Stabilization of a general linear heat‐ODE system coupling at an intermediate point

Abstract

Summary This paper considers the stabilization of a general linear heat‐ODE system coupling at an intermediate point which is an extension from our previous work on stabilizing a coupled system of a heat PDE and a second‐order ODE. A novel backstepping transformation is proposed which can overcome the difficulty of using the one proposed in our previous work to a more general system considered in the current paper. Existence of smooth kernels in both the forward and inverse transformations is proved. Also, we show that forward and inverse transformations are a mutually inverse pair governed by the kernels equations. Finally, the effectiveness of the controller design is demonstrated with some numerical examples for heat‐ODE systems. Copyright © 2017 John Wiley & Sons, Ltd.

Bilateral Boundary Control Design for a Cascaded Diffusion-ODE System Coupled at an Arbitrary Interior Point

“…The system in question in this paper involves an unstable linear ODE coupled not at a boundary, but rather, at an interior point. Previous work by Zhou et al (2017) has studied this problem in the context of unilateral control design, employing a nontraditional Fredholm transformation technique with separable kernels. This is in contrast to the work proposed in this paper, which utilizes a methodology of bilateral control design called folding.…”

“…Certain results exist for these systems in the unilateral sense -for example, for ODEs "sandwiched" between first-order hyperbolic PDEs as in work by Yu et al (2019). This idea exists for ODEs sandwiched by parabolic equations in the work by Zhou et al (2017). The "opposite" case of a parabolic equation sandwiched by ODEs is considered by Wang and Krstic (2019).…”