Boundary Output Feedback Stabilization for a Cascaded-Wave PDE-ODE System with Velocity Recirculation

'is paper considers the output feedback stabilization for a cascaded-wave PDE-ODE system with velocity recirculation by boundary control. First, we choose a well-known exponentially stable system as its target system and find a backstepping transformation to design a state feedback controller for the original system. Second, we attempt to give an output feedback controller for the original system by introducing the observer. 'e resulting closed-loop system admits a unique solution which is proved to be exponentially stable. Finally, we give some numerical examples to prove the validity for the theoretical results.


Introduction
In control engineering, ordinary differential equations (ODEs) and partial differential equations (PDEs) are widely used to model such problems. Many researchers have done detailed research on this content in the past decades. Systems modeled by ODEs are common in this respect. On the other hand, there are more and more works that contributed to stabilization for systems described by PDEs in recent years ( [1][2][3][4][5][6][7]). For instance, aiming at solving the problem of system with variable coefficients and Neumann boundary actuation, Wu et al. used the backstepping approach that converted the wave equation into a same type equation in [8]. Not only that, the coupled PDE-ODE problem was also studied in control systems, where PDEs mainly included heat equations and wave equations. With the development of research on such problems, more results of PDE-ODE emerged. For example, in [9][10][11][12][13][14][15][16], the authors solved the stabilization for heat PDE-ODE by boundary control systems which used the PDE backstepping method. In [10,13], Tang and Xie considered the systems where ζ(l, t) � U(t) in boundary of "PDE part." e system of ζ(l, t) � U(t − d) in "PDE part" was considered in [12]. e stabilization for wave PDE-ODE systems by the PDE backstepping method was solved in [6,7,17,18]. But beyond that, in [19], Meglio solved the problem of stabilizing a linear ODE coupled first-order hyperbolic PDE, which was also based on the backstepping approach. is paper proposed a unified framework of iterative learning control for typical flexible structures under spatiotemporally varying disturbances in [20]. In [21], the control problem was addressed for a hybrid PDE-ODE system that described a nonuniform gantry crane system with constrained tension. In addition, the PDE-ODE models also played a significant role in practical application, such as 3D-printing [22], oil drilling [23], cable elevator [24], battery management [25], traffic [26], and so on.
is approach was dedicated to finding an invertible Volterra integral transformation which mapped the considered plant with the boundary feedback law into a known stable target system. About some PDE systems, Jin and Guo also used the backstepping method to solve the PDEs system of nonlocal terms in [31]. In [32], Hasan and Tang used the backstepping method to model boundary stabilization of the Korteweg-De Vries (KdV) equation with sensors and actuators. For some PDE-ODE systems, the authors also used the backstepping approach to design the controller in [6,7,[9][10][11][12][13][14][15][16][17][18]. In [17], Zhou et al. used directly the known backstepping method (from [30]) to achieve the stabilization of system matched disturbance with boundary control. It can be said that backstepping is a significant method and key move to deal with such PDE and PDE-ODE problems.
In this paper, our main focus is on the output feedback stabilization for a cascaded-wave PDE-ODE system subject to boundary control with velocity recirculation: On some special cases, we consider the following. For system (1) without "X part," Su et al. gave the backstepping method and the state feedback controller in [33], which was ζ tt (x, t) � ζ xx (x, t) + qζ t (0, t). When q � 0, by the backstepping method shown in [30], one can easily design a control law to stabilize system (1). And in [16], Zhou and Xu used backstepping transformation to settle the problem of coupled PDE-ODE where there was no non-local term in heat equation. On such coupling problems, there was few research on PDE-ODE problem with non-local term. Nonlocal terms, including both boundary terms and strict feedback/Volterra terms, have been the mainstay of the applications of PDE backstepping methods to parabolic PDEs. Equation ζ tt (x, t) � ζ xx (x, t) + qu t (x 0 , t) models the string vibration of an electric guitar with a pickup at the location x 0 ∈ [0, 1]. e move of x 0 leads to the location change of the pickup. Inspired by it, we let q � 0 in ζ tt (x, t) � ζ xx (x, t) + qu t (x 0 , t). Compared with [30], we add a non-local term qζ t (0, t) to the right end of the wave equation. is makes it more difficult for us to research this problem. In [30], the form of backstepping transformation is ; it does not work to solve the problem in their way. Inspired by (4) in [34], we obtain the new form of backstepping transformation. Using the backstepping and introducing the exponentially stable target system to design state feedback controller are also nice methods in this paper. e paper is organized as follows. We design the state feedback controller for system (1) in Section 2. Section 3 is contributed to designing the output feedback control and proving the exponential stability of the closed-loop system.
In Section 4, we give some numerical examples to prove the validity of the proposed controller.

State Feedback Controller Design
In this section, we are devoted to designing a state feedback controller for system (1). First, we introduce the backstepping approach: where the kernel functions k(x, y), l(x, y), μ(x), and c(x) will be defined later. Compared with the backstepping in [30], we add an additional term "− μ(x)ζ(0, t)" in transformation (2). And we hope that turns system (1) into the target system where c 1 , c 2 > 0 are tuning parameters and G is chosen such that D + FG is Hurwitz. It is well known that there exists a exponentially stable solution for "v part" in the state space H 1 � H 1 (0, 1) × L 2 (0, 1), which is equipped with the norm induced by inner product In order to get these kernel functions, we make the following calculation. Taking derivative of (2) with respect to Taking derivative of (2) with respect to x twice, we find In the same way, taking derivative of (2) with respect to t and taking derivative of (2) with respect to t twice and using the partial integral method, we can get the following: By the boundary condition v x (0, t) � 0 in (3) and (5), we obtain Complexity 3 Hence, we choose the functions k(x, y), l(x, y), We can obtain μ(0) � 0 by q x 0 l(x, y)dy + μ(x) � 0. In [30], we find that the solution of c(x) is as follows: By (2) and (10), Next we can let k(x, y) � k 1 (x − y), l(x, y) � l 1 (x − y); by "k part" in (10), we can have According to the above formulas, we obtain According to k(x, y) � k 1 (x − y), l(x, y) � l 1 (x − y), simplify the above three formulas in (14) By differentiating (17) with respect to x twice, we transform (17) into a second-order linear ODE and get the solution of k 1 (x): We substitute (20) for (18) and obtain the solution of l 1 (x). In the same way, the solution of μ(x) can also be obtained. Because We suppose that the inverse transformation of (2) exists in the form where m(x, y), n(x, y), h(x) are the kernel functions.
Derive (21) with respect to x, and we obtain Differentiating (21) with respect to x twice, we have 4 Complexity We differentiate (21) with respect to t and get Differentiating (21) twice on both sides with respect to t yields en, we have Complexity 5 e boundary condition at x � 0 in (1) implies Hence, we choose the functions m(x, y), n(x, y), l(x) to satisfy We obtain the solution of h(x) in [30]: Let m(x, y) � m 1 (x − y), n(x, y) � n 1 (x − y); then, Combining the forth equations in (1) and (3), we can obtain the controller U for (1) For convenience, we describe system (1) under controller (31) as

Moreover, closed-loop system (32) is exponentially stable:
there exist two positive constants M, ε depending on c 1 , c 2 and G such that Proof. It is a direct result from the equivalence between system (3) and (32). Obviously, we define the operator A 1 : D(A 1 ) ⟶ H 1 for the "v part" of target system (3): en the "v part" of system (3) can be written as an abstract evolutionary equation in As we all know, the operator A 1 can generate an exponentially stable C 0 -semigroup e A 1 t on H 1 [35]. In other words, there exist M 1 , α > 0 such that Next we consider the "X part" of target system (3) on R n . Because v(0, t) ∈ C(0, ∞), using the constant variational formula, the solution of "X part" is given by Because D + FG is Hurwitz, there exist positive constants C 5 , θ > 0 such that From eorem 1 in [17], the solution of "X part" is exponentially stable. So, system (3) admits a unique solution and the solution is exponentially stable. According to transformations (2) and (21), we define a bounded invertible operator P: (X, f, g) ⟶ (X, φ, ψ) in H: where (14)(16) and (30), respectively. By the formula above, we have the following expression:

Complexity 7 and k(x, y), l(x, y), μ(x), c(x), m(x, y), n(x, y), h(x) are defined in
Since the operator P is bounded and invertible, there exists M 2 > 0 such that ‖P − 1 ‖ < M 2 . Besides, the "v part" admits a unique solution and the solution is exponentially stable, so the solution of closed-loop system (32) is exponentially stable by (42). ere exist M, ε > 0 such that (34) holds.

Observer and Output Feedback Controller Design
In this section, we focus on designing output feedback controller for system (1) on the grounds of the observer. Next, we give the observer for system (1).

Complexity
Proof. As we all know, the "ζ part" of system (44) admits a unique exponentially stable solution from [33] and D + PQ is Hurwitz, so system (44) has a unique exponentially stable solution.
Notice that the energy (E(t) − |X| 2 R n ) is equivalent to ‖(ζ(·, t), ζ t (·, t)) ⊤ ‖ 2 H 1 . We only prove (48) and (49). By Cauchy inequality and Hölder inequality, we have for some constant C > 0. It is obvious that Integrating above equation from t 1 to t 2 , we obtain Define another function It is clear that |ρ(t)| ≤ E(t). Taking derivative of (54), we get Integrating above equation from t 1 to t 2 , we have for some constant C 2 > 0. Hence, (48) and (49) hold. Especially, when t 1 � 0, f(t) ∈ L 2 (0, ∞). We give the output feedback controller about system (1): In terms of control law (57), closed-loop system (1) can be described as We consider system (58) in the state space H 2 .

Concluding Remarks
is paper considers the boundary output feedback stabilization for a cascaded-wave PDE-ODE system with velocity recirculation. Inspired by Krstic and Su et al. [30,33], we consider the problem of stabilization of a cascaded-wave PDE-ODE system with non-local term. We have used the backstepping method to turn system (1) into target system (3). Next we designed the state feedback controller and the output feedback controller for (1). And eorems 1-3 have been proved. Finally, as shown in Figures 1 and 2, it is obvious that the output feedback control law (57) can stabilize system (1) exponentially by the measurements. Illustrative simulation results are given to demonstrate the validity of the theoretical analysis and show the performance of the proposed method. We hope to have more innovation and development about stabilization of PDE-ODE systems with more non-local terms in this field afterwards.

Data Availability
e data used to support the findings of this study are included within the article.