Stabilization of a spatially non‐causal reaction–diffusion equation by boundary control

Then, one remains to design the associated kernel functions (such as p ( x , y ), k ( x , y ) and ϕ ( x )) in to convert the system into an exponentially stable target system

({, \begin{matrix} array \dot{X} (t) = (A + B K) X (t) + B w (x_{0}, t), t > 0, \\ array w_{t} (x, t) = w_{x x} (x, t), x \in (0, 1), t > 0, \\ array w_{x} (0, t) = 0, w (1, t) = 0, t > 0, \\ array w (x, 0) = w_{0} (x), X (0) = X_{0}, \end{matrix})

where K is any chosen feedback matrix such that A + B K is Hurwitz. By w (1, t )=0, we can construct a state feedback form

\begin{matrix} right U (t) & left = \int_{0}^{x_{0}} p (1, y) u (y, t) d y & right \\ right & left \end{matrix}

…”

Section: Backstepping Transformationmentioning

confidence: 99%

“…By taking derivative of (2) with respect to x, we obtain . Similarly, taking derivative of (2) with respect to t , according to the system (1) and integration by parts, then we have For system (9), we can obtain the following C 2 classical solutions for p.x; y/; k.x; y/ and .x/.…”

Section: Solutions Of the Kernel Equationsmentioning

confidence: 99%

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

Zhou

Ren

2017

Intl J Robust & Nonlinear

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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.

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“…The backstepping method, in particular, has several advantages in feedback controller design, such as being easy to understand and implement (see [4,5,7,9,10,12]). For the stabilization of PDEs with integral terms, several backstepping methods are available (see [1,2,6,11,13] backstepping for ODE-wave and ODE-heat systems with integral terms. Recently, in [13], the stabilization of an ODEheat system in which the heat equation and ODE are coupled at an intermediate point has been considered.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of a coupled second order ODE-wave system

Zhen

Ke²,

Zhou

et al. 2016

2016 35th Chinese Control Conference (CCC)

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This paper considers the stabilization of a coupled second order ODE-wave system, where the ODE dynamics contain the solution of the wave equation at an intermediate point. We design a stabilizing feedback controller by choosing a suitable target system and backstepping transformation. The backstepping transformation is defined in terms of several kernel functions, for which we establish existence, uniqueness and smoothness properties. We also prove exponential stability for the resulting closed-loop system. Finally, the effectiveness of the proposed feedback controller is verified via a numerical example.

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Stabilization of a spatially non‐causal reaction–diffusion equation by boundary control

Cited by 20 publications

References 14 publications

Stabilization of a second order ODE–heat system coupling at intermediate point

Stabilization of a second order ODE–heat system coupling at intermediate point

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

Stabilization of a coupled second order ODE-wave system

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