Singular Perturbation Approach for Linear Coupled ODE-PDE Systems

Abstract

This paper focuses on a class of linear coupled ODE-PDE systems whose dynamics evolve in two time scales. The fast time scale modeled by a small positive perturbation parameter is introduced to the dynamics either of the ODE or of the PDE. By setting the perturbation parameter to zero, two subsystems, namely the reduced and the boundary-layer subsystems, are formally computed. Firstly, we propose a sufficient stability condition for the full coupled system. This stability condition implies the stability of both subsystems. Then, we state an approximation of the full coupled ODE-PDE systems by the subsystems based on the singular perturbation method. The error between the solution of the full system and that of the subsystems is the order of the perturbation parameter. Finally, numerical simulations on academic examples illustrate the theoretical results.

1 Introduction

Systems modeled by coupled ordinary differential equations (ODEs)-partial differential equations (PDEs) have been studied in many research works [2, 5, 11]. It is interesting to analyze such kind of systems due to their significant physical applications. For instance, elastic beams linked to rigid bodies in [12], power converters connected to transmission lines in [4] etc.

Singular perturbation theory has been widely used in control engineering from late 1960s. It is a powerful tool for analysis and design of control systems thanks to the reduction of the system’s order by neglecting the fast transitions [7, 9, 10]. This theory is effective in many applications, such as semiconductors, electrical chains and so on.

Tikhonov approximation, which describes the limiting behavior of system’s solutions, is an important method for analysis of singularly perturbed systems. Tikhonov approach for finite dimensional systems modeled by ODEs has been considered in [6]. In [14], a Tikhonov theorem for infinite dimensional systems governed by hyperbolic PDEs has been established by means of a $L^2$ Lyapunov function. The approximation for linear hyperbolic systems has been improved by using a $H^2$ Lyapunov function in [15].

The present work is concerned with a class of coupled singularly perturbed linear ODE and linear hyperbolic PDE systems. Firstly, a sufficient stability condition is proposed for both coupled ODE-fast PDE and PDE-fast ODE systems. The stability of the full system implies the stability of both subsystems. Secondly, the Tikhonov approximation for such systems is achieved by Lyapunov method. Under the stability conditions of both subsystems, the coupled ODE-fast PDE system is approximated by the two subsystems for $\varepsilon$ sufficiently small. However, for PDE-fast ODE system, the approximation is valid if the full system is stable. The error between the full system and the subsystems is estimated as the order of the perturbation parameter $\varepsilon$.

The paper is organized as follows. The coupled ODE-PDE systems under consideration are given in Sect. 2. The reduced and the boundary-layer subsystems are formally computed in the same section. Section 3 provides sufficient stability conditions for the full system and both subsystems. The Tikhonov approximation for such systems is stated in Sect. 4. Numerical simulations on academic examples are shown in Sect. 5. Concluding remarks end of this paper.

Notation. Given a matrix G in $\mathbb {R}^{n\times n}$, $G^{-1}$ and $G^{\top }$ represent the inverse and the transpose matrix of G respectively. The minimum and maximum eigenvalues of a symmetric matrix G are denoted by $\underline{\lambda }(G)$ and $\overline{\lambda }(G)$. The symbol $\star$ in partitioned symmetric matrix stands for the symmetric block. For a positive integer n, $I_{n}$ is the identity matrix in $\mathbb {R}^{n\times n}$. $\vert \cdot \vert$ denotes the usual Euclidean norm in $\mathbb {R}^n$ and $\Vert \cdot \Vert$ is associated with the usual 2-norm of matrices in $\mathbb {R}^{n\times n}$. $\Vert \cdot \Vert _{L^2}$ denotes the associated norm in $L^2(0,1)$ space, defined by $\Vert f\Vert _{L^2}=\sqrt{\left( \int _0^1 |f(x)|^2 dx\right) }$ for all functions $f\in L^2(0,1)$. Similarly, the associated norm in $H^2(0,1)$ space is denoted by $\Vert \cdot \Vert _{H^2}$, defined for all functions $f\in H^2(0,1)$, by $\Vert f\Vert _{H^2}=\sqrt{\left( \int _0^1 |f(x)|^2+|f^{\prime }(x)|^2+|f^{\prime \prime }(x)|^2 dx\right) }$. Given a real interval I and a normed space J, $C^0(I,J)$ denotes the set of continuous functions from I to J.

2 Singularly Perturbed Linear Coupled ODE-PDE Systems

In this section, the coupled ODE-fast PDE and PDE-fast ODE systems under consideration are given respectively.

2.1 Coupled ODE-Fast PDE System

We consider the following linear ODE-fast hyperbolic PDE system

where $x\in [0,1]$, $t\in [0,+\infty )$, $Z:[0,+\infty ) \rightarrow \mathbb {R}^n$, $y:[0,1]\times [0,+\infty ) \rightarrow \mathbb {R}^m$. The perturbation parameter $\varepsilon >0$ is a small constant and $\varLambda$ is a diagonal positive matrix in $\mathbb {R}^{m\times m}$. The matrices A and B are of appropriate dimensions. The boundary condition matrices $K_1$ and $K_2$ are constant matrices of appropriate dimensions.

Adopting the computations of the subsystems for singularly perturbed ODEs [8], the reduced and the boundary-layer subsystems for system (1) are formally computed as follows. By setting $\varepsilon =0$ in Eq. (1b), we obtain

$$\begin{aligned} y_x(x,t)=0. \end{aligned}$$

(2)

It implies $y(.,t)=y(1,t)$. Assuming $(I_m-K_1)$ invertible, the boundary condition (1c) becomes

$$\begin{aligned} y(.,t)=K_rZ(t), \end{aligned}$$

(3)

where $K_r=(I_m-K_1)^{-1}K_2$. Using the right-hand side of (3) to replace y(1, t) in (1a), the reduced subsystem is computed as

where $A_r=A+BK_r$. The bar indicates that the variables belong to the system with $\varepsilon =0$. Using the following change of variable $\bar{y}=y-K_rZ$ and a new time scale $\tau =t/\varepsilon$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \bar{y}_{\tau }(x,\tau )+\varLambda \bar{y}_x(x,\tau )=-\varepsilon K_r(AZ(\tau )+By(1,\tau )),\\ \bar{y}(0,\tau )=K_1\bar{y}(1,\tau ). \end{array}\right. } \end{aligned}$$

The boundary-layer subsystem is formally computed with $\varepsilon =0$

2.2 Coupled PDE-Fast ODE System

Similar to system (1), we consider the following linear hyperbolic PDE-fast ODE system

The two subsystems are computed as follows. By formally setting $\varepsilon =0$ in (6a) and assuming A invertible, we have

$$\begin{aligned} Z=-A^{-1}By(1). \end{aligned}$$

(7)

Substituting (7) into (6c), the reduced subsystem is

where $K_r=K_1-K_2A^{-1}B$. Performing a change of variable $\bar{Z}=Z+A^{-1}By\,(1)$ and using a new time scale $\tau =t/\varepsilon$, we get

$$\begin{aligned} \frac{d\bar{Z}(\tau )}{d \tau }&=A\bar{Z}(\tau )-\varepsilon A^{-1}B\varLambda y_x(1,\tau ). \end{aligned}$$

The boundary-layer subsystem is formally computed with $\varepsilon =0$

Remark 1

Due to [1, Theorem A.6.], the Cauchy problems (1) and (6) are well-posed, that is, for every $Z_0\in \mathbb {R}^n$, for every $y_0\in L^2(0,1)$, systems (1) and (6) have a unique solution $Z\in C^0([0,+\infty ),\mathbb {R}^n)$, $y\in C^0([0,+\infty ), L^2((0,1),\mathbb {R}^m))$.

3 Stability Condition of Coupled ODE-PDE Systems

We first provide a sufficient stability condition for both coupled ODE-PDE systems (1) and (6). Then, we study the link of the stability between the full system and both subsystems.

Proposition 1

Systems (1) and (6) are exponentially stable for all $\varepsilon >0$ if there exist diagonal positive matrix Q, symmetric positive matrix P and positive constant $\mu$ such that the following holds

$$\begin{aligned} \left( {\begin{matrix} e^{-\mu } Q\varLambda -K^{\top }_1 Q\varLambda K_1&{}-(K^{\top }_1 Q\varLambda K_2+B^{\top } P)\\ \star &{}-(A^{\top } P+ PA)-K^{\top }_2 Q\varLambda K_2 \end{matrix}}\right) >0. \end{aligned}$$

(10)

The next two propositions show that condition (10) implies the stability of the reduced and the boundary-layer subsystems.

Proposition 2

Condition (10) implies

$$\begin{aligned} A^{\top }_rP+PA_r<0, \end{aligned}$$

(11)

which is equivalent to the stability of the reduced subsystem (4), and

$$\begin{aligned} e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1>0, \end{aligned}$$

(12)

which implies the stability of the boundary-layer subsystem (5).

Proposition 3

Condition (10) implies

$$\begin{aligned} e^{-\mu }Q\varLambda -K^{\top }_rQ\varLambda K_r>0, \end{aligned}$$

(13)

which implies the stability of the reduced subsystem (8), and

$$\begin{aligned} A^{\top }P+PA<0, \end{aligned}$$

(14)

which is equivalent to the stability of the boundary-layer subsystem (9).

In view of Proposition 1, the stability of the full systems (1) and (6) is guaranteed for all positive $\varepsilon$ under condition (10). For $\varepsilon$ sufficiently small, the stability of both subsystems (4) and (5) implies the stability of the coupled ODE-fast PDE system (1), even though condition (10) is not satisfied. However, this result is not valid in the context for PDE-fast ODE system (6). That is, system (6) could be unstable even though the two subsystems (8) and (9) are stable. We refer the readers to [13] for more details.

4 Tikhonov Approximation of Coupled ODE-PDE Systems

We deal with the Tikhonov approximation of the coupled systems when $\varepsilon$ is sufficiently small as follows. If the two subsystems are stable, the coupled ODE-fast PDE system can be approximated by the subsystems. The approach for coupled PDE-fast ODE system is valid if the full system is stable.

4.1 Tikhonov Theorem for Linear ODE-Fast PDE System

Let us state Tikhonov theorem for system (1) in the next theorem.

Theorem 1

Consider system (1). If (11) and (12) are satisfied, there exist positive values $C_1$, $C_2$, $\theta$, $\varepsilon ^*$, such that for all $0<\varepsilon <\varepsilon ^*$, and for any initial conditions $Z_0\in \mathbb {R}^n$, $y_0=K_rZ_0$, it holds for $t\geqslant 0$

$$\begin{aligned} \vert Z(t)-\bar{Z}(t)\vert ^2\leqslant & {} \varepsilon C_1e^{-\theta t}\vert \bar{Z}_0\vert ^2,\end{aligned}$$

(15)

$$\begin{aligned} \Vert y(.,t)-K_r\bar{Z}(t)\Vert ^2_{L^2(0,1)}\leqslant & {} \varepsilon C_2e^{-\theta t}\vert \bar{Z}_0\vert ^2. \end{aligned}$$

(16)

Before proving this theorem, we first write the error system of (1). Let us perform the following change of variables

$$\begin{aligned} \eta&= Z-\bar{Z},\end{aligned}$$

(17a)

$$\begin{aligned} \delta&= y-K_r \bar{Z}, \end{aligned}$$

(17b)

where $\eta$ represents the error between the slow dynamics of the full system and the reduced subsystem while $\delta$ is the error between the fast dynamics of the full system and its equilibrium point. Due to (1a) and (4), we write

$$\begin{aligned} \dot{\eta }&=\dot{Z}-\dot{\bar{Z}}=AZ+By(1)-A_r\bar{Z}\\&=A(Z-\bar{Z})+B(y(1)-K_r\bar{Z}). \end{aligned}$$

Due to (1b) and (4), we compute

$$\begin{aligned} \delta _t&=y_t-K_r\dot{\bar{Z}}=y_t-K_r(A_r\bar{Z}),\\ \delta _x&=y_x. \end{aligned}$$

Due to (1c), we have

$$\begin{aligned} \delta (0)= & {} y(0)-K_r\bar{Z}=K_1y(1)+K_2Z-K_r\bar{Z}\\ {}= & {} K_1\biggl (y(1)-K_r\bar{Z}\biggr )+K_2Z-(I_m-K_1)K_r\bar{Z}\\= & {} K_1\biggl (y(1)-K_r\bar{Z}\biggr )+K_2 Z-(I_m-K_1)(I_m-K_1)^{-1}K_2\bar{Z}\\= & {} K_1\biggl (y(1)-K_r\bar{Z}\biggr )+K_2(Z-\bar{Z}). \end{aligned}$$

Thus, the error system is written as follows

Based on the above error system we are ready to prove Theorem 1.

Proof

Let us consider the following candidate Lyapunov function for system (18)

$$\begin{aligned} W(\eta ,\delta )=\eta ^{\top }P\eta +\int _0^1 e^{-\mu x}(\delta -K_r\eta )^{\top }Q(\delta -K_r\eta )\;dx, \end{aligned}$$

(19)

with $\mu >0$, matrices P and Q are specified later.

We rewrite $W(\eta ,\delta )=W_1+W_2$, with $W_1=\eta ^{\top }P\eta$ and $W_2=\int _0^1 e^{-\mu x}(\delta -K_r\eta )^{\top }Q(\delta -K_r\eta )\;dx$. The time derivative of $W_1$ along the solution to system (18a) is computed as

$$\begin{aligned} \dot{W}_1= & {} 2\eta ^{\top }P\dot{\eta }\\= & {} 2\eta ^{\top }P(A\eta +B\delta (1))\\= & {} \eta ^{\top }\biggl (PA_r+A_r^{\top }P\biggr ) \eta +2\eta ^{\top }PB\biggl (\delta (1)-K_r\eta \biggr ). \end{aligned}$$

According to (11), there exists a symmetric positive matrix P such that

$$\begin{aligned} PA_r+A_r^{\top }P<-I_n. \end{aligned}$$

(20)

Due to Cauchy Schwarz inequality, it holds

$$\begin{aligned} \dot{W}_1\leqslant -\vert \eta \vert ^2+2\Vert P B\Vert \;\vert \delta (1)-K_r\eta \vert \;\vert \eta \vert . \end{aligned}$$

(21)

The time derivative of $W_2$ along the solution to system (18b) is

$$\begin{aligned} \dot{W}_2= & {} 2\int ^1_0 e^{-\mu x}(\delta -K_r\eta )^{\top }Q(\delta _t-K_r \dot{\eta })\;dx\nonumber \\= & {} -\frac{2}{\varepsilon }\int ^1_0 e^{-\mu x}(\delta -K_r\eta )^{\top }Q\varLambda \delta _x\;dx-2\int ^1_0 e^{-\mu x}(\delta -K_r\eta )^{\top }QK_r(A\eta +B\delta (1))\;dx\nonumber \\&-2\int _0^1 e^{-\mu x}(\delta -K_r\eta )^{\top }QK_rA_r\bar{Z}\;dx. \end{aligned}$$

(22)

Performing an integration by parts on the first integral in the right-hand side of (22), $\dot{W}_2$ follows

$$\begin{aligned} \dot{W}_2= & {} -\frac{1}{\varepsilon }\biggl [e^{-\mu x}(\delta -K_r\eta )^{\top }Q\varLambda (\delta -K_r\eta )\biggr ]^{x=1}_{x=0}-\frac{\mu }{\varepsilon }\int ^1_0 e^{-\mu x} (\delta -K_r\eta )^{\top }Q\varLambda (\delta -K_r\eta )\;dx\nonumber \\&-2\int ^1_0 e^{-\mu x}(\delta -K_r\eta )^{\top }QK_r(A\eta +B\delta (1))\;dx-2\int _0^1 e^{-\mu x}(\delta -K_r\eta )^{\top }QK_rA_r\bar{Z}\;dx. \nonumber \\ \end{aligned}$$

(23)

Let the first term in (23) be $W_{21}$. Under the boundary condition (18c), we have

$$\begin{aligned} W_{21}= & {} -\frac{1}{\varepsilon }\biggl [e^{-\mu }(\delta (1)-K_r\eta )^{\top }Q\varLambda (\delta (1)-K_r\eta )-(\delta (0)-K_r\eta )^{\top }Q\varLambda (\delta (0)-K_r\eta )\biggr ]\nonumber \\= & {} -\frac{1}{\varepsilon }\biggl [e^{-\mu }(\delta (1)-K_r\eta )^{\top }Q\varLambda (\delta (1)-K_r\eta )\nonumber \\&\;\;\;\;\;\;\;\;\;\;-(K_1\delta (1)+K_2\eta -K_r\eta )^{\top }Q\varLambda (K_1\delta (1)+K_2\eta -K_r\eta )\biggr ]. \end{aligned}$$

(24)

We write

$$\begin{aligned} K_2-K_r= & {} K_2-(I_m-K_1)^{-1}K_2=(I_m-K_1)(I_m-K_1)^{-1}K_2-(I_m-K_1)^{-1}K_2\nonumber \\= & {} (I_m-K_1-I_m)\;(I_m-K_1)^{-1}K_2=-K_1K_r. \end{aligned}$$

(25)

Using the right-hand side of (25) to replace $K_2-K_r$ in (24), we obtain

$$\begin{aligned} W_{21}=-\frac{1}{\varepsilon }\biggl [(\delta (1)-K_r\eta )^{\top }(e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1)(\delta (1)-K_r\eta )\biggl ]. \end{aligned}$$

By using (12), there exists a diagonal positive matrix Q such that

$$\begin{aligned} e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1>\underline{\lambda }(e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1)>0. \end{aligned}$$

(26)

Thus

$$\begin{aligned} W_{21}\leqslant -\frac{\underline{\lambda }(e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1)}{\varepsilon }\vert \delta (1)-K_r\eta \vert ^2. \end{aligned}$$

(27)

Let $W_{22}$ denote the second term in (23), it follows

$$\begin{aligned} W_{22}\leqslant & {} -\frac{\mu e^{-\mu }\underline{\lambda }(Q\varLambda )}{\varepsilon } \Vert \delta -K_r\eta \Vert ^2_{L^2(0,1)}. \end{aligned}$$

(28)

Let the third term in (23) be $W_{23}$, it follows

$$\begin{aligned} W_{23}= & {} -2\int ^1_0 e^{-\mu x}(\delta -K_r\eta )^{\top }QK_r\biggl (A_r\eta +B(\delta (1)-K_r\eta )\biggr )\;dx. \end{aligned}$$

Due to Cauchy Schwarz inequality, $W_{23}$ follows

$$\begin{aligned} W_{23}\leqslant & {} 2\Vert Q K_r A_r\Vert \; \vert \eta \vert \;\Vert \delta -K_r\eta \Vert _{L^2(0,1)}+2\Vert QK_rB\Vert \; \vert \delta (1)-K_r\eta \vert \;\Vert \delta -K_r\eta \Vert _{L^2(0,1)}.\nonumber \\ \end{aligned}$$

(29)

We denote the last term in (23) as $W_{24}$. It holds

$$\begin{aligned} W_{24}\leqslant & {} 2\Vert Q K_r A_r\Vert \; \Vert \delta -K_r\eta \Vert _{L^2(0,1)} \; \vert \bar{Z}\vert . \end{aligned}$$

(30)

Combining (27), (28), (29) and (30), the following hold for all $\kappa >0$,

$$\begin{aligned} \dot{W}_2\leqslant & {} -\frac{\underline{\lambda }(e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1)}{\varepsilon }\vert \delta (1)-K_r\eta \vert ^2-\frac{\mu e^{-\mu }\underline{\lambda }(Q\varLambda )}{\varepsilon }\Vert \delta -K_r\eta \Vert ^2_{L^2(0,1)}\nonumber \\&+2\Vert Q K_r A_r\Vert \; \vert \eta \vert \;\Vert \delta -K_r\eta \Vert _{L^2(0,1)}+2\Vert QK_rB\Vert \; \vert \delta (1)-K_r\eta \vert \;\Vert \delta -K_r\eta \Vert _{L^2(0,1)}\nonumber \\&+\kappa \Vert QK_rA_r\Vert \;\vert \bar{Z}\vert ^2+\frac{\Vert QK_rA_r\Vert }{\kappa }\Vert \delta -K_r\eta \Vert ^2_{L^2(0,1)}. \end{aligned}$$

(31)

Combining (21) and (31), $\dot{W}$ follows

$$\begin{aligned} \dot{W}\leqslant & {} -\begin{pmatrix} \vert \delta (1)-K_r\eta \vert \\ \vert \eta \vert \\ \Vert \delta -K_r\eta \Vert _{L^2(0,1)} \end{pmatrix}^{\top }M\begin{pmatrix} \vert \delta (1)-K_r\eta \vert \\ \vert \eta \vert \\ \Vert \delta -K_r\eta \Vert _{L^2(0,1)} \end{pmatrix}+\kappa \Vert QK_rA_r\Vert \;\vert \bar{Z}\vert ^2, \end{aligned}$$

where $M=\begin{pmatrix} M_1&{}M_2\\ \star &{}{M}_4 \end{pmatrix}$, with $M_1=\begin{pmatrix} M_{11}&{}M_{12}\\ \star &{}M_{14} \end{pmatrix}=\begin{pmatrix} \frac{\underline{\lambda }(e^{-\mu }Q\varLambda -K^{\top }_1Q\varLambda K_1)}{\varepsilon }&{}-\Vert P B\Vert \\ \star &{}1 \end{pmatrix}$, $M_2=\begin{pmatrix} -\Vert QK_rB\Vert \\ Vert QK_rA_r\Vert \end{pmatrix}$, $M_4=\begin{pmatrix} \frac{\mu e^{-\mu }\underline{\lambda }(Q\varLambda )}{\varepsilon }-\frac{\Vert QK_rA_r\Vert }{\kappa } \end{pmatrix}$.

Since $M_{14}>0$, there exists $\varepsilon ^*_1>0$ such that for $\varepsilon \in (0,\varepsilon ^*_1)$, $M_{11}-M_{12}M^{-1}_{14}M^{\top }_{12}>0$. Due to the Schur complement, it holds $M_1>0$. There exists $\sigma >0$ sufficiently large such that $M_4>0$ with $\kappa =\sigma \varepsilon$. Then, there exists $\varepsilon ^*_2>0$, such that for all $0<\varepsilon <\min (\varepsilon ^*_1,\varepsilon ^*_2)$, we have $M_1-M_2M^{-1}_4M^{\top }_2>0$. Using again the Schur complement, it holds $M>0$. Hence, the following holds

$$\begin{aligned} \dot{W}\leqslant -\theta W+\sigma \varepsilon \Vert QK_rA_r\Vert \;\vert \bar{Z}\vert ^2, \end{aligned}$$

for any $0<\theta \leqslant \min \left\{ \frac{\underline{\lambda }(M)}{\overline{\lambda }(P)}, \frac{\underline{\lambda }(M)}{\overline{\lambda }(Q)}\right\}$. Condition (11) implies the exponential stability of the reduced subsystem (4a), that is, there exist positive constants $\bar{C}$ and r, such that for all $t\geqslant 0$,

$$\begin{aligned} \vert \bar{Z}(t)\vert ^2 \leqslant \bar{C} e^{-rt}\vert \bar{Z}_0\vert ^2. \end{aligned}$$

Thus we get

$$\begin{aligned} \dot{W}\leqslant -\theta W+\bar{C}\sigma \varepsilon e^{-rt} \Vert QK_rA_r\Vert \;\vert \bar{Z}_0\vert ^2. \end{aligned}$$

It holds

$$\begin{aligned} W\leqslant & {} e^{-\theta t}W(\eta _0,\delta _0)+\bar{C}\sigma \varepsilon \Vert QK_rA_r\Vert \;\vert \bar{Z}_0\vert ^2\int ^t_0 e^{-\theta (t-s)}e^{-rs}ds\\\leqslant & {} e^{-\theta t}W(\eta _0,\delta _0)+\frac{\bar{C}\sigma \varepsilon \Vert QK_rA_r\Vert e^{-\theta t}\left( 1-e^{(\theta -r)t}\right) }{r-\theta }\vert \bar{Z}_0\vert ^2. \end{aligned}$$

We may assume that $r>\theta$, the above inequality becomes

$$\begin{aligned} W\leqslant & {} e^{-\theta t}\left( W(\eta _0,\delta _0)+\frac{\bar{C}\sigma \varepsilon \Vert QK_rA_r\Vert }{r-\theta }\vert \bar{Z}_0\vert ^2\right) . \end{aligned}$$

The function W is lower and upper bounded by

$$\begin{aligned} \underline{\lambda }(P)\;\vert \eta \vert ^2+e^{-\mu }\underline{\lambda }(Q)\;\Vert \delta -K_r\eta \Vert ^2_{L^2(0,1)}\leqslant & {} W\leqslant \Vert P\Vert \;\vert \eta \vert ^2+\Vert Q\Vert \;\Vert \delta -K_r\eta \Vert ^2_{L^2(0,1)}. \end{aligned}$$

Since the initial conditions are $\eta _0=\delta _0=0$, we obtain

$$\begin{aligned} \vert \eta \vert ^2&\leqslant \varepsilon C_1e^{-\theta t}\vert \bar{Z}_0\vert ^2, \end{aligned}$$

where $C_1>0$. Moreover, it holds

$$\begin{aligned} \Vert \delta \Vert _{L^2(0,1)}&=\Vert \delta -K_r\eta +K_r\eta \Vert _{L^2(0,1)}\leqslant \Vert \delta -K_r\eta \Vert _{L^2(0,1)}+K_r\vert \eta \vert . \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \Vert \delta \Vert ^2_{L^2(0,1)}&\leqslant \varepsilon C_2e^{-\theta t}\vert \bar{Z}_0\vert ^2, \end{aligned}$$

where $C_2>0$. This concludes the proof of Theorem 1. $\blacksquare$

4.2 Tikhonov Theorem for Linear Hyperbolic PDE-Fast ODE System

Following the similar computation in Sect. 4.1, the error system of (6) is written as follows

where $\tilde{\eta }=y-\bar{y}, \tilde{\delta }=Z+A^{-1}B\bar{y}(1)$.

Theorem 2

Consider system (6). If (10) is satisfied, there exist positive values $C_1$, $\gamma$, $\varepsilon ^*$, such that for all $0<\varepsilon <\varepsilon ^*$, for any initial condition $y_0\in H^2(0,1)$ satisfying the compatibility conditions $y_0(0)=K_r y_0(1)$ and $y_{0x}(0)=\varLambda ^{-1}K_r\varLambda y_{0x}(1)$, with $\bar{y}_0=y_0$, and for $Z_0\in \mathbb {R}^n$, it holds for $t\geqslant 0$

$$\begin{aligned} \Vert y(.,t)-\bar{y}(.,t)\Vert ^2_{L^2(0,1)}\leqslant & {} \varepsilon C_1e^{-\gamma t}\biggl (\Vert \bar{y}_0\Vert ^2_{H^2(0,1)}+\vert Z_0+A^{-1}B\bar{y}_0(1)\vert ^2\biggr ). \end{aligned}$$

(33)

Proof

We consider the following candidate Lyapunov function for system (32).

$$ L_{\varepsilon }(\tilde{\eta },\tilde{\delta })=\varepsilon \tilde{\delta }^{\top } P\tilde{\delta }+\int ^1_0 e^{-\mu x}\;\tilde{\eta }^{\top } Q\tilde{\eta }\;dx. $$

Adopting the similar computations in the proof of Theorem 1, the time derivative of $L_{\varepsilon }(\tilde{\eta },\tilde{\delta })$ along the solution to system (32) is

$$\begin{aligned} \dot{L}_{\varepsilon }(\tilde{\eta },\tilde{\delta })= & {} -\begin{pmatrix} \tilde{\eta }(1)\\ \tilde{\delta }\end{pmatrix}^{\top }T\begin{pmatrix} \tilde{\eta }(1)\\ \tilde{\delta }\end{pmatrix}-\mu \int ^1_0 e^{-\mu x}\tilde{\eta }^{\top } Q\varLambda \tilde{\eta }\;dx+2\varepsilon \tilde{\delta }^{\top } P(A^{-1}B\varLambda )\bar{y}_x(1), \end{aligned}$$

where $T=\left( {\begin{matrix}e^{-\mu } Q\varLambda -K^{\top }_1 Q\varLambda K_1&{}-(K^{\top }_1 Q\varLambda K_2+B^{\top } P)\\ \star &{}-(A^{\top } P+ PA)-K^{\top }_2 Q\varLambda K_2\end{matrix}}\right)$.

According to (10), using Cauchy Schwarz inequality and Young’s inequality, the above inequality holds for all $\kappa >0$

$$\begin{aligned} \dot{L}_{\varepsilon }(\tilde{\eta },\tilde{\delta })\leqslant & {} -\underline{\lambda }(T)\vert \tilde{\delta }\vert ^2-\mu e^{-\mu }\underline{\lambda }(Q\varLambda )\Vert \tilde{\eta }\Vert ^2_{L^2(0,1)}\nonumber \\&+\kappa \varepsilon \Vert P(A^{-1}B\varLambda )\Vert \vert \tilde{\delta }\vert ^2+\frac{\varepsilon \Vert P(A^{-1}B\varLambda )\Vert }{\kappa }\vert \bar{y}_x(1)\vert ^2. \end{aligned}$$

(34)

The function $L_\varepsilon (\tilde{\eta },\tilde{\delta })$ is upper and lower bounded as

$$\begin{aligned} e^{-\mu }\underline{\lambda }(Q)\;\Vert \tilde{\eta }\Vert ^2_{L^2(0,1)}+\varepsilon \underline{\lambda }(P)\;\vert \tilde{\delta }\vert ^2\leqslant L_{\varepsilon }(\tilde{\eta },\tilde{\delta })\leqslant \Vert Q\Vert \; \Vert \tilde{\eta }\Vert ^2_{L^2(0,1)}+\varepsilon \Vert P\Vert \;\vert \tilde{\delta }\vert ^2.\qquad \end{aligned}$$

(35)

By choosing $\kappa =1$, there exist $\varepsilon ^*, \gamma >0$, such that for all $\varepsilon \in (0,\varepsilon ^*)$, the following holds from (34)

$$\begin{aligned} \dot{L}_{\varepsilon }(\tilde{\eta },\tilde{\delta })\leqslant & {} -\gamma L_{\varepsilon }(\tilde{\eta },\tilde{\delta })+\varepsilon \Vert P(A^{-1}B\varLambda )\Vert \vert \bar{y}_x(1)\vert ^2. \end{aligned}$$

(36)

Condition (10) implies that $e^{-\mu }Q\varLambda -K^{\top }_rQ\varLambda K_r>0$. Let $\Delta$ be a diagonal positive matrix and $Q=\Delta ^2\varLambda ^{-1}$. It holds $\Delta ^2-K^{\top }_r\Delta ^2K_r>0$, which is equivalent to $\Vert \Delta K_r\Delta ^{-1}\Vert <1$. Then according to [3, Theorem 2.3], the reduced subsystem (8) is exponentially stable in $H^2$-norm. Thus, we deduce from (36)

$$\begin{aligned} \dot{L}_{\varepsilon }(\tilde{\eta },\tilde{\delta })\leqslant & {} -\gamma L_{\varepsilon }(\tilde{\eta },\tilde{\delta })+C_r\varepsilon e^{-ct}\Vert P(A^{-1}B\varLambda )\Vert \; \Vert \bar{y}_0\Vert ^2_{H^2(0,1)}, \end{aligned}$$

(37)

where $C_r$ and c are positive values.

The following holds

$$\begin{aligned} L_{\varepsilon }(\tilde{\eta },\tilde{\delta })\leqslant & {} e^{-\gamma t}L_{\varepsilon }(\tilde{\eta }_0,\tilde{\delta }_0)+C_r\varepsilon \Vert P(A^{-1}B\varLambda )\Vert \; \Vert \bar{y}_0\Vert ^2_{H^2(0,1)}\int ^t_0 e^{-\gamma (t-s)}e^{-cs}ds. \end{aligned}$$

We may assume that $\gamma <c$, the above inequality becomes

$$\begin{aligned} L_{\varepsilon }(\tilde{\eta },\tilde{\delta })\leqslant & {} e^{-\gamma t}L_{\varepsilon }(\tilde{\eta }_0,\tilde{\delta }_0)+e^{-\gamma t}\frac{C_r\varepsilon }{c-\gamma } \Vert P(A^{-1}B\varLambda )\Vert \; \Vert \bar{y}_0\Vert ^2_{H^2(0,1)}. \end{aligned}$$

Using (35), we get

$$\begin{aligned} \Vert \tilde{\eta }\Vert ^2_{L^2(0,1)}\leqslant & {} C_1e^{-\gamma t}\biggl (\Vert \tilde{\eta }_0\Vert ^2_{L^2(0,1)}+\varepsilon \vert \tilde{\delta }_0\vert ^2+\varepsilon \Vert \bar{y}_0\Vert ^2_{H^2(0,1)}\biggr ), \end{aligned}$$

where $C_1$ is positive constant. Since $\tilde{\eta }_0=0$, the inequality (33) holds for all $t\geqslant 0$. This concludes the proof of Theorem 2. $\blacksquare$

5 Numerical Simulations

Let us first show the numerical simulations on an ODE-fast PDE system (1), in which condition (10) is not satisfied. The full system is approximated by both subsystems under the stability conditions of the subsystems for $\varepsilon$ sufficiently small. We next provide the numerical simulations on an academic example of PDE-fast ODE system (6). The Tikhonov approximation is achieved under the full system’s stability condition (10).

5.1 Numerical Simulations on an ODE-Fast PDE System

Let us consider system (1) with $A=1$, $B=-1$, $\varLambda =1$, $K_1=\frac{1}{2}$, $K_2=1$. The initial conditions (1d)–(1e) are selected as $Z_0=0.5$ and $y_0(x)=\cos (4\pi x)-1$. The perturbation parameter $\varepsilon$ is chosen as $\varepsilon =0.01$. It is computed $A_r=-1$ for the reduced subsystem (4). The initial condition (4b) is chosen as the same as for the full system $\bar{Z}_0=Z_0=0.5$. The boundary condition for the boundary-layer subsystem is $K_1=\frac{1}{2}$. The initial condition is chosen as $\bar{y}_0=\cos (4\pi x)-\frac{3}{2}$. In view of $A_r$, condition (11) is satisfied for any $P>0$. By choosing $Q=1$, condition (12) holds. Figure 1 shows that the reduced and the boundary-layer subsystems converge to the origin as time increasing.

Fig. 1

Time evolutions of subsystems (4) and (5)

Let us choose $\varepsilon =\lbrace 0.03,0.02,0.01\rbrace$, the initial condition $y_0$ is selected as the equilibrium point $y_0=K_rZ_0$. Table 1 shows that the errors between the full system and the reduced subsystem decrease as $\varepsilon$ decreasing, as expected from Theorem 1.

Table 1 Estimates of the error for different values of $\varepsilon$ with the initial condition $y_0=K_rZ_0$

5.2 Numerical Simulations on a PDE-Fast ODE System

We consider system (6) with $A=-1$, $B=\frac{1}{2}$ and $\varLambda =1$. The boundary condition (6c) is given by $K_1=-\frac{1}{4}, K_2=-\frac{1}{2}$. The initial conditions (6d)–(6e) are selected as $Z_0=0.2$ and $y_0(x)=\cos (4\pi x)-1$. The perturbation parameter is $\varepsilon =0.01$. By choosing $P=Q=1$, the stability condition (10) is satisfied. Therefore, Proposition 1 applies. In Fig. 2, the solutions of the slow and the fast dynamics of the full system tend to zero when time increases, as expected from Proposition 1.

Fig. 2

Time evolutions of the full system (6)

Moreover, we compute the boundary condition matrix for the reduced subsystem as $K_r=-\frac{1}{2}$. The initial condition (8c) is chosen as $\bar{y}_0=y_0(x)=\cos (4\pi x)-1$. It is observed in Fig. 3 that the solutions of both subsystems converge to the origin as time increasing.

Fig. 3

Time evolutions of subsystems (8) and (9)

Let us choose $\varepsilon =\lbrace 0.03,0.02,0.01\rbrace$. Table 2 shows that the errors between the slow dynamics of the full system and the reduced subsystem decrease as $\varepsilon$ decreasing, as expected from Theorem 2.

Table 2 Estimates of the error for different values of $\varepsilon$

6 Conclusions

A class of singularly perturbed linear ODE coupled with linear hyperbolic PDE systems has been considered in this work. The two subsystems have been formally computed based on the singular perturbation method. A general sufficient stability condition has been provided, which guarantees the stability of the full coupled ODE-PDE system and both subsystems. The Tikhonov approximation for full systems has been established by Lyapunov method. More precisely, based on the stability of both subsystems, the full ODE-fast PDE system is approximated by the subsystems. The estimate error is the order of the perturbation parameter $\varepsilon$. However, for PDE-fast ODE system, the approximation is valid if the general sufficient stability condition is satisfied.