Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 80, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.80 OUTPUT TRACKING FOR A 1-D WAVE EQUATIONS WITH SPATIALLY VARYING COEFFICIENTS AND SUBJECT TO UNKNOWN DISTURBANCES YAN-NA JIA, CAN JIN, XIU-FANG YU Abstract. In this article, we study the output tracking problem for a wave equation with variable coefficients, and subject to boundary control matched disturbances. Both the disturbances and the reference signal are unknown harmonic signal. The performance output is non-collocated with the control input. Initially, we establish an undisturbed auxiliary system and devise an appropriate internal model dynamic to reformulate the tracking error. Sub- sequently, we introduce an error-based feedback controller, leveraging an in- vertible transformation to achieve output tracking. The well-posedness and stability of the closed-loop system are established by applying semigroup the- ory approach. Finally, we illustrate the effectiveness of these theoretical results with numerical simulations. 1. Introduction Output tracking is one of the most fundamental challenges in control theory. In numerous engineering scenarios, the primary focus lies in ensuring that the out- put signal of the control system asymptotically converges to the desired references, even in the presence of disturbances. Additionally, it is imperative that all internal loop states remain within a suitable range. Over the past few decades, extensive re- search has been conducted on output tracking within the context of beam equations [11, 14, 19], heat equations [12, 17, 29], wave equations [4, 5, 31], and various other partial differential equations (PDEs) [15, 16]. A highly effective method for ad- dressing the output tracking problem is the internal model principle (IMP), which has been established in the literature since [3, 8, 24]. Through the application of the IMP, the task of achieving robust output tracking is significantly streamlined by constructing a dynamic tracking error feedback control system that incorporates a p-copy of the exosystem, where p ∈ N+ represents the dimension of the output [21]. Furthermore, several classic results have been extended to infinite-dimensional systems, as demonstrated in studies such as [1, 4, 5, 18, 24], among many others. On the other hand, the most challenging aspect of output tracking lies in man- aging disturbances. Various strategies have been devised to tackle disturbances or uncertain parameters in PDEs control problems. These methods include active 2020 Mathematics Subject Classification. 93C20, 35L05. Key words and phrases. Output tracking; internal model dynamic; error feedback; wave equation. ©2024. This work is licensed under a CC BY 4.0 license. Submitted June 14, 2024. Published December 3. 2024. 1 2 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 disturbance rejection control [9, 32], sliding mode control [10, 26], adaptive control [13, 30], and IMP [8, 24]. In [5, 6], the authors address regulation problems by identifying unique solutions to regulator equations, from which compensators can be formulated based on kernel equations. Recently, in [7], a control system based on an observer framework was developed for a one-dimensional wave equation, incor- porating non-collocated disturbances through the application of approach. In [28], the issue of output tracking for one-dimensional wave equations, which are sub- ject to unknown harmonic disturbances and reference signals, is tackled through the utilization of an adaptive internal model and an adaptive frequency estimation technique. In this paper, we investigate the output tracking for a wave equation with variable coefficients subject to boundary control matched harmonic disturbance. The system is governed by the PDEs Γtt(x, t) = (a(x)Γx(x, t))x, x ∈ (0, 1), t > 0, Γx(0, t) = mΓt(0, t), t ≥ 0, Γx(1, t) = U(t) + d(t), t ≥ 0, Γ(x, 0) = Γ0(x), Γt(x, 0) = Γ1(x), x ∈ [0, 1], yp(t) = Γ(0, t), t ≥ 0, (1.1) where Γ(·, t) represents the state of the entire system, yp(t) is the performance output, U(t) is the control input. m is a known positive constant. Define a(·) ∈ C1[0, 1] as follows a(x) = g1x+ g2, g1 > 0, g2 > 1. (1.2) For a given reference signal r(t), we aim at finding a controller U(t) so that lim t→∞ e(t) = lim t→∞ (yp(t)− r(t)) = 0. (1.3) The disturbance d(t) and reference signal r(t) are generated by the following finite- dimensional exosystem v̇(t) = Qv(t), v(0) = v0, t > 0, d(t) = F1v(t), r(t) = F2v(t), t > 0, (1.4) where Q ∈ R2q×2q is known but the initial value v0 and Fj ∈ R1×2q, j = 1, 2 are unknown, which makes the disturbance d(t) and reference signal r(t) unknown. To achieve output tracking, the following assumptions are required: Assumption 1.1. The tracking error e(t) and its derivative ė(t) are available measurements. Assumption 1.2. The spectrum of Q is {±iωj , j = 1, 2, . . . , q}, where 0 < ω1 < ω2 < · · · < ωq are distinct known parameters. Under Assumption 1.2, the disturbance and reference signal can be rewritten as d(t) = q∑ j=1 (A1j sinωjt+B1j cosωjt) , (1.5) r(t) = q∑ j=1 (A2j sinωjt+B2j cosωjt) , (1.6) EJDE-2024/80 OUTPUT TRACKING OF 1-D WAVE EQUATION 3 where {ωj} represent known frequencies and {A1j}, {B1j}, {A2j}, {B2j} are un- known amplitudes. The rest of thisarticle is organized as follows. In Section 2, we reconstruct the measurable tracking error using the undisturbed auxiliary system and the proper internal model dynamic. In Section 3, we focus on the construction of error-based feedback controller. In Section 4, we prove the well-posedness and stability of the closed-loop system. Finally, we present some numerical simulations in Section 5, and our conclusions are concluded in Section 6. 2. Estimation Before we design the controller to reject unknown disturbances and achieve out- put tracking, it is necessary to estimate these disturbances. Frist we construct an auxiliary system based on the measurable tracking error and its derivative Γ̂tt(x, t) = (a(x)Γ̂x(x, t))x, Γ̂x(0, t) = −k1(e(t)− Γ̂(0, t)) + k2Γ̂t(0, t) + (m− k2)ė(t), Γ̂x(1, t) = U(t), (2.1) where k1, k2 > 0. For simplicity, in system (2.1) and hereafter we omit the initial value when there is no confusion. Let Γ̃(x, t) = Γ(x, t)− Γ̂(x, t). (2.2) Then Γ̃tt(x, t) = (a(x)Γ̃x(x, t))x, Γ̃x(0, t) = k1Γ̃(0, t) + k2Γ̃t(0, t)− (k1F2 + k2F2Q+mF2Q)v(t), Γ̃x(1, t) = F1v(t). (2.3) Now we introduce a new system wtt(x, t) = (a(x)wx(x, t))x, wx(0, t) = k1w(0, t) + k2wt(0, t), wx(1, t) = 0, (2.4) where k1, k2 > 0 and a(x) satisfies (1.2). We consider system (2.4) in the state space H = H1(0, 1)× L2(0, 1) equipped with the inner product ⟨(µ1, ν1), (µ2, ν2)⟩H = ∫ 1 0 [a(x)µ′ 1(x)µ ′ 2(x) + ν1(x)ν2(x)] dx+ k1µ1(0)µ2(0). (2.5) System (2.4) can be easily rewritten as an evolution equation in H: d dt (w(·, t), wt(·, t)) = A (w(·, t), wt(·, t)) , (2.6) where operator A : D(A) ⊂ H → H is defined by A(f, g) = (g, (af ′)′), ∀(f, g) ∈ D(A), D(A) = { (f, g) ∈ H2(0, 1)×H1(0, 1) | f ′(1) = 0, f ′(0) = k1f(0) + k2g(0) } . . (2.7) According to [23], we can see thatA generates an exponentially stable C0-semigroup on H, that is, there exist two constants ζ0, ι0 > 0 such that ∥eAt∥H ≤ ζ0e −ι0t, ∀t ≥ 0. (2.8) 4 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 Theorem 2.1. Let k1, k2 > 0. Then for any initial state (Γ̃(·, 0), Γ̃t(·, 0)) ∈ H, system (2.3) admits a unique solution (Γ̃, Γ̃t) ∈ C([0,∞);H). Moreover, there exists a vector γ ∈ R1×2q such that χ1(t) = Γ̃(0, t)− γv(t), (2.9) satisfying lim t→∞ χ1(t) = 0, (2.10) exponentially. Furthermore, the following hidden regularity holds eι1·χ̇1(·) ∈ L2([0,∞);R), (2.11) where ι1 is a positive constant. Proof. A direct computation shows that the adjoint operator A∗ of A satisfies A∗(f, g) = (−g,−(af ′)′) , ∀(f, g) ∈ D(A∗), D(A∗) = { (f, g) ∈ H2(0, 1)×H1(0, 1) : f ′(1) = 0, f ′(0) = −k1f(0)− k2g(0) } . (2.12) System (2.3) can be rewritten as d dt ( Γ̃(·, t), Γ̃t(·, t) ) = A ( Γ̃(·, t), Γ̃t(·, t) ) + Bv(t), (2.13) where B = (0, −(k1F2 + k2F2Q+mF2Q)δ(·) + F1δ(· − 1)) with the Dirac distribu- tion δ(·). Since v(t) satisfies (1.4), we have Bv(t) ∈ H1 loc([0,∞); [D(A∗)]′). Hence, by [25], system (2.3) admits a unique solution ( Γ̃, Γ̃t ) ∈ C([0,∞);H). Now we show (2.9)-(2.11). Let Ω(x) = (Ω1(x), Ω2(x)) T ∈ R2×2q, x ∈ [0, 1]. (2.14) By recalling (1.4) and (2.13), we have d dt [ Γ̃(·, t)− Ω1(·)v(t) Γ̃t(·, t)− Ω2(·)v(t) ] = A [ Γ̃(·, t)− Ω1(·)v(t) Γ̃t(·, t)− Ω2(·)v(t) ] + (AΩ(·)− Ω(·)Q+ B)v(t). (2.15) Since eAt is exponentially stable, it must have Reλ < 0 for any λ ∈ σ(A). This together with Assumption 1.2 implies that σ(A) ∩ σ(Q) = ∅. Furthermore, from A ∈ L(X1, [D(A∗)]′) and B ∈ L(R, [D(A∗)]′), it follows from [22] that the Sylvester equation AΩ − ΩQ = −B admits a solution Ω ∈ L(R2q, [D(A∗)]′). Then, the exponential stability of eAt implies that limt→∞ Γ̃(·, t) = Ω1(·)v(t) exponentially. Letting γ := Ω1(0), we conclude that (2.9)-(2.11) hold. The proof is complete. □ Combining equations (2.2) and (2.9), we have e(t) = Γ̂(0, t) + p(t) + χ1(t), (2.16) where p(t) = (γ−F2)v(t). By Assumption 1.2, the term p(t) = (γ−F2)v(t) contains the sinusoids of no more than q distinct frequencies. Then p(t) can be expressed as p(t) = q∑ j=1 (A3j sinωjt+B3j cosωjt), (2.17) where {A3j} and {B3j} are uncertain parameters. EJDE-2024/80 OUTPUT TRACKING OF 1-D WAVE EQUATION 5 Lemma 2.2. The p(t) = (γ − F2)v(t) can be generated by the exosystem Ż(t) = ΥZ(t), Z(0) = Z0, p(t) = F̂Z(t), (2.18) where Z(t) = (z1(t), z2(t), . . . , z2q(t)) ⊤ ∈ R2q, Υ = ( 0(2q−1)×1 I(2q−1)×(2q−1) −ϖ1 0 −ϖ2 0 . . . −ϖq 0 ) , F̂ = (1, 0, 0, . . . , 0), (2.19) the I(2q−1)×(2q−1) denotes (2q − 1) × (2q − 1) identity matrix, and ϖ1, . . . , ϖq are chosen so that l2q +ϖql 2q−2 +ϖq−1l 2q−4 + · · ·+ϖ1 = (l2 + ω2 1) . . . (l 2 + ω2 q ). (2.20) Proof. We choose ϖ1, . . . , ϖq to satisfy (2.20). By (2.17), we obtain ϖ1p(t) +ϖ2p ′′(t) + · · ·+ϖqp (2q−2)(t) + p(2q)(t) = 0. Let Z(t) = ( p(t), ṗ(t), · · ·, p(2q−1)(t) )⊤ , and then we show that Z(t) satisfies (2.18). The proof is complete. □ From (2.16) and (2.18), the output tracking of system (1.1) is converted into a new output tracking problem for system (2.1) and (2.18). However, p(t) is not suitable for the controller design since the initial state Z(0) of the exosystem (2.18) is unknown. To overcome this, we introduce the internal model dynamic ˙̂ Φ(t) = ΞΦ̂(t) + K̂(e(t)− Γ̂(0, t)), (2.21) where Φ̂(t) = [ ϕ̂(t), ϕ̂′(t), . . . , ϕ̂(2q−1)(t) ]⊤ ∈ R2q×1, Ξ = ( 0(2q−1)×1 I(2q−1)×(2q−1) −η1 −η2 −η3 . . . −η2q−1 −η2q ) , K̂ = (0, 0, . . . , ξ)⊤, (2.22) where the parameter η = (η1, η2, η3, . . . , η2q) is chosen so that Ξ is Hurwitz and ξ > 0. Before we proceed, we need the following Lemma 2.3 from [2]. Lemma 2.3. Suppose that (S, F ) is observable and (Ξ, K̂) is controllable, where S, Ξ ∈ R2q×2q and F ∈ R1×2q, K̂ ∈ R2q×1. The Sylvester equation PS − ΞP = K̂F, (2.23) admits a unique invertible matrix P . Lemma 2.4. Consider system (2.21) with the controller pair (Ξ, K̂), where Ξ ∈ R2q×2q and K̂ ∈ R2q×1 are given by (2.22). Then, there exists a unique nonsingular matrix P such that χ2(t) = p(t)− F̂P−1Φ̂(t) (2.24) tends to zero exponentially as t goes to infinity. 6 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 Proof. Since (Υ, F̂ ) is observable, we can replace (S, F ) of Lemma 2.3 by the (Υ, F̂ ). Then, the Sylvester equation PΥ− ΞP = K̂F̂ , (2.25) has a unique invertible matrix P by Lemma 2.3. We define the error Z̃(t) = PZ(t)− Φ̂(t). (2.26) From (2.16), (2.18) and (2.21), we have ˙̃Z(t) = ΞZ̃(t)− K̂χ1(t). (2.27) Since P has its invertible matrix, Ξ is a Hurwitz, and χ1(t) satisfies Theorem 2.1, we have that Z̃(t) tends exponentially to zero and so does for χ2(t) = F̂P−1Z̃(t). Furthermore, by differentiating (2.27) with respect to t, we obtain Z̃ ′′(t) = Ξ ˙̃Z(t)− K̂χ̇1(t). Owing to Ξ is Hurwitz and χ1(t) satisfies (2.11), there exists ι2 > 0 such that eι2· ˙̃Z(·) ∈ L2([0,∞);R), eι2·χ̇2(·) ∈ L2([0,∞);R). (2.28) The proof is complete. □ Substituting (2.24) into (2.16), we obtain e(t) = Γ̂(0, t) + F̂P−1Φ̂(t) + χ(t), (2.29) where χ(t) = χ1(t)+χ2(t) converges exponentially to zero as t → ∞. In particular, from (2.11) and (2.28), we deduce that eι·χ̇(·) ∈ L2([0,∞);R), ι > 0. (2.30) Next, we determine the matrix P of (2.25). Let Pj , j = 1, 2, . . . , 2q denote the jth row of P . Then expanding PΥ− ΞP = K̂F̂ gives P1Υ− P2 P2Υ− P3 ... P2q−1Υ− P2q P2qΥ+ ∑2q j=1 ηjPj  =  0 0 0 . . . 0 0 0 0 . . . 0 ... ... ... . . . ... 0 0 0 . . . 0 ξ 0 0 . . . 0  . (2.31) By a straightforward calculation, one gets P = ξ ( η1I + η2Υ+ · · ·+ η2qΥ 2q−1 +Υ2q )−1 . (2.32) Substituting (2.29) into (2.21) and using (2.25), we obtain ˙̂ Φ(t) = PΥP−1Φ̂(t) + K̂χ(t). (2.33) Noting that ΥP−1 = P−1Υ, we rewrite (2.33) as ˙̂ Φ(t) = ΥΦ̂(t) + K̂χ(t). (2.34) Remark 2.5. Solving (2.34) gives Φ̂(t) = eΥtΦ̂(0) + ∫ t 0 eΥ(t−τ)K̂χ(τ)dτ. Since χ(t) converges exponentially to zero and Υ has eigenvalues ±iω1, ±iω2,. . . , ±iωq, we can conclude that Φ̂(t) is bounded, i.e., ∥Φ̂(t)∥ < ∞. EJDE-2024/80 OUTPUT TRACKING OF 1-D WAVE EQUATION 7 3. Controller design From Section 2, the tracking error e(t) can be represented by the state Φ̂, Γ̂ and Γ̂t. Since the state Φ̂, Γ̂ and Γ̂t are available for the controller design and χ(t) converges exponentially to zero, we can design a feedback control for system (2.1) and (2.21) such that lim t→∞ ê(t) = lim t→∞ ( Γ̂(0, t) + F̂P−1Φ̂(t) ) = 0. (3.1) Then, this controller has the same control effect on system (1.1). Motivated by [20], we introduce the transformation y(x, t) = Γ̂(x, t)− Σ(x)Φ̂(t), (3.2) where Σ(x) satisfies (a(x)Σ′(x))′ = Σ(x)Υ2, Σ′(0) = −F̂P−1(k1I + (k2 −m)Υ), Σ(0) = −F̂P−1. (3.3) Substituting (3.2) into (2.1) and using (2.29) and (2.34), we obtain ytt(x, t) = (a(x)yx(x, t))x − Σ(x)ΥK̂χ(t)− Σ(x)K̂χ̇(t), yx(0, t) = myt(0, t)− k1χ(t)− k2(F̂P−1K̂χ(t) + χ̇(t)) +mχ̇(t), yx(1, t) = U(t)− Σ′(1)Φ̂(t), y(0, t) = ê(t). (3.4) Evidently, the output regulation problem of Γ̂(0, t)− p(t) → 0 has been converted into a stabilization problem y(·, t) → 0 when t goes to infinity. Then, we propose the output feedback control as follows: U(t) = −kyt(1, t)− y(1, t) + Σ′(1)Φ̂(t) = −kΓ̂t(1, t)− Γ(1, t) + kΣ(1)ΥΦ̂(t)− Σ(1)Φ̂(t) + Σ′(1)Φ̂(t), (3.5) where k is a positive constant. Under the controller (3.5), system (3.4) becomes ytt(x, t) = (a(x)yx(x, t))x − Σ(x)ΥK̂χ(t)− Σ(x)K̂χ̇(t), yx(0, t) = myt(0, t)− k1χ(t)− k2(F̂P−1K̂χ(t) + χ̇(t)) +mχ̇(t), yx(1, t) = −kyt(1, t)− y(1, t), y(0, t) = ê(t). (3.6) Lemma 3.1. Suppose that m, k > 0. When χ(t) ≡ 0, system (3.6) is exponentially stable in H. Proof. When χ(t) ≡ 0, system (3.6) reads ytt(x, t) = (a(x)yx(x, t))x, yx(0, t) = myt(0, t), yx(1, t) = −kyt(1, t)− y(1, t). (3.7) 8 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 Let ỹ(x, t) = y(1− x, t). Then, ỹ(x, t) satisfies ỹtt(x, t) = (a(x)ỹx(x, t))x, ỹx(0, t) = kỹt(0, t) + ỹ(0, t), ỹx(1, t) = −mỹt(1, t). (3.8) We define the energy of system (3.8) as E(t) = 1 2 ∫ 1 0 a(x)ỹ2x(x, t) + ỹ2t (x, t)dx+ a(0) 2 ỹ2(0, t). (3.9) The derivative of E(t) along (3.8) satisfies Ė(t) = −ma(1)ỹ2t (1, t)− ka(0)ỹ2t (0, t). (3.10) We establish the energy multiplier as follows: φ(t) = ∫ 1 0 (x−1)ỹx(x, t)ỹt(x, t)dx+2sỹ(0, t) ∫ 1 0 (1−x)ỹt(x, t)dx+ lỹ2(0, t), (3.11) where s, l > 0 to be determined later. Obviously, |φ(t)| ≤ ME(t), M > 0. A direct computation shows that φ̇(t) = 1 2 ỹ2t (0, t) + 1 2 a2(0)ỹ2x(0, t)− 1 2 ∫ 1 0 [ a(x)ỹ2x(x, t) + ỹ2t (x, t) ] dx + 1 2 ∫ 2 0 a′(2)(x− 1)ỹ2x(x, t)dx+ 2sỹt(0, t) ∫ 1 0 (1− x)ỹt(x, t)dx + 2lỹ(0, t)ỹt(0, t)− 2sa(0)ỹ(0, t)ỹx(0, t) + 2sỹ(0, t) ∫ 1 0 a(x)ỹx(x, t)dx ≤ (1 2 − 2sa(0)k + lr ) ỹ2t (0, t) + ( a2(0) + s− 2sa(0) + l r ) ỹ2(0, t) − 1 2 ∫ 1 0 [ a(x)ỹ2x(x, t) + ỹ2t (x, t) ] dx, where r > 0 is chosen so that 1 2 −2sa(0)k+ lr > 0. Since a(0) > 1/2, we can choose sufficiently large s such that a2(0) + s − 2sa(0) + l r < 0. Therefore, there exists M0 > 0 such that φ̇(t) ≤ ( 12 − 2sa(0)k + lr)ỹ2t (0, t)−M0E(t). Let Π(t) = E(t) + µ M φ(t), µ > 0. (3.12) Then (1− µ)E(t) ≤ Π(t) ≤ (1 + µ)E(t), Π̇(t) ≤ − M0µ M(1 + µ) Π(t), (3.13) for all sufficiently small µ > 0. This shows that E(t) ≤ 1 + µ 1− µ e− M0µ M(1+µ) tE(0). (3.14) The proof is complete. □ Lemma 3.2. Suppose that k > 0. For any initial state (y(·, 0), yt(·, 0)) ∈ H, system (3.6) admits a unique solution (y, yt) ∈ C([0,∞);H), which is exponentially stable. Moreover, limt→∞ |ê(t)| = 0. EJDE-2024/80 OUTPUT TRACKING OF 1-D WAVE EQUATION 9 Proof. System (3.6) can be written abstractly as d dt (y(·, t), yt(·, t)) = A1(y(·, t), yt(·, t)) +B1χ(t) +B2χ̇(t), (3.15) where B1 = ( 0,−Σ(·)ΥK̂ − k1δ(·)− k2F̂P−1K̂δ(·) ) , B2 = ( 0,−Σ(·)K̂ − (k2 −m)δ(·) ) with the Dirac distribution δ(·). Thanks to the result of Lemma 3.1, A1 gener- ates an exponentially stable C0-semigroup on H, and B1, B2 are admissible for eA1t (see [25]). Therefore, by [27] and Lemma 2.4, system (3.15) is exponentially stable, which admits a unique solution (y, yt) ∈ C([0,∞);H). Besides, we obtain limt→∞ |ê(t)| = 0. □ Lemma 3.3. Equation (3.3) admits a unique solution Σ(·) ∈ C∞([0, 1];R2q). Proof. It is clear that (3.3) is an initial value problem, so we prove the existence of its solution. □ 4. Closed-loop system In this section, we consider the following closed-loop system which is composed of (1.1), (2.1) and (2.21): Γtt(x, t)− (a(x)Γx(x, t))x = 0, Γx(0, t) = mΓt(0, t), Γx(1, t) = U(t) + d(t), Γ̂tt(x, t) = (a(x)Γ̂x(x, t))x, Γ̂x(0, t) = −k1(e(t)− Γ̂(0, t)) + k2Γ̂t(0, t) + (m− k2)ė(t), Γ̂x(1, t) = U(t), ˙̂ Φ(t) = ΞΦ̂(t) + K̂(e(t)− Γ̂(0, t)), e(t) = yp(t)− r(t), U(t) = −kΓ̂t(1, t)− Γ̂(1, t) + kΣ(1)ΥΦ̂(t)− Σ(1)Φ̂(t) + Σ′(1)Φ̂(t). (4.1) Now we study system (4.1) in the Hilbert space X = H×H× R2q. Theorem 4.1. Let k, k1, k2 > 0. Suppose that d(t) and r(t) satisfy (1.5) and (1.6), respectively. Then, for any initial state( Γ(·, 0),Γt(·, 0), Γ̂(·, 0), Γ̂t(·, 0), Φ̂(0) ) ∈ X , (4.2) the closed-loop system (4.1) admits a unique solution( Γ,Γt, Γ̂, Γ̂t, Φ̂ ) ∈ C([0,∞);X ), (4.3) satisfying |e(t)| < Le−ωt, (4.4) for some constants L > 0 and ω > 0. Moreover, (i) the state of the closed-loop (4.1) is uniformly bounded sup t∈[0,∞) ∥∥(Γ(·, t),Γt(·, t), Γ̂(·, t), Γ̂t(·, t), Φ̂(t) )∥∥ < ∞; (4.5) 10 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 (ii) when d(t) ≡ 0 and r(t) ≡ 0, there exist positive constants ζ1 and ι1 such that∥∥(Γ(·, t),Γt(·, t), Γ̂(·, t), Γ̂t(·, t), Φ̂(t) )∥∥ H×H×R2q ≤ ζ1e −ι1t, ∀t > 0. (4.6) Proof. According to (2.2) and (2.16), we just need to consider that Γ̂tt(x, t) = (a(x)Γ̂x(x, t))x, Γ̂x(0, t) = −k1(p(t) + χ1(t))− k2(ṗ(t) + χ̇1(t)) +mė(t), Γ̂x(1, t) = −kΓ̂t(1, t)− Γ(1, t) + kΣ(1)ΥΦ̂(t)− Σ(1)Φ̂(t) + Σ′(1)Φ̂(t), (4.7) and ˙̂ Φ(t) = ΞΦ̂(t) + K̂(p(t) + χ1(t)), (4.8) where χ1(t) is given by (2.9) and p(t) = (γ − F2)v(t). By Remark 2.5 and the transformation (3.2), we obtain that ˙̂ Φ(t) = ΥΦ̂(t) + K̂χ(t), ytt(x, t) = (a(x)yx(x, t))x − Σ(x)ΥK̂χ(t)− Σ(x)K̂χ̇(t), yx(0, t) = myt(0, t)− k1χ(t)− k2(F̂P−1K̂χ(t) + χ̇(t)) +mχ̇(t), yx(1, t) = −kyt(1, t)− y(1, t), (4.9) where Σ(x) is given by (3.3). The well-posedness and exponential stability of (y, yt)- part has been shown in Lemma 3.2. Therefore, the state of the system (4.1) is uniformly bounded. According to (2.29) and transformation (3.2), we see that e(t) = y(0, t) + χ(t). This implies that e(t) exponentially converges to zero as time goes to infinity. When d(t) ≡ 0 and r(t) ≡ 0. Φ̂(t) is given by ˙̂ Φ(t) = ΞΦ̂(t) + K̂χ1(1). (4.10) Since Ξ is Hurwitz and χ1(t) is exponentially stable, we obtain Φ̂(t) is exponentially stable. Similarly to Lemma 3.2, the (Γ̂, Γ̂t)-part of system admits a unique solution (Γ̂, Γ̂t) ∈ C([0,∞);H), and it follows from transformation (3.2) that (Γ̂, Γ̂t) is exponentially stable. By (2.16), we conclude that e(t) decays exponentially to zero. Therefore, the estimate (4.4) holds. The proof is complete. □ 5. Numerical simulations In this section, we present some numerical simulations to validate the theoretical results. The numerical results are programmed in MATLAB. The space step and time step are taken as dx = 0.001 and dt = 0.999dx, respectively. Let a(x) = 1 and the initial states are chosen as Γ(x, 0) = cos(2πx)− 1, Γt(x, 0) = Γ̂(x, 0) = Γ̂t(x, 0) = 0, Φ̂(0) = 0, v0 = (1, 1, 1, 1)⊤. (5.1) The parameters are chosen as m = 0.5, k = 1, k1 = 1, k2 = 0.9, η = (4, 6, 4, 1), (5.2) EJDE-2024/80 OUTPUT TRACKING OF 1-D WAVE EQUATION 11 and the matrix Q is Q = bdiag {( 0 1 −1 0 ) , ( 0 4 −1 0 )} . (5.3) In this case, the solution of equation (1.4) is v(t) = (sin t, cos t, sin 2t, cos 2t)⊤. The corresponding parameters are chosen as F1 = (1, 1, 2, 1), F2 = (1, 2, 1, 2). (5.4) The solution of PDE-part of the closed-loop system (4.1) is depicted in Figure 1. The control law Φ(t) and the state U(t) are plotted in Figure 2. It is seen that all states of the closed-loop system are uniformly bounded. Figure 3 shows that tracking error e(t) converges to zero as time goes to infinity, and F̂P−1Φ̂(t) can gradually synchronize with p(t), respectively. Both of them converge effectively. (a) state of Γ(x, t) (b) state of Γ̂(x, t) Figure 1. Solution of closed-loop system (4.1) 0 5 10 15 20 25 30 35 40 t -6 -4 -2 0 2 4 6 8 10 0 5 10 15 20 25 30 35 40 t -30 -20 -10 0 10 20 30 (a) state of Φ(t) (b) state of U(t) Figure 2. state of Φ(t) and U(t) 12 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 0 5 10 15 20 25 30 35 40 t -5 0 5 10 15 20 0 5 10 15 20 25 30 35 40 t -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (a) tracking error e(t) (b) state estimation F̂P−1Φ̂(t) → p(t) Figure 3. Tracking performance 6. Conclusions In this article, we address the output tracking problem of a wave equation with variable coefficients subjected to boundary control matched with harmonic distur- bances. The performance output is non-collocated with the control input. First, we establish an auxiliary system using measurable tracking error and its derivative. Subsequently, we construct an internal model dynamics to estimate the unknown disturbances. 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Li; Adaptive stabilization for a class of PDE-ODE cascade systems with uncertain harmonic disturbances, ESAIM Control Optim. Calc. Var. 23 (2017), no. 2, 497– 515. [31] C. T. Yilmaz, H. I. Basturk; Adaptive output regulator for wave PDEs with unknown har- monic disturbance, Automatica J. IFAC 113 (2020), 108808, 9. [32] Z. L. Zhao, B. Z. Guo; On active disturbance rejection control for nonlinear systems using time-varying gain, Eur. J. Control 23 (2015), 62–70. 14 Y.-N. JIA, C. JIN, X.-F. YU EJDE-2024/80 Yan-Na Jia School of Mathematics and Statistics, Taiyuan Normal University, Taiyuan Shanxi 030619, China Email address: jiayanna1224@126.com Can Jin School of Mathematics and Statistics, Taiyuan Normal University, Taiyuan Shanxi 030619, China Email address: jclllnnn@163.com Xiu-Fang Yu (corresponding author) College of Mathematics, Taiyuan University of Technology, Taiyuan Shanxi 030600, China Email address: yuxiufang@tyut.edu.cn 1. Introduction 2. Estimation 3. Controller design 4. Closed-loop system 5. Numerical simulations 6. Conclusions Acknowledgements References