Stabilization of an ODE–Schrödinger Cascade

Abstract

We consider the problem of stabilization of a linear ODE with input dynamics governed by the linearized Schrödinger equation. The interconnection between the ODE and Schrödinger equation is bi-directional at a single point. We construct an explicit feedback law that compensates the Schrödinger dynamics at the inputs of the ODE and stabilizes the overall system. Our design is based on a two-step backstepping transformation by introducing an intermediate system and an intermediate control. By adopting the Riesz basis approach, the exponential stability of the closed-loop system is built with the pre-designed decay rate and the spectrum-determined growth condition is obtained. A numerical simulation is provided to illustrate the effectiveness of the proposed design.

Introduction

The backstepping approach, which was originally developed for parabolic PDEs [1], [2], has recently been applied to control problems for PDE–ODE cascades [3], [4], [5], [6], [7] etc., with applications of interest including fluids, structures, thermal, chemically-reacting, and plasma systems. This method uses an invertible Volterra integral transformation together with the boundary feedback to convert the unstable plant into a well-damped target system. The kernel of this transformation satisfies a certain PDE and ODE, which turn out to be solvable in closed form. In [3], the backstepping method was applied to a system which consists of the first-order hyperbolic PDE coupled with a second-order (in space) ODE. This system resembles the Korteweg–de Vries equation which describes shallow water waves and ion acoustic waves in plasma. In [5], predictor-like feedback laws and observers have been developed for diffusion PDE–ODE cascades. In [6], an explicit feedback law was developed that compensates the wave PDE dynamics at the input of an linear time-invariant ODE and stabilizes the overall system. In [7], PDE–ODE cascades considered were extended from Dirichlet type interconnections to Neumann type interconnections. For PDE–ODE cascades considered above, the interconnection between the PDE and the ODE is one directional. For example, in [5], the dynamics at the input of the ODE is governed by a heat equation, whereas the ODE does not act on the PDE. In some cases, the interconnection between them could be bi-directional, i.e., the PDE and ODE are coupled with each other as discussed in [8].

In this paper, we consider a problem of stabilization of an ODE–Schrödinger equation cascade as in (1), (2), (3), (4), where the interconnection between them is bi-directional at a single point:

$$\dot{X}\left(t\right)=AX\left(t\right)+Bu(0,t)\text{,}t>0\text{,}$$$$u_t(x,t)=-i{u}_{xx}(x,t)\text{,}x\in (0,1),t>0\text{,}$$$$u_x(0,t)=CX\left(t\right)\text{,}$$$$u(1,t)=U\left(t\right)$$

where $A\in {\mathbb{C}}^{n\times n}$, $B\in {\mathbb{C}}^{n\times 1}$, $C\in {\mathbb{C}}^{1\times n}$; $X\left(t\right)\in {\mathbb{C}}^{n\times 1}$ is the state of ordinary differential equation; $u(x,t)\in \mathbb{C}$ is the state of Schrödinger equation; and $U\left(t\right)\in \mathbb{C}$ is the control force to the entire system. The whole system is depicted in Fig. 1. Both states of ODE and Schrödinger equation are complex valued. The motivation for this kind of problem can be provided in the context of various applications in quantum mechanics, chemical process control, and other areas.

The stabilization of the linearized Schrödinger equation using boundary control was investigated in [9], [10], [11]. For exact controllability and observability results, see [12], [13], [14]. The interconnection of Schrödinger–heat equation with boundary coupling was considered in [15] and the Schrödinger equation has the Gevrey regularity under the compensation of the heat equation. As in [11], [16], the Schrödinger equation is usually considered as a complex-valued heat equation such that the backstepping method developed for parabolic PDEs could be applied. Motivated by two-step backstepping transformations in [5], which was adopted to improve performance and achieve exponential stability with arbitrarily fast decay rate, we adopt two-step backstepping control design to make the system stable in this paper.

Compared with our previous works using backstepping designs, the main contributions of the results in this paper lie in:

• Instead of one-step backstepping control, which results in difficulty in finding the kernels for the stabilization of the ODE–Schrödinger cascade considered in this paper, the design developed in this paper is based on a two-step backstepping transformation by introducing an intermediate system and an intermediate control.

• Instead of the Lyapunov synthesis, the Riesz basis approach is adopted in this paper, through which the exponential stability of the closed-loop system was built with the pre-designed decay rate and the spectrum-determined growth condition was obtained.

The paper is organized as follows. In Section 2, the two-step backstepping design is developed using the bounded and invertible operators. First, we design the comprehensive control to convert the original system into the intermediate system. Secondly, we design the intermediate control to convert the intermediate system into the final target system. Section 3 is devoted to the spectral analysis of the target system. In Section 4, Riesz spectral method is adopted for the stability analysis of the closed-loop system. Finally, a simulation example and the concluding remarks are provided in Section 5.

In this paper, we consider the following energy Hilbert space

$$\mathcal{H}={\mathbb{C}}^n\times L^2(0,1)$$

with inner product

$${〈f_1,f_2〉}_{\mathcal{H}}=\overline{X_1^T}{X}_2+{\int }_0^1\overline{z_1\left(x\right)}{z}_2\left(x\right)dx\text{,}{f}_i=(X_i,z_i)\in \mathcal{H}\text{,}i=1,2\text{,}$$

and $\mathcal{H}$-norm

$${\Vert f_i\Vert }_{\mathcal{H}}={({\Vert X_i\Vert }_{{\mathbb{C}}^n}^2+{\Vert z_i\Vert }_{L^2(0,1)}^2)}^{\frac{1}{2}}\text{.}$$