A journey in numerical linear algebra: a workshop in honor of Michele Benzi’s 60th birthday

Computing the most stable switching laws of a linear system.

11 Jun 2022, 11:45

30m

Aula Magna (Dipartimento di Matematica, Università di Pisa)

Speaker

Nicola Guglielmi (Gran Sasso Science Institute)

Description

When considering the stability under arbitrary switching of a discrete-time linear switched system
$$x(n+1)=A_{\sigma (n)}x(n),\sigma :\mathbb{N}\to \{1,...,m\},$$ where $x(0)\in {\mathbb{R}}^d$, the matrix $A_{\sigma (n)}\in {\mathbb{R}}^{d\times d}$ belongs to a finite family $F=\{A_i{\}}_{1\le i\le m}$ 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).