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This paper deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations.

Motivated by the essence of spatial-temporal evolutions in nature, control of systems governed by partial differential equations (PDEs), or distributed parameter system, has been studied for decades (e.g., [

We consider the stabilization problem of a hyperbolic PDE system with a spatial-temporal actuation over the full physical domain. Our approach in this work relies on the backstepping method, which was originally developed in the 1990s for adaptive and robust control of nonlinear lumped parameter systems governed by ordinary differential equations (ODEs) [

Backstepping can be used to achieve the stabilization of unstable PDEs in a physically appealing way where the destabilizing terms are eliminated by means of an invertible integral transformation of the PDE together with the boundary feedback. In addition, unlike the linear quadratic regulator (LQR) approaches for boundary control [

The backstepping approach is so far a systematic method and gives a rather straightforward way for boundary feedback controller design for PDE systems. More recently, the authors in [

The problem of boundary feedback stabilization of first-order hyperbolic PDEs has been studied in [

The rest of this paper is organized as follows. In Section

We consider the following 1-dimensional hyperbolic PDE of the form

Since (

In the following, the design procedure is presented to the state feedback controller for the system (

We first introduce the following state transformation for system (

Now, we apply the backstepping transformation

Now, we will discuss the boundary condition that

We introduce the following differential transformation:

In this section, we describe the invertibility of the differential transformation defined by (

Assuming that

To verify the existence of

Next, we will prove (

From (

We note that

Assume that

Under the feedback control law given in Theorem

Assume that

Define another linear operator

By Lemma

If (

Let

We first show that, for any

Let

To prove this theorem, we need to show that the control law (

Now, we will show that, for any

By making use of the transformations (

Assume that

For any given

Let

Now we consider the following 1-dimensional hyperbolic PDE (i.e., by taking

We choose

The response of the open-loop system.

The response of the closed-loop system (

The temporal function

The shape function

The interior actuation function

By Theorem

The

The

The

We can realize that the spatial-temporal actuation

As shown in Figure

The response of the closed-loop system control acting on

The response of the closed-loop system with control acting on

The response of the closed-loop system with control acting on

The shape function

In this paper, the stabilization problem using backstepping techniques for first-order hyperbolic PDEs with internal actuation has been investigated. For this, backstepping integral transformations together with a differential transformation are used to determine the state feedback controller. In future work, several directions can be pursued. First, just as said in Section

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by the National Natural Science Foundation of China (61374096 and 61104048) and the Natural Science Foundation of Zhejiang (Y6110751).