Solving the regulator problem for the one-dimensional Schrodinger equation via backstepping

Stabilization of higher order Schrödinger equations on a finite interval: Part I

We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this…

Regulator Design for a Congested Continuum Traffic Model with App-Routing Instability

In this paper, we propose a control design methodology for a linearized continuum traffic model in the congested regime. The continuum traffic flow on a highway is modeled using a linearized quasilinear hyperbolic partial differential equation model known as the Aw-Rascle-Zhang (ARZ) model. The linear traffic model is augmented with a novel non-local boundary condition representing car influx due to the use of routing apps such as Google Maps and Waze. The routing apps act as real-time previews for highway traffic, introducing potentially destabilizing feedback in the app-based navigation deci…

Stabilization of higher order Schrödinger equations on a finite interval: Part II

Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in Part I of this study where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to s…

The exponential tracking and disturbance rejection for the unstable Burgers' equation with general references and disturbances

In solving the problem of asymptotic tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected must be generated by an exosystem such as a finite dimensional exosystem with pure imaginary eigenvalues. The objective of this paper is to solve such a tracking problem for the unstable Burgers' equation without this assumption. Our treatment of this problem is straightforward. Using the method of variable transform, the tracking problem is split into two separate problems: a simple Neumann boundary stabilization problem a…

具有输出延迟和单一测量的不稳定波动方程的输出调节

Output regulation for an unstable wave equation with output delay and one measurement only

本文针对具有输出延迟和未知扰动的不稳定波动方程，通过反步法设计前馈调节器，并利用单一测量构建误差反馈控制器，实现跟踪误差指数收敛，为分布式参数系统输出调节提供开创性方案。

Robust Cooperative Output Regulation for a Network of Parabolic PDE Systems

This paper considers the robust cooperative output regulation for a network of parabolic PDE systems. The solution of this problem is obtained by extending the cooperative internal model principle from finite to infinite dimensions. For a time-invariant digraph describing the communication topology, a two-step backstepping approach is presented to systematically design cooperative state feedback regulators. They allow to solve both the leader-follower and the leaderless output synchronization problem in the presence of disturbances and model uncertainty for a finite-dimensional leader. Solvabi…