Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE

Abstract

This paper deals with the exponential stability of systems made of a hyperbolic PDE coupled with an ODE with different time scales, the dynamics of the PDE being much faster than that of the ODE. Such a difference of time scales is modeled though a small parameter $\varepsilon$ multiplying the time derivative in the PDE, and our stability analysis relies on the singular perturbation method. More precisely, we define two subsystems: a reduced order system, representing the dynamics of the full system in the limit $\varepsilon = 0$, and a boundary-layer system, which represents the dynamics of the PDE in the fast time scale. Our main result shows that, if both the reduced order and the boundary-layer systems are exponentially stable, then the full system is also exponentially stable for $\varepsilon$ small enough, and our strategy is based on a spectral analysis of the systems under consideration. Our main result improves a previous result in the literature, which was proved using a Lyapunov approach and required a stronger assumption on the boundary-layer system to obtain the same conclusion.

Figures (2)

• Figure 1: Solution of the ODE in \ref{['transport-ode']} for various values of $\varepsilon$ and solution of the reduced order system \ref{['ros']}.
• Figure 2: (a) Trace of the solution of the PDE in \ref{['transport-ode']} at $x = 0$ for various values of $\varepsilon$, compared with the function $\bar{y}_\ast(t) = (I_m - G_1)^{-1} G_2 \bar{z}(t)$. (b) Solution of the PDE in \ref{['transport-ode']} with $\varepsilon = 0.01$, in the time intervals $[0, 8\varepsilon]$ (left) and $[0, 5]$ (right). In both (a) and (b), the top and bottom figures represent the components $y_1$ and $y_2$ of $y$, respectively.

Theorems & Definitions (17)

• Definition II.1
• Proposition II.2
• Proposition II.3
• Theorem III.1
• Proposition III.2
• Lemma III.3
• proof
• Lemma III.4
• proof
• Lemma III.5