| Russian Mathematical Surveys |
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This article is cited in 2 scientific papers (total in 2 papers) Brief communications Chattering extremals in control-affine stabilization problems N. B. Melnikova, M. I. Ronzhinab a Lomonosov Moscow State University, Moscow, Russia b Gubkin Russian State University of Oil and Gas "Gubkin University", Moscow, Russia
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    In stabilization problems, under sufficiently broad assumptions the Hamiltonian of Pontryagin’s maximum principle $H(\psi,x,u)=H_0(\psi,x)+u H_1(\psi,x)$ is a function affine in the control $|u|\leqslant 1$ (see, for instance, [1]–[5]). Therefore, the problem can have singular extremals $(\psi(t),x(t))$ such that $H_1(\psi(t),x(t))\equiv0$. A singular extremal $(\psi(t),x(t))$ has the intrinsic $l$th order if the relations
$$
\begin{equation*}
\frac{\partial}{\partial u}\, \frac{d^i}{dt^i} H_1(\psi,x)=0, \quad i=1,\dots,2l-1,\quad\text{and} \quad \frac{\partial}{\partial u}\, \frac{d^{2l}}{dt^{2l}}H_1(\psi,x) \ne 0
\end{equation*}
\notag
$$
are valid for all $(\psi,x)$ in a neighbourhood of the singular extremal, and it has the local $l$th order if they are valid only at the points of the singular extremal itself.
In control problems for mechanical systems singular extremals of the second order exist for a generic Hamiltonian [6], [7]. If a singular extremal has the intrinsic second order and satisfies the strict Kelley condition, then any non-singular extremal that hits the singular one must be a chattering extremal, that is, it must have an infinite number of control discontinuities on a finite interval of time. The existence of chattering extremals in a neighbourhood of a singular extremal of the local second order has been proved for a Hamiltonian system that can be reduced to the form [6]
$$
\begin{equation*}
\begin{alignedat}{3} \dot{z}_1 &=z_2+f_1(z,w,u), &\quad \dot{z}_3 &=z_4+f_3(z,w,u), &\quad \dot{w} &=F(z,w,u), \\ \dot{z}_2 &=z_3+f_2(z,w,u), &\quad \dot{z}_4 &=\alpha(w)+\beta(w)u+f_4(z,w,u), &\quad u &= \operatorname{sgn} z_1, \end{alignedat}
\end{equation*}
\notag
$$
where the functions $f_i(z,w,u)$ are of small order: $f_i (g_\lambda(z),w,u)=O(\lambda^{5-i})$ as $\lambda\to 0$ with respect to the action of the Fuller group $g_\lambda(z)=(\lambda^4 z_1,\lambda^3 z_2,\lambda^2 z_3,\lambda z_4)$, $\lambda > 0$. For the variables $(z,w)$ defined by Poisson brackets of a special form, the conditions on $f_i(z,w,u)$ can be weakened [7]. But these Poisson brackets can turn out to be functionally dependent. For the stabilization problem at the origin of $K+1$ degrees of freedom in a system with $N$ degrees of freedom the following holds.
Theorem 1. Let $(z,w,v)\in\mathbb{R}^{4N}$, where $w=(w_1,\dots,w_K)\in\mathbb{R}^{4K}$ and $v\in\mathbb{R}^{4L}$, be a coordinate system such that $(z_0,w_0,v_0)=(0,0,v_0)$ is a point on a singular extremal of the second order, and in some neighbourhood the Hamiltonian system can be written in the form We outline the proof for the case of two degrees of freedom ($K=1$ and $L=0$). For system (1) we define the Poincaré map $\Phi\colon \Sigma\to\Sigma$ of the control switching surface $\Sigma=\{(z,w) \mid z_1=0\}$ onto itself. We resolve the singularity of the two-fold Poincaré map $\Phi^2$ using the change of variables
$$
\begin{equation*}
\begin{alignedat}{4} z_1 &= \xi_4^4 \xi_1, &\quad z_2 &= \xi_4^3 \xi_2, &\quad z_3 &= \xi_4^2 \xi_3, &\quad z_4 &= \xi_4, \\ w_1 &= \xi_4^4 \eta_1, &\quad w_2 &= \xi_4^3 \eta_2, &\quad w_3 &= \xi_4^2 \eta_3, &\quad w_4 &= \xi_4 \eta_4. \end{alignedat}
\end{equation*}
\notag
$$
We find the fixed point $(\xi^*,\eta^*)=(0,\xi^{*}_2,\xi^{*}_3,0,0,\xi^{*}_2,\xi^{*}_3,1)$ of $\Phi^2$ on the surface $\xi_4=0$, which corresponds to the chattering extremals reaching zero. We calculate the differential of $\Phi^2$ at this fixed point:
$$
\begin{equation*}
D\Phi^2(\xi^{*},\eta^{*})\,{=}\! \begin{pmatrix} \dfrac{\partial\xi^{(2)}_2}{\partial\xi_2} & \dfrac{\partial\xi^{(2)}_2}{\partial\xi_3} & * & 0 & 0 & 0 & 0 \\ \dfrac{\partial\xi^{(2)\vphantom{\sum^0}}_3}{\partial\xi_2}\! & \dfrac{\partial\xi^{(2)}_3}{\partial\xi_3}\! & * & 0 & 0 & 0 & 0 \\ 0 & 0 & \!\!(1+\tau_1^*)^2\!\! & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \!(1+\tau_1^*)^{-8}\!\! & * & * & * \\ 0 & 0 & 0 & 0 & \!(1+\tau_1^*)^{-6}\!\! & * & * \\ 0 & 0 & 0 & 0 & 0 & \!(1+\tau_1^*)^{-4}\!\! & * \\ 0 & 0 & 0 & 0 & 0 & 0 & \!(1+\tau_1^*)^{-2}\! \end{pmatrix}.
\end{equation*}
\notag
$$
We show that the fixed point $(\xi^*,\eta^*)$ of $\Phi^2$ is hyperbolic. Applying the invariant manifold theorem (see [8]) we obtain that the stable submanifold is one-dimensional, and the eigenvalue $(1+ \tau_1^*)^2\in(0,1)$ of the map $D\Phi^2(\xi^*,\eta^*)$ corresponds to an eigenvector transversal to the plane $\xi_4=0$. Returning to the coordinates $(z,w)$, we write the invariant curve in the form $z_1=0$, $z_2=\xi_2^* z_4^3+o(z_4^3)$, $z_3=\xi_3^* z_4^2+o(z_4^2)$, $w=z$. This curve defines a two-dimensional integral submanifold of system (1), which is filled by chattering extremals going to the origin.
As an example we consider the stabilization problem for a nonlinear system consisting of a beam pivoted in the middle and a ball that can roll along this beam [9]. The chattering-bundle theorems [6], [7] are not applicable here, but Theorem 1 ensures the existence of chattering extremals. 
Citation in format AMSBIB
\Bibitem{MelRon24} |
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