Lyapunov techniques for stabilization of switched linear systems of conservation laws

Abstract

Abstract-In this paper, the exponential stability in L 2 -norm is investigated for a class of switched linear systems of conservation laws. The state equations and the boundary conditions are both subject to switching. We consider the problem of synthesizing stabilizing switching controllers. By means of Lyapunov techniques, three control strategies are developed based on steepest descent selection, possibly combined with a hysteresis and a low-pass filter. Some numerical examples are considered to illustrate our approach and to show the merits of the proposed strategies.

Switching Rules for Stabilization of Linear Systems of Conservation Laws

“…The limitation of this rule is a possible fast switching behavior (see [13] and the example therein for an illustration of such behavior), and from a theoretical point of view leading to the absence of solution beyond a given time. From a pratical point of view this fast switching is undesirable.…”

“…For a discussion on the impact of the μ parameter on the rate of convergence, and the number of switches per time unit, see [13]. slope.…”

“…The merits of the three switching rules have been illustrated in [13], though the velocities were constant and ε(t) ≡ 0. To illustrate Propositions 11 and 14, let us consider system (1) Thanks to a result of [3], boundary matrices G i which destabilize the unswitched system must satisfy ρ(|G i |) > 1.…”

New stability and exact observability conditions for semilinear wave equations

“…The LMI approach to distributed parameter systems is capable of utilizing nonlinearities and of providing the desired system performance (see e.g. [4,7,12]). For 1-D wave equations, several control problems were solved by using the direct Lyapunov method in terms of LMIs [8,5].…”