Output feedback stabilization of cascaded ODE‐Wave equations with time delay in observation

Abstract

In this paper, we are concerned with the stabilization of a cascaded ODE‐wave system subject to boundary observation with time delay. Well‐posedness of the open‐loop system is observable to show first. Both observer and predictor systems are designed to estimate the state variable on the time interval [0,t−τ] when the observation is available, and to predict the state variable on the time interval [t−τ,t] when the observation is not available respectively. Next, we transform the original system to the well‐posed and exponentially stable system by the backstepping method. Based on the estimated state variable and the output feedback stabilizing controller using the backstepping method. It is shown that the closed‐loop system is exponentially stable. The numerical simulation is presented to illustrate the effect of the stabilizing controller.

Infinite‐dimensional extended state observer based stabilization for an ODE‐wave cascaded system with control matched disturbance

“…Since (A, B) is stabilizable, there exists a matrix K ∈ C n×1 such that A + BK is a Hurwitz matrix. When there is no disturbance, Than and Wang [13] verified that, the state feedback based control law…”

“…In previous works [20, 23], the wave part of the considered ODE‐wave cascaded systems with disturbances had the Neumann boundary at the left end ( $$$ {u}_x\left(0,t\right)=0 $$$), whereas the wave part of our system () has Dirichlet boundary condition at the left end ( $$$ u\left(0,t\right)=0 $$$). The exponentially stability of system () without disturbance ( $$$ F(t)=0 $$$) was firstly discussed by Than and Wang [13]. Suppose that $$$ k>0 $$$ is a constant.…”

“…Since $$$ \left(A,B\right) $$$ is stabilizable, there exists a matrix $$$ K\in {\mathbb{C}}^{n\times 1} $$$ such that $$$ A+ BK $$$ is a Hurwitz matrix. When there is no disturbance, Than and Wang [13] verified that, the state feedback based control law $$$$ {\displaystyle \begin{array}{ll}U(t)=& \kern0.2em k\left[ KBu\left(1,t\right)-{u}_t\left(1,t\right)\right]+\rho (1)X(t)\\ {}& +{\int}_0^1\rho \left(1-x\right)\left[ Au\left(x,t\right)+B{u}_t\left(x,t\right)\right] dx,\end{array}} $$$$ where $$$$ \rho (x)=K\left(0\kern0.5em I\right){e}^{\left(\begin{array}{cc}0& {A}^2\\ {}I& 0\end{array}\right)x}\left(\begin{array}{c} kA\\ {}I\end{array}\right), $$$$ exponentially stabilizes system () without disturban...…”

“…Recent years, the control design for ODE-PDE cascaded system receives more and more attentions [6][7][8]. Backstepping method, developed by Krstic, is a powerful tool for the design of exponential stability control law for ODE-PDE cascaded systems, such as ODE-Schrödinger equations by Ren, Wang and Krstic [9], Tang and Xie's ODE-heat equations [10,11], and ODE-wave equations [12][13][14].…”

Boundary output feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems

“…More recently, an output feedback control was designed for fractional diffusion systems with spatially varying parameters in [11], while the stability robustness against small diffusivity perturbations was further studied in [12]. For more about the stabilization of fractional systems and the boundary control of PDEs, we refer to several works [13][14][15][16][17][18][19][20][21]. As far as we know, most of the existing works only considered about the state feedback design with scalar state.…”