Section: New Results

New results: switched systems

• In [6] we consider a family of linear control systems $\dot{x}=Ax+\alpha Bu$ on ${ℝ}^d$, where $\alpha$ belongs to a given class of persistently exciting signals. We seek maximal $\alpha$-uniform stabilization and destabilization by means of linear feedbacks $u=Kx$. We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if there exists at least one $K$ such that the Lie algebra generated by $A$ and $BK$ is equal to the set of all $d\times d$ matrices, then the maximal rate of convergence of $(A,B)$ is equal to the maximal rate of divergence of $(A,B)$. We also provide more precise results in the general single-input case, where the above result is obtained under the simpler assumption of controllability of the pair $(A,B)$.

• The paper [10] considers the stabilization to the origin of a persistently excited linear system by means of a linear state feedback, where we suppose that the feedback law is not applied instantaneously, but after a certain positive delay (not necessarily constant). The main result is that, under certain spectral hypotheses on the linear system, stabilization by means of a linear delayed feedback is indeed possible, generalizing a previous result already known for non-delayed feedback laws.

• In [16] and [26] 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 the linear retarded functional differential equations. This is carried out using the switched system transformation approach.

• 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. In [23] we deal with 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 $\{A,A+b{c}^T\}$. After pointing out a fatal gap in Barabanov's proof we partially recover his result by alternative arguments.

• In [24] we address the exponential stability of a system of transport equations with intermittent damping on a network of $N\ge 2$ circles intersecting at a single point $O$. The $N$ equations are coupled through a linear mixing of their values at $O$, described by a matrix $M$. The activity of the intermittent damping is determined by persistently exciting signals, all belonging to a fixed class. The main result is that, under suitable hypotheses on $M$ and on the rationality of the ratios between the lengths of the circles, such a system is exponentially stable, uniformly with respect to the persistently exciting signals. The proof relies on an explicit formula for the solutions of this system, which allows one to track down the effects of the intermittent damping.