,0)^T,
&\ldots
\\
f_n&= \frac{\partial{f_{n-1}}}{\partial{q}}f_1 -
\frac{\partial{f_1}}{\partial{q}}f_{n-1} =[f_1,f_{n-1}]=(0,0,\dots,0,1)^T.
\end{align*}
Nilpotent Lie algebra is generated by $f_1, f_2$:
$$
Lie(f_1, f_2) = span (f_1, f_2, \dots , f_n),
$$
multiplication table being of the form:
$$
[f_1,f_2] = f_3, \ [f_1,f_3] = f_4, \ \dots \ , \ [f_1,f_{n-1}] = f_n.
$$
All the others are equal to zero. These relations define the so-called Goursat distribution ([1], [2]).
$ $ Using the Pontryagin maximum principle, one can construct the Hamiltonian
$$
H(q,p,u) = \frac{u^2_1 +u^2_2}{2}=\frac{\dot{x_1}^2+\dot{x_2}^2}{2}=\frac{\langle p,f_1 \rangle ^2 + \langle p,f_2 \rangle ^2}{2},
$$
thus obtaining the following system:
$$
\tag{1}
\begin{cases}
\dot{x_1}=p_1-x_2 p_3 - \dots -x_{n-1} p_n, \\
\dot{x_2}=p_2, \\
\dot{x_3}=-x_2 \dot{x_1}, \\
\ldots \\
\dot{x_n}=-x_{n-1} \dot{x_1}, \\
\dot{p_1}=0, \\
\dot{p_2}=p_3 \dot{x_1}, \\
\ldots \\
\dot{p_{n-1}}=p_n \dot{x_1}, \\
\dot{p_n}=0. \\
\end{cases}
$$
Let us introduce the new coordinates by the following way:
\begin{equation*}
\begin{cases}
P_1=p_1-x_2 p_3-x_3 p_4-\dots-x_{n-1} p_n, \\
P_n=p_n, \\
P_{n-1}=p_{n-1}-P_n x_1, \\
P_{n-2}=p_{n-2}-P_{n-1} x_1-P_n \frac{x_1^2}{2!}, \\
\dots \\
P_3=p_3-P_4 x_1-P_5 \frac{x_1^2}{2!}-\dots-P_n \frac{x_1^{n-3}}{(n-3)!}, \\
P_2=p_2. \\
\end{cases}
\end{equation*}
The following theorem holds.
$ $ Theorem.
(1) is the completely integrable system (in the Liouville sense).
The whole set of the first integrals is as follows: \begin{equation*}
\begin{cases}
F_n=P_n, \\
F_{n-1}=P_{n-1}, \\
\dots \\
F_3=P_3, \\
F_2=P_2-P_3 x_1-\dots-P_n \frac{x_1^{n-2}}{(n-2)!}, \\
F_1=H=\frac{1}{2} (P_1^2+P_2^2), \\
\end{cases}
\end{equation*}
Thus one can consider the following “moment map”:
$$\Phi: (x, P) \rightarrow
\begin{pmatrix}
F_n \\
\ldots \\
F_1 \\
\end{pmatrix}. $$
The primary aim here is to study critical points of this mapping and its properties. That's what we are keep working on.
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