Section: New Results

New results: switched systems

• In [5] we consider a continuous-time linear switched system on ${ℝ}^n$ associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a Barabanov norm associated with the system. We look at two types of issues: $\left(a\right)$ properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; $\left(b\right)$ asymptotic behaviour of the extremal solutions of the linear switched system. Regarding Issue $\left(a\right)$, we provide partial answers and propose four related open problems. As for Issue $\left(b\right)$, we establish, when $n=3$, a Poincaré–Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behaviour of linear switched system on ${ℝ}^3$ associated with a pair of Hurwitz matrices $\{A,A+b{c}^T\}$.

• Motivated by an open problem posed by J.P. Hespanha, in [23] we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one.

• In [14] in the totally observed case and in [23] in the general case, we answer an open problem posed by J.P. Hespanha in 2003. We first extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the $L_2$-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one.

• In [24] we address the stability of non-autonomous difference equations by providing a suitable representation of the solution at time $t$ in terms of the initial condition and time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of that of such coefficients. As a consequence, we obtain necessary and sufficient stability criteria for non-autonomous linear difference equations. In the case of difference equations with arbitrary switching, we obtain a generalization of the well-known criterion for autonomous systems due to Hale and Silkowski, which, as the latter, is delay-independent. These results are applied to transport and wave propagation on networks. In particular, we show that the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but one.

• For linear systems in continuous time with random switching, we characterize in [25] the Lyapunov exponents using the Multiplicative Ergodic Theorem for an associated system in discrete time. An application to control systems shows that here a controllability condition implies that arbitrary exponential decay rates for almost sure stabilization can be obtained.

A result related to switched system is the one obtained in [6] and [15] : we study the stability of linear time-varying delay differential equations where the delay enters as a switching parameter. In [6] we give a collection of converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly-degenerate Lyapunov–Krasovskii functional is a necessary and sufficient condition for the global exponential stability of linear retarded functional differential equations. In [15] the fundamental question that we consider is the following: assuming that every individual (constant-delay) subsystem is exponentially stable, can we characterize the cases when the system is not exponentially stable? This is nothing else than the so-called Markus-Yamabe instability and we give new conditions ensuring it.