Stefan M. Grünvogel
Lyapunov Exponents and Control Sets new Singular Points
Abstract: We consider control affine systems of the form
\[
\begin{array}[c]{l}
\dot{x} = f_{0}(x) + \sum_{i=1}^{m} u_{i}(t) f_{i}(x), \\
u \in \mathcal{U} = \{ u: R \rightarrow U: \mbox{loc. integrable} \}
\end{array}
\]
on $R^{d}$ with compact control range $U$ and a singular point $x^{\ast} \in R^{d}$, i.e. $f_{i}(x^{\ast}) = 0, i=0, \ldots, m$. Control sets are maximal subsets of $R^{d}$ with nonvoid interior where the system is approximately controllable. We suppose that there is a periodic control functions $u^h$ such that linearized the system has positive and negative Lyapunov exponents for $u^h$ and a periodic control function $u^{s}$ such that the linearized system has only negative Lyapunov exponents for $u^{s}$. Under an inner pair condition we show the existence of control sets near the singular point, by using local stable and unstable manifolds of the system corresponding to $u^{h}$ and its asymptotic phase.
\[
\begin{array}[c]{l}
\dot{x} = f_{0}(x) + \sum_{i=1}^{m} u_{i}(t) f_{i}(x), \\
u \in \mathcal{U} = \{ u: R \rightarrow U: \mbox{loc. integrable} \}
\end{array}
\]
on $R^{d}$ with compact control range $U$ and a singular point $x^{\ast} \in R^{d}$, i.e. $f_{i}(x^{\ast}) = 0, i=0, \ldots, m$. Control sets are maximal subsets of $R^{d}$ with nonvoid interior where the system is approximately controllable. We suppose that there is a periodic control functions $u^h$ such that linearized the system has positive and negative Lyapunov exponents for $u^h$ and a periodic control function $u^{s}$ such that the linearized system has only negative Lyapunov exponents for $u^{s}$. Under an inner pair condition we show the existence of control sets near the singular point, by using local stable and unstable manifolds of the system corresponding to $u^{h}$ and its asymptotic phase.