Abstract

This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.

中文翻译：

---

具有边界输入时滞的ODE-线性KdV级联系统的指数稳定性

摘要

This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.