\documentclass{abstract}
\def\R{\mathbb R}

\begin{document}

\title{Bang--bang trajectories with  a double switching time in the minimum time problem}
\author{Laura Poggiolini}
\institute{Universit\`a degli Studi di Firenze}
\email{laura.poggiolini@unifi.it}
\coauthor{Marco Spadini} % comment out if not needed

\mymaketitle

\section{Abstract}
We consider the minimum time problem with fixed end-points on a finite
dimensional manifold $M$ in the case when the dynamics is affine with
respect to the control and the control set is a box in $\R^m$.
Namely, we consider the following optimal control problem:
\begin{subequations}
  \label{eq:problema0}
  \begin{align}
    & T \to \min, \label{eq:costo0} \\
    & \dot\xi(t) = f_{0}(\xi(t)) %%%
    + \displaystyle\sum_{s=1}^m u_{s}(t) f_{s}(\xi(t)), \label{eq:dinamica0} \\
    & \xi(0) = x_{0}, \qquad \xi(T) = x_{f}, \label{eq:estremi0} \\
    & \left\vert u_{s}(t)\right\vert \leq 1 \quad s = 1,2,\ldots, m
    \qquad \rm{a.e.~} t \in [0, T]. \label{eq:box0}
  \end{align}
\end{subequations}
For such problem , we say that $(T, \xi, u)$ is an {\em admissible
  triple} if $T > 0$ and the couple $(\xi, u) \in W^{1, \infty}([0,
T], M) \times L^\infty([0, T], \R^m)$ satisfies \eqref{eq:dinamica0},
\eqref{eq:estremi0} and \eqref{eq:box0}.  We assume we are given a
reference triple $\begin{pmatrix} \widehat T, \widehat\xi,\widehat
  u \end{pmatrix} $ which satisfies Pontryagin Maximum Principle (PMP)
with an associated covector $\widehat\lambda$, and where the reference
control $\widehat u$ is a regular bang-bang control with a double
switching time $\widehat\tau$ and a finite number of simple switching
times.

We are interested in giving sufficient conditions for {\em strong
  local optimality} of the triplet, according to the following
definition:


{\em The trajectory $\widehat\xi$ is a state-local minimiser if there
  is a neighborhood ${\mathcal U}$ of its range in $M$ such that
  $\widehat\xi$ is a minimiser among the admissible trajectories whose
  range is in ${\mathcal U}$.  }

Our sufficient conditions are given using Hamiltonian methods, following the
same lines of the case where only simple switches occur (see
\cite{PS04}) but here the presence of a double switch makes the proof
of the local invertibility of the maximised Hamiltonian flow much more
tricky.  To keep the notation to the minimum in this talk I will
consider the case where $m=2$ and only the double switch occurs. In
fact this case already contains most of the mathematical difficulties
of the proof. Namely, the presence of a double switch gives rise to a
piecewise $C^1$ ($PC^1$) maximised Hamiltonian flow where the number
of {\em smoooth pieces} around $\widehat\lambda(\widehat\tau)$ is
five, thus requiring a non trivial proof of the local invertibility of
such flow, see \cite{PSp13}.

More precisely, the definition of $PC^1$ maps is the following: {\em
  Given two finite dimensional manifolds $N_1$ and $N_2$, we say that
  a function $\gamma \colon N_1 \to N_2$ is a continuous selection of
  $C^1$ functions if $\gamma$ is continuous and there exists a finite
  number of $C^1$ functions $\gamma_1$, \ldots, $\gamma_k$ from $N_1$
  in $N_2$ such that the {\em active index set} $I := \left\{i \in
    \{1, 2, \ldots, k\} \colon \gamma(x) = \gamma_i(x)\right\}$ is
  nonempty for each $x \in N_1$. The functions
  $\gamma_i$'s are called {\em selection functions} of $\gamma$. \\
  A continuous function $\gamma$ is called a $PC^1$ function if at
  every point $x \in N_1$ there exists a neighborhood $V$ such that
  the restriction of $\gamma$ to $V$ is a continuous selection of
  $C^1$ functions.  }





% \paragraph{Acknowledgments}
% Funding acknowledgments (if required)

\begin{thebibliography}{9}
\bibitem{PS04} Laura Poggiolini and Gianna Stefani.  \newblock
  State-local optimality of a bang-bang trajectory: a {H}amiltonian
  approach.  \newblock {\em Systems \& {C}ontrol {L}etters},
  53:269--279, 2004.

\bibitem{PSp13} Laura Poggiolini and Marco Spadini.  \newblock Local
  inversion of planar maps with nice nondifferentiability structure.
  \newblock {\em Advanced Nonlinear Studies}, 13:411--430, 2013.
\end{thebibliography}

\end{document}
