Exponential stabilization of cascade ODE-linearized KdV system by boundary Dirichlet actuation

Abstract

In this paper, we solve the problem of exponential stabilization for a class of cascade ODE-PDE system governed by a linear ordinary differential equation and a 1 − d linearized Korteweg-de Vries equation (KdV) posed on a bounded interval. The control for the entire system acts on the left boundary with Dirichlet condition of the KdV equation whereas the KdV acts in the linear ODE by a Dirichlet connection. We use the socalled backstepping design in infinite dimension to convert the system under consideration into an exponentially stable cascade ODE-PDE system.Then, we use the invertibility of such design to achieve the exponential stability for the ODE-PDE cascade system under 2010 Mathematics Subject Classification. 93D05, 93D20, 34D05, 34D20.

Stabilization of the coupled ODE-linearized KdV system

“…Proof: Combined with the well-posedness analysis in [5] and [6], we know system (1) has a unique solution , ∈ 0, ∞ , 0,1 . Next, we prove the stability of target system, then show the stability of the closed-loop system in terms of the invertibility of transformation.…”

“…To prove the stability of the closed-loop system (1) with the control law (6), we need to show that transformation (2) is bounded and invertible.…”

“…People have carried out a lot of research on the KdV equation, see [1][2] and reference therein. For changing the state of KdV for satisfying our purpose, its control problem is introduced, for example, kinds of stability and controllability results for KdV can be found in [3][4][5][6][7] and reference therein.…”

“…For researching more complex shallow water behavior coupled with its surrounding, the coupled ODE-KdV system are presented to describe it. In [6], the authors considered the stabilization of cascaded ODE-linearized KdV with lower-order term interaction, but not the case of higher-order terms. In fact, stability issues are more difficult when ODE-linearized KdV is coupled to higher-order terms.…”

Output feedback stabilization of an ODE-Reaction–Diffusion PDE cascade with a long interconnection delay

“…The state-feedback stabilization and the observer design of a diffusion PDE cascaded with an ODE was solved using backstepping design in [23,36]. These control design approaches, which can be seen as the generalization of predictor feedback techniques [19] as they aim at compensating an infinite-dimensional input dynamics [24], have then been extended to the statefeedback stabilization of other systems described by PDEs such as strings [22,36] and linearized Korteweg-de Vries equation [3]. They have also been extended to the state-feedback stabilization and observer design of multi-input-multi-output LTI systems with actuator or sensor dynamics governed by diffusion [7] and wave [6] PDEs, the output feedback stabilization of a diffusion PDE coupled with an ODE [37,38], and the output feedback stabilization of either a diffusion PDE [39] or hyperbolic PDEs [14,41] sandwiched between two ODEs.…”