Boundary Stabilization and Disturbance Rejection for an Unstable Time Fractional Diﬀusion-Wave Equation

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diﬀusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leﬄer stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two inﬁnite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leﬄer stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in [ Nonlinear Dynam. , 38(2004), 339-354] where all results were veriﬁed by simulations only.


Introduction
In this paper, we consider boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Neumann boundary control governed by u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), 0 ≤ x ≤ 1, where α ∈ (1, 2) is the order of the fractional derivative, u(x, t) is the displacement of wave propagation, λ(x) is a continuous function on [0, 1], U (t) is the control input, and C 0 D α t u(x, t) stands for the Caputo derivative with respect to time variable t. We consider system (1.1) in the state space L 2 (0, 1). The investigation on boundary stabilization problem of this class of fractional partial differential equations (PDEs) can be traced back to the 2004 paper [19], where a special case of system (1.1) with λ(x) ≡ 0 was studied. System (1.1) can be considered as a "special" cable, possibly made by some special smart materials, fixed at one end, and stabilized by a boundary controller at the other end. In [19], a boundary controller for system (1.1) with λ(x) ≡ 0 was designed as U (t) = −ku t (1, t) with k > 0, which is a classical negative velocity feedback control law based on the passive principle for the wave equation. In [20], the performance and properties of a fractional order boundary controller given by U (t) = −k C 0 D µ t u (1, t) with k > 0 for system (1.1) with λ(x) ≡ 0 were verified. However, we notice that the results in both [19] and [20] were only illustrated by numerical simulations without any theoretical proof. For system (1.1) with λ(x) ≡ 0, (1, t), k > 0 in [28], the asymptotic stability was proved by LaSalle's invariance principle and the fact that the input-output relationship of a class of generalized diffusion equation can be shown by fractional integrals. In the last a few years, stabilization for fractional time derivative PDEs has been attracted much attention. Recently, both the state feedback and output feedback for unstable time fractional reaction diffusion equations have been developed in [40] by the Riesz basis and backstepping methods. The output stabilization for a fractional reaction diffusion equation subject to spatially-varying diffusion coefficient has been developed in [4].
The Mittag-Leffler convergent observer based feedback control of a coupled semilinear subdiffusion system has been explored in [13]. For controllability aspect of fractional PDEs, an interesting result that fractional in time systems fail to be null controllable due to the long-tail effect of fractional derivative has been presented in [24]. However, the key techniques obtaining the stability in [40,4,13] are based on the fractional Lyapunov method established initially in [22] where the concept of Mittag-Leffler stability was introduced. A useful fractional derivative inequality: for α ∈ (0, 1) which is not known until 2014 in [1] is another powerful tool in establishing the stability. For the fractional Halanay inequality with time-varying delay, we refer to [17]. However, when α ∈ (1, 2), the inequality C 0 D α t x 2 (t) ≤ 2x(t) C 0 D α t x(t) fails, which can be seen from a simple counter-example that when α = 1.5 and x(t) = t, C 0 D α t x 2 (t) = 2 Γ(1.5) t 1.5 and x(t) C 0 D α t x(t) ≡ 0, which implies that C 0 D α t x 2 (t) > 2x(t) C 0 D α t x(t) = 0 for t > 0. Therefore, the method used in [4,14,40] is not applicable for the fractional wave equation.
In most practical situations, systems operate in the environment suffering from different kinds of external disturbances or uncertainties. To maintain the performance of systems, the designed control law must be required to be robust against the disturbance or uncertainty in some extent.
There have been many effective approaches to the control of systems with external disturbance or uncertainty, such as the adaptive control strategy for systems with unknown constants [5] or unknown amplitudes of harmonic disturbance [12], the sliding model control for one dimensional heat equation [37], wave equation with nonlinear boundary [23] and fractional order systems [2] that are all subject to boundary control matched disturbance, and the uncertainty and disturbance estimator (UDE)-based control for filtering the system input and state information [6]. Another remarkable powerful approach applied widely in dealing with external disturbance and uncertainty is the so-called active disturbance rejection control (ADRC) initially proposed by Han in the later 1980s [16]. Analogous to the adaptive control where the estimation/cancellation strategy is adopted, the ADRC can handle much more general disturbance that is estimated in real time by an extended state observer (ESO) and is compensated in the closed-loop by an ESO-based feedback control, which makes the control energy significantly reduced in practice [41]. Very recently, this emerging control technology has been shown to be highly efficient in stabilizing the one dimensional partial differential equations (PDEs) [9,7], multi-dimensional PDEs [10,38] and stochastic differential equations [11,36], which are all subject to external disturbance, among many others. The investigation on ADRC for nonlinear fractional-order systems and fractional PDEs with fractional derivative order α ∈ (0, 1) also has been available in [15] and in our recent work [39], but ADRC for the case of α ∈ (1, 2) has not been yet studied. Disturbance rejection for fractional PDEs with α ∈ (1, 2) has been verified in [19] by numerical simulations without mathematical proof, which motivates this paper to establish the mathematical basis for this case.
We proceed as follows. The problem formulation and some preliminaries are presented in Section 2. The boundary state feedback control and boundary output feedback control for system (1.1) without disturbance are included in Sections 3 and 4 respectively. In Section 5, the boundary controller for system (1.1) subject to external disturbance is explored. A numerical simulation is presented in Section 6 followed up concluding remarks in Section 7.
Next, we present some preliminaries. The Caputo derivative of w(x, t) with respect to time variable t is defined by where I 2−α is the Riemann-Liouville fractional integral operator given by The time fractional diffusion-wave equation is perhaps one of the most important fractional order linear partial differential equations, which is expected to describe generalized processes which interpolate or extrapolate the classical phenomenon of diffusion [34]. In a physical model presented in [29], the fractional wave equation describes the propagation of mechanical diffusive waves in viscoelastic media. It is well known from the definition of the Caputo derivative that This means that when α = 2, the system (1.1) is reduced to the classic wave equations. To analyze the stability, we recall the two-parameter Mittag-Leffler function E δ,γ (z) defined as where δ, γ > 0 and Γ(·) is the Gamma function. Taking γ = 1, we define one-parameter Mittag- , and System (1.1) without control input is unstable when λ(x) is sufficiently large, which can be seen from the following simple Example 2.1.
It is seen from Example 2.1 that an appropriate control must be imposed at the control end to guarantee the asymptotic stability of the closed-loop of system (1.1). Generally speaking, by [27], the eigenvalues of the operator ∂ 2 ∂x 2 + λ(x) with the certain domain lying outside the closed angular sector |arg( ∂ 2 ∂x 2 + λ(x))| > π 2 α will leads to instability.
The first aim of this paper is to design a control law U (t) to stabilize (1.1) and to present a mathematical proof. Obviously, once this is achieved, by taking λ(x) = 0, we give a complete solution for problem raised in [19].
Significantly, two long-standing unsolved problems left in the paper [19] are just the special cases of Problems I and II, where the plant is the system (1.1) with λ(x) ≡ 0 and the stability results have only been checked by numerical simulations without theoretical proofs.

Boundary stabilization via state feedback
In this section, a backstepping approach is adopted to construct a state feedback stabilizing control for system (1.1). Motivated by [32,33,40], we introduce an invertible transformation u → w: to transform system (1.1) into the following equivalent system: which is asymptotically stable. Now we determine the kernel function k(x, y) of (3.1). For this purpose, finding Caputo's fractional derivative for (3.1) and utilizing the first equation of (1.1), via performing the integration by parts, we obtain and  {(x, y) ∈ R 2 : 0 ≤ y ≤ x ≤ 1}. In particular, if λ(x) = λ is a constant, then its unique explicit solution is given by k(x, y) = −λy where I 1 (x) is a first-order modified Bessel function of the first kind given by In order to find the inverse transformation of (3.1), let Similarly, finding Caputo's fractional derivative for (3.6) and utilizing the first equation of (3.2) by performing the integration by parts, we obtain and  Once again, by [32, Theorem 2.2], the PDE (3.9) admits a unique solution l ∈ C 2 (F). In particular, when λ(x) = λ is a constant function, the unique solution is given explicitly by l(x, y) = −λy 2 2n+1 n!(n+1)! . Next, we show the asymptotic stability of the system (3.2). Lemma 3.1. Let α ∈ (1, 2). For any initial value (w 0 , w 1 ) ∈ [L 2 (0, 1)] 2 , system (3.2) admits a unique solution w ∈ C(0, ∞; L 2 (0, 1)), and there exists a constant M > 0 such that Moreover, when (w 0 , w 1 ) ∈ [H 2 (0, 1)] 2 , there exists a constant M ′ > 0 such that Proof. Define the operator A : D(A)(⊂ L 2 (0, 1)) → L 2 (0, 1) as follows: (3.12) A direct computation shows that A is self-adjoint in L 2 (0, 1) with the eigen-pairs {µ j , e j (x)} given by Since {e j (x)} forms an orthnormal basis for L 2 (0, 1), we can express the solution of (3.2) as (3.14) should satisfy the following linear fractional differential equation: In view of [18,Theorem 4.3], the solution of (3.15) is found to be Hence, the solution of (3.2) is finally derived by (3.17) Since {e j (x)} forms an orthnormal basis for L 2 (0, 1), it follows from (3.16) and (3.17) that In the last step of (3.18), we used the fact that (3.23) Since lim t→∞ 2 which, together with (3.18), implies that (3.11) holds with M ′ = M + M 1 .

Remark 3.1. From Lemma 3.1, for the stabilization of system (1.1) with λ(x) ≡ 0 considered
in [19,20], the controller can be taken as U (t) ≡ 0 if there is no disturbance. In other words, system (1.1) with λ(x) ≡ 0 is automatically asymptotically stable without control. In [21] where the controller for system (1.1) with λ(x) ≡ 0 was designed based on the Smith predictor by using delayed boundary measurement. By Lemma 3.1 again, there is no need to measure the boundary because system (1.1) with λ(x) ≡ 0 without disturbance is asymptotically stable. We can therefore say from Lemma 3.1 that the stability of system (1.1) with λ(x) ≡ 0 raised in [19] has been rigorously proved.

Remark 3.2.
It is worth emphasizing that the asymptotic stability of (3.2) does not hold when the which is a classical wave equation. Since Mittag-Leffler function has the properties:

from (3.16), the solution of (3.24) is explicitly solved by
This implies the fact that in fractional case the influence of the natural frequency of the system vanishes over time, which reveals a damping characteristic and differs from the integer-order case We propose the following state feedback control law: under which, the closed-loop system of (1.1) becomes (3.26) On the other hand, by (3.1) and (3.6), there exist two constants (3.29) The (3.27) then follows from (3.28) and (3.29) By the well known inequality of Mittag-Leffler function [35, Theorem 4] below which, together with the definition of Mittag-Leffler stability [22], leads to Mittag-Leffler stability of (3.26). It is worth noting that Mittag-Leffler stability implies the asymptotical stability and not conversely.

Observer-based output feedback for system (1.1)
In this section, assume the output measurement of (1.1) without disturbance is u(1, t), we are devoted to design an output feedback control for system (1.1) and do stability analysis. Note that the output feedback stabilization for the system without disturbance is also interesting and nontrivial. To this purpose, an observer for system (1.1) is firstly proposed as follows: where x ∈ [0, 1] and u 0 (·), u 1 (·) ∈ H 2 (0, 1) are the initial states; p 1 (x) and p 0 are the observer gains to be chosen to guarantee that the observer error u(x, t) − u(x, t) approaches zero as t goes to infinity.

Disturbance rejection via state feedback
In this section, as addressed in [19], it is assumed that a disturbance force d(t) comes from the same end where the boundary control input is imposed, that is, the stabilization and disturbance rejection is considered for a system described by where α ∈ (1, 2) is the order of the fractional derivative, u(x, t) is the displacement of wave propa- is the control input, and d(t) is the boundary disturbance. Now, we come to the second objective of the paper to seek a control law U (t) to stabilize (5.1) and reject the disturbance d(t). Obviously, once this is done, by taking λ(x) = 0, we give a complete solution to problem raised in [19].
To achieve Mittag-Leffler stability for system (5.1) and reject the disturbance d(t), the following assumption about d(t) is assumed.
The examples of such kinds of disturbances satisfying Assumption 5.1 include all finite sum of harmonic disturbances like d(t) = a j sin(ω j t) with unknown amplitude a j and unknown frequency ω j . To see this, for any given frequency ω, we prove that C 0 D α t sin(ωt), C 0 D α t cos(ωt) are uniformly bounded for all t ≥ 0, i.e., sup t≥0 | C 0 D α t sin(ωt)| < +∞ and sup t≥0 | C 0 D α t cos(ωt)| < +∞.
Indeed, by Caputo's derivative, we have It follows from [8, p. 284, A.4.11] that and hold for α ∈ (1, 2), which infers that both ∫ t 0 s 1−α cos(ωs)ds and ∫ t 0 s 1−α sin(ωs)ds are bounded over [0, ∞). According to the boundedness of sin(ωt) and cos(ωt), we know that sup t≥0 | C 0 D α t sin(ωt)| < +∞ and thus C 0 D α t sin(ωt) ∈ L ∞ (0, ∞). Similarly, one can show that C 0 D α t cos(ω·) ∈ L ∞ (0, ∞). Since any periodic signal can be expanded through Fourier decomposition and can be approximated by the finite sum of periodic harmonic signal [3], the disturbance signal satisfying Assumption 5.1 is much general, and Assumption 5.1 seems to be reasonable. It is worth pointing out that the noise d(t) in [19] is assumed to be a sinusoidal disturbance signal with an unknown amplitude and phase but with a known frequency ω. Clearly, Assumption 5.1 includes these sinusoidal signals and is much more general than the one of [19].
It should be noted that for the case of no disturbance, the collocated feedback control (3.25) will stabilize Mittag-Leffler the system (5.1). Nevertheless, this designed stabilizer (3.25) is not robust to the external disturbance. For instance, when d(t) ≡ C ̸ = 0 is a constant, from the transformation (3.1) and (3.6), it is easy to verify that the system (5.1) under the control (3.25) has a solution: with l(x, y) being the solution of (3.9). So, when there is the external disturbance, the control must be re-designed.
Since a stabilizing control law for system without disturbance force d(t) has been designed in Sections 3 and 4, it is natural to find a disturbance estimator to estimate the disturbance force d(t) and compensate for the disturbance in the closed-loop.
To estimate the disturbance, following [7], we propose a disturbance estimator as follows: which is an infinite-dimensional system with the state consisting of the functions v, z, defined on (0, 1). It is seen that system (5.3) is completely determined by the displacement u(x, t) and input U (t). In other words, system (5.3) is a completely known system. Since the disturbance estimator (5.3) proposed here looks complicated, now let us explain why we make such a construction.
Proof. Since the disturbance d(t) in (5.4) is at the boundary, we define a function g(x) = x, x ∈ [0, 1] and introduce the variable v(x, t) = v(x, t) + g(x)d(t) to shift d(t) from the boundary into the in-domain. Indeed, we can verify that v(x, t) is governed by where where the Fourier's coefficient a j is given by a j = ∫ 1 0 e j (x)xdx. Since {e j (x)} forms an orthnormal basis for L 2 (0, 1), we write the solution of (5.5) as v(x, t) = ∞ ∑ j=0 ϕ j (t)e j (x). (5.6) The initial states are v 0 (x) = and ϕ j (0) = b j , ϕ ′ j (0) = c j . In view of [18,Theorem 4.3], the solution of (5.7) is calculated to be Since {e j (x)} is an orthonormal basis for L 2 (0, 1), by the inequality for any a, b, c ∈ R, it follows from (5.6), (5.8) and (4.26) that (1−µ 0 t α ) 2 = 0, it follows from (4.33) and (5.9) that This, together witgh Assumption 5.1 and v( it suffices to show that Indeed, for any given ε > 0, since lim t→∞ C 0 D α t d(t) = 0, we can take T > 0 sufficiently large so that which implies that (5.10) holds. Since Actually, from (5.8),φ This ends the proof of the lemma.
From the proof of Lemma 5.1, it is concluded that in order to ensure lim t→∞ ∥ v(·, t)∥ L 2 (0,1) = 0, the condition lim t→∞ d(t) = 0 cannot be removed.
Then, from the boundary condition v(0, t) = 0, v x (1, t) = 0, it is easy to verify that system (5.12) is a conservation system, i.e., for all t ≥ 0. Moreover, from Remark 3.2, the solution of (5.12) with d(t) ≡ 0 is given by Secondly, the "z"-part of system (5.3) is to estimate the disturbance d(t). Actually, let z( When (w 0 , w 1 ) ∈ [H 2 (0, 1)] 2 , there exists a constant M ′ > 0 such that Proof. Since the proof is similar to the proof of Lemma 3.1, we omit the details.
By (3.25), we propose the following control law t)). (5.14) The purpose of this control law is essentially just cancelling the disturbance by its estimate. The closed-loop system is governed by  (5.15), and there exist two positive constants M > 0 such that 16) and there exists a constant M ′ > 0 such that

(5.22)
The disturbance estimator for (5.22) is given by  (1, t), t ≥ 0, which is completely determined by output u(1, t) and input U (t) and thus is an output-based disturbance estimator. By Theorem 5.1, the control U (t) can be designed as U (t) = z x (1, t) that is based only on the the disturbance estimator. Since the disturbance estimator depends only on u(1, t) and U (t), the control U (t) = z x (1, t) is an output feedback controller. Moreover, Theorem 5.1 also provides a mathematical proof for the stability of the closed-loop system of system (5.22).
Hence, the Problem II arised in [19] is completely solved.

Numerical simulation
In this section, we present some numerical simulations results for the closed-loop system (5.15).
For numerical computations, we take fractional order α = 1.5 and α = 2. The disturbance is taken as d(t) = sin(t), and the initial values are u(x, 0) = 2 sin x, u t (x, 0) = (x + 1) sin x, v(x, 0) = 2x 2 + 2 sin x, v t (x, 0) = 3x(x − 1) + (x + 1) sin x, z(x, 0) = cos x − 2x + 2x 2 , z t (x, 0) = 2x cos x + 3x(x − 1), λ(x) = 4. The numerical algorithm is based on the combination of the L1 scheme [30] in time and the second order centred difference scheme [25] in space. We take the spacial step as dx = 0.001 and the time step as dt = 0.01. Figure 1 displays the state (u(·, t), v(·, t), z(·, t)) of system (5.15) with α = 1.5. It can be observed that the control law (5.14) is very effective to make the "u-part" of system (5.15) convergent to zero and make the "(v, z)-part" of system (5.15) being bounded. Figure 2 shows the state (u(·, t), v(·, t), z(·, t)) of system (5.15) with α = 2. It is seen that the "(u, v, z)-part" of system (5.15) with α = 2 is bounded, where the "u-part" of system (5.15) does not converge to zero. This is exactly as that pointed out in Remark 5.5: The control law (5.14) does not work when α = 2. Figure 3(a) and Figure 3(b) demonstrate the disturbance d(t), its estimate −z x (1, t) and the control input U (t) for system (5.15) with α = 1.5 and for system (5.15) with α = 2, respectively. It is seen that for α = 1.5 the disturbance is fast estimated and the control approximates the value −d(t) as the time t is sufficiently large while the disturbance cannot be estimated for α = 2. This is similarly as that pointed out in Remark 3.2: The solution of the z-part of system (5.18) with α = 2 is not convergent to zero.

Concluding remarks
In this paper, the boundary stabilization of an unstable time fractional diffusion-wave equation system with or without boundary external disturbance is considered. For the system free from boundary external disturbance, we achieve the Mittag-Leffler stability via state feedback and asymptotic stability via output feedback, by using the backstepping method and Riesz basis approach to the analytic solution of the closed-loop system. For the system subject to boundary external disturbance, we construct an infinite dimensional disturbance estimator consisting of two subsystems to estimate the disturbance and compensate the disturbance in the feedback loop. It is shown that the u-part of the resulting closed-loop system is asymptotically sable, while the (v, z)-part is bounded.
As a result, the paper solves completely two long-standing mathematical control problems raised in [Nonlinear Dynam., 38(2004), 339-354]. As future works, designing an output feedback controller  for system (5.1) by the active disturbance rejection control or sliding mode control method would be an interesting problem. It is worth pointing that the difficulty in designing output feedback for system (5.1) is because the system (5.1) has the term λ(x)u(x, t).