Input-to-state stability of a scalar conservation law with nonlocal velocity

In this paper, we study input-to-state stability (ISS) of an equilibrium for a scalar conservation law with nonlocal velocity and measurement error arising in a highly re-entrant manufacturing system. By using a suitable Lyapunov function, we prove sufficient and necessary conditions on ISS. We also analyze the numerical discretization of ISS for a discrete scalar conservation law with nonlocal velocity and measurement error. A suitable discretized Lyapunov function is also analyzed to provide ISS of an equilibrium for the numerical approximation. Finally, we show numerical simulations to validate the theoretical findings.

1. Introduction. The nature of modern high-volume production is characterized by a large number of items passing through many production steps. This type of production system has fluid-like properties and has been modelled successfully by continuum models [1,2,12,18,25]. In these models, the product at different production stages and the speed of production are the quantities of interest.
Specifically, in the manufacturing system of a factory that involves a highly reentrant system where products visit machines multiple times, such as the production of semiconductor devices, a continuum model has been introduced in [2] that is inspired by the Lighthill -Whitham traffic model [24]. The dynamics of this model is mathematically given by hyperbolic partial differential equation of the form (1.1) ∂ t ρ(t, x) + λ(W (t))∂ x ρ(t, x) = 0, t ∈ [0, +∞), x ∈ [0, 1], where ρ(t, x) is the product density which describes the total mass W (t) at the time t and the production stage x, x)dx, t ∈ (0, +∞).
Contrary to classical traffic flow models the differential equation depends on the nonlocal quantity (1.2). The function λ(W (t)) is a velocity. In production systems, it is natural to assume that the velocity function is positive and decreasing as the total mass is increasing. In the manufacturing system, the initial density of products at production stage x is taken as the initial data 1], and the influx is used to control the system or stabilize the system at an equilibrium. Since the velocity is positive, we only require boundary conditions at x = 0 i.e. the influx (1.4) ρ(t, 0)λ(W (t)) = U (t), t ∈ [0, +∞).
Under suitable assumptions on λ, ρ 0 and U , the existence and uniqueness of a classical solution of the Cauchy problem for the scalar conservation law (1.1) with (1.3) and (1.4) is proven in [13,14,17,28]. General stabilization problems with boundary controls have been studied in the past years in [5-9, 11, 15, 16, 19, 21, 27, 29, 32] for hyperbolic systems and recently in [13,17] for scalar conservation laws with nonlocal velocity. The focus is to derive an asymptotic stability around a given equilibrium such that solutions to the conservation laws reach the equilibrium state as time tends to infinity. Such a property is attained by an exponential stability result e.g. in [6]. However, when boundary controls are subjected to unknown disturbances, solutions reaching the given equilibrium point are influenced by the disturbances and a notion of asymptotic stability is required. This property is covered in an input-to-state stability (ISS) [26,27,29]. Concerning an asymptotic behavior of classical solutions, the Lyapunov method is used to investigate sufficient conditions to achieve an exponential stability in [8,16,19] for hyperbolic systems and in [13,17] for scalar conservation laws with nonlocal velocity. The Lyapunov method is also used for ISS of (local) hyperbolic systems in [27,29]. For the numerical analysis of asymptotic behavior of numerical solutions discretized by a first-order finite volume scheme, a discrete Lyapunov function is used to prove exponential stability results for hyperbolic systems in [3,4,20,22,23] and for scalar conservation laws with nonlocal velocity in [13], and ISS results for (local) hyperbolic systems could be established recently in [10,30,31].
In connection with a scalar conservation law with nonlocal velocity, in [13], the authors have studied global feedback stabilization of the closed-loop system (1.1) under the feedback law where k ∈ R is the feedback parameter and ρ * ∈ R is a given equilibrium. They generalize the stabilization results of [17] by using a Lyapunov function. In particular, for a given equilibrium ρ * = 0 and a general velocity function λ ∈ C 1 ([0, +∞); [0, +∞)), the global stabilization result in L 2 for the closed-loop system (1.1), (1.3), and (1.5) is generalized to L p (p ≥ 1). Then, the global stabilization result in L 2 for the closed-loop system (1.1), (1.3), and (1.5) with a family of velocity functions is obtained for a given equilibrium ρ * > 0. By using a discrete Lyapunov function, they also established stabilization results for a discrete scalar conservation law with nonlocal velocity and using a first-order finite volume scheme.
In this paper, we study ISS for the closed-loop system (1.1) and (1.3) under the feedback law defined by where d ∈ R is a bounded perturbation in the measurement. In particular, we use an ISS-Lyapunov function to investigate sufficient and necessary conditions for ISS in L 2 for an equilibrium ρ * ≥ 0 and the velocity function defined by (1.6). The numerical analysis of sufficient and necessary conditions for ISS is performed by using a discrete ISS-Lyapunov function for numerical solution obtained by a first-order finite volume scheme. Moreover, we provide numerical simulations to illustrate theoretical results for some velocity functions of type (1.6).
The paper is organized as follows: In section 2, we present stabilization results of ISS for a scalar conservation law with nonlocal velocity and measurement error. The numerical discretization of stabilization results of ISS for the scalar conservation law with nonlocal velocity and measurement error is presented in section 3. Finally, in section 4, we show numerical simulations for the scalar conservation law with nonlocal velocity and measurement error to illustrate the theoretical results.
2. Asymptotic stability of a scalar conservation law with nonlocal velocity and measurement error. We study ISS of a closed-loop system of scalar conservation laws with nonlocal velocity and measurement error of the form: where ρ(t, x) is the product density, λ(·) ∈ C 1 ([0, +∞), (0, +∞)) is the velocity function, W (t) is total mass defined by (1.2), U (t) is the controller or the input function defined by (1.4), k ∈ [0, 1] is a feedback parameter, ρ * ≥ 0 is an equilibrium solution and d(t) ∈ R is a bounded (known) perturbation in the measurement. A weak solution of the closed-loop system (2.1) is defined below.
We now analyze ISS for the system (2.1) with ρ * ≥ 0 in the sense of the following definitions:

Definition 2.3 (ISS-Lyapunov function).
A continuously differentiable function L : [0, ∞) → R + is said to be an ISS-Lyapunov function for the closed-loop system This manuscript is for review purposes only.
Before we begin the proof of Theorem 2.4, we consider the following transformation around the equilibrium ρ * , Then, the system (2.1) with (1.6) can be rewritten as follows for t ∈ (0, +∞): . By using the velocity function (1.6) in (2.6), we have For convenience, until the end of this proof, we omit the symbol "˜". Then, the system (2.6) with (2.7) can be rewritten in the following form for t ∈ (0, +∞): With the above notation, the assumption (2.5) in Theorem 2.4 is then Proof. The following proof of Theorem 2.4 is an extension of the proof of Theorem 3.2 in [13]. Since C 1 −functions are dense in L 2 (0, 1), we can analyze ISS for the system (2.8) with non-negative weak solution ρ ∈ C 0 ([0, +∞); L 2 (0, 1)) as follows: We first define a candidate ISS-Lyapunov function by where β > 0 and a ∈ R are constants. If In particular, for ρ * = 0, we take a = 0 in (2.10). Then, the time derivative of the candidate ISS-Lyapunov function (2.10) is computed as follows: 13) where the boundary term is defined as The boundary term can be further simplified as This manuscript is for review purposes only.
3. Numerical study of asymptotic stability of a scalar conservation law with nonlocal velocity and measurement error. In order to study ISS of the closed-loop system (2.1), we first divide the spatial domain [0, 1] with the cells centers and boundary points are denoted by x j = (j − 1 2 )∆x, j ∈ {1, . . . , J} and, x 0 and x J+1 , respectively such that ∆xJ = 1, J ∈ N. Moreover, we approximate W (t) as with the point wise values of the solution ρ n j = ρ(t n , x j ) and the discrete velocity function λ n is given by This manuscript is for review purposes only.
We use (3.23) in (3.21) to obtain  We consider the closed-loop system (2.1) with the given velocity function and rate of measurement error

Example 1.
Given an equilibrium solution ρ * = 0, we set an initial condition ρ 0 (x) = 1 + sin(2πx) for x ∈ [0, 1]. Moreover, for d ≡ 0 and in the sense of Definition 3.2 the decay rate of the Lyapunov function is obtained as follows λ(ρ * + W (t)) and where θ = ρ * 1+ρ * , and k ∈ [0, 1) and β > 0 are taken such that stability conditions hold. Besides, in the sense of Definition 3.1, the discrete decay rates of the solution is given by γ 1 := 0.5η. Then, we show the L 2 −error of the solution of the system (2.1) and the discrete decay rates for two given CFL conditions 0.5 and 0.9 in Table 1, respectively. Due to the artificial diffusion and the disturbance we observe only approximately the first-order of the numerical scheme. Furthermore, Figure 1 shows the convergence of the solution of the system (2.1) to the equilibrium for different values of k. In Figure 1, we observe that as k increases the rate of decay of the Lyapunov function decreases due to the weaker control action. Furthermore, we observe that below the mesh accuracy of ∆x = 10 −3 no further decay is observed.

Example 2.
We take an equilibrium solution ρ * = 1 and an initial condition ρ 0 (x) = 2 + 2 sin(2πx) x ∈ [0, 1]. We show similar results as above for the system (2.1) with equilibrium ρ * = 1 which are presented in Table 2 and Figure 2. Here, the first-order convergence of the scheme is visible. The observed decay rate γ 1 is smaller possibly due to the different equilibrium.