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__A novel method to compute Lyapunov exponents of switched linear systems (II)__

M. Zennaro and N. Guglielmi

**Abstract**

For a given finite set of matrices $\mathcal{G} = \{ C_1,\ldots,C_m\}$, where
$C_i \in \mathbb{R}^{d,d}$, $i=1,\ldots,m$, we consider the switched
linear system of ODEs
$\dot{x}(t) = C( u(t)) x(t), x(0) = x_0$,
where $x(t) \in \mathbb{R}^d$, $u: \mathbb{R} \longrightarrow \{1,\ldots,m\}$ is a piecewise constant control function and $C(i)=C_i$.
We propose a novel method for the approximation of the upper Lyapunov exponent
and, under suitable assumptions, of the lower Lyapunov exponent which is based on the computation of the joint spectral radius and lower spectral radius of a sequence of discretized systems obtained by forcing the switching instants
to be multiple of $\Delta^{(k)} t$, where $\Delta^{(k)} t \to 0$ as $k \to \infty$.
In the first talk we shall give general ideas to bound
the Lyapunov exponents.
In the second talk we shall provide details about the approximation of the joint (lower) spectral radius.