Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

Abstract

Abstract. In this paper we study asymptotic behaviour of distributed parameter systems governed by partial differential equations (abbreviated to PDE). We first review some recently developed results on the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionals we prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDE systems. Then we apply the result to establish exponential stability of various chemical engineering processes and, in particular, exponential stability of heat exchangers. Through concrete examples we show how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE. Meanwhile we explain how the method is adapted to the framework of Banach spaces L p , 1 < p ≤ ∞.

Stability and L 2 ‐gain performance for non‐linear switched impulsive systems

“…If conditions (7) and (9) in Lemma 3 of [6] hold, then we can find α and T a = τ a for β = μ > 1. In fact…”

“…Then (9) yields (8) with := −N 0 ln β. Thus, under (4)-(7), the system is GUAS over AII satisfying (9).…”

“…While in this paper, we do not require the Lyapunov functions be always decreasing during the continuous portion, and jumping increase is allowed at the impulsive instants. The switched Lyapunov functions only need to satisfy an aggregated form (9). Moreover, compared with the results in [20], we do not need the Lipschitz continuation of the non-linear terms φ σ (t) (t, x(t)), and the condition ρ k = ρ(I + D k ) for the non-linear terms in the impulses is also relaxed.…”

“…System (1) can achieve L 2 -gain γ = max{μ exp(−N 0 ln β), μ exp(N 0 ln β)} with AII defined by (9) if there exist diagonal positive definite matrices P i ∈ R n×n , a ∈ R, η, ζ , δ k > 0 and switching signal σ (t) such that and (11) …”

“…Recently, great development of stability theory for dynamic systems including switched systems has been made in the control community, see [1][2][3][4][5][6][7][8][9]. When switched systems are not stable under arbitrary switching, it is necessary to search for effective tools to guarantee the stability of the systems.…”