Speaker
Description
When considering the stability under arbitrary switching of a discrete-time linear switched system
$$
x(n + 1) = A_{\sigma(n)} x(n), \qquad
\sigma : \mathbb N \to \left\{ 1, . . . , m \right\},
$$
where $x(0) \in \mathbb{R}^d$, the matrix $A_{\sigma(n)} \in \mathbb{R}^{d \times d}$ belongs to a finite family
$F = \{ A_i \}_{1 \leq i \leq m}$ and $\sigma$ denotes the switching law, it is known that the most stable switching laws are associated to the so-called spectrum-minimizing products of the family $F$. The minimal rate of growth of its trajectories is called lower spectral radius and is denoted as $\hat \rho(F)$. For a normalized family $F$ of matrices (i.e., with lower spectral radius $\hat \rho(F) = 1$) that share an invariant cone $K$ (the most typical case is the positive orthant for nonnegative matrices), all the most stable trajectories starting from the
interior of $K$ lie on the boundary of a certain invariant set (called Barabanov antinorm), for which a canonical constructive procedure is available [4], that is based on the invariant polytope algorithm developed in [3]. After presenting the main results for this case, I will consider families of matrices $F$ that share an invariant multicone $K_{\mathrm{mul}}$ [1, 2] and show how to generalize some of the known results to this more general setting (Guglielmi & Zennaro, in progress [5]). These generalizations are of interest because common invariant multicones may well exist when common invariant cones do not.
References
[1] M. Brundu, M. Zennaro Multicones, duality and matrix invariance, J. Convex Anal. 26 (2019), pp. 1021–1052.
[2] M. Brundu, M. Zennaro Invariant multicones for families of matrices, Ann. Mat. Pur. Appl. 198 (2019), 571–614.
[3] N. Guglielmi, V.Yu. Protasov, Exact computation of joint spectral characteristics of linear operators, Found. Comput. Math., 13 (2013), pp. 37–97.
[4] N. Guglielmi, M. Zennaro Canonical construction of polytope Barabanov norms and antinorms for sets of matrices, SIAM J. Matrix Anal. Appl. 36 (2015), pp. 634–655.
[5] N. Guglielmi, M. Zennaro Antinorms for sets of matrices sharing an invariant multicone, in preparation (2022).
