Analysis of a conservation law modeling a highly re-entrant manufacturing system

This article studies a hyperbolic conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. Characteristic features are the nonlocal character of the velocity and that the influx and outflux constitute the control and output signal, respectively. We prove the existence and uniqueness of solutions for $L^1$-data, and study their regularity properties. We also prove the existence of optimal controls that minimizes in the $L^2$-sense the mismatch between the actual and a desired output signal. Finally, the time-optimal control for a step between equilibrium states is identified and proven to be optimal.

1. Introduction and prior work. This article studies optimal control problems governed by the scalar hyperbolic conservation law ∂ t ρ(t, x) + ∂ x (λ(W (t)) ρ(t, x)) = 0 where W (t) = 1 0 ρ(t, x) dx, on a rectangular domain [0, T ] × [0, 1] or the semi-infinite strip [0, ∞) × [0, 1]. We assume that λ(·) ∈ C 1 ([0, +∞); (0, +∞)) in the whole paper. This work is motivated by problems arising in the control of semiconductor manufacturing systems which are characterized by their highly re-entrant character, see below for more details. In the manufacturing system the natural control input is the influx, which suggests the boundary conditions 2 JEAN-MICHEL CORON, MATTHIAS KAWSKI, AND ZHIQIANG WANG ρ(0, x) = ρ 0 (x), for 0 ≤ x ≤ 1, and ρ(t, 0)λ(W (t)) = u(t), for t ≥ 0. (2) Various different choices of the space of admissible controls are of both practical and mathematical interest, each leading to distinct mathematical problems. Motivated by this application from manufacturing systems, natural control objectives are to minimize the error signal that is the difference between a given demand forecast y d and the actual out-flux y(t) = λ(W (t))ρ(t, 1). An alternative to this problem modeling a perishable demand, is the similar problem that permits backlogs. In that case, the objective is to minimize in a suitable sense the size of the different error signal while keeping the state ρ(·, x) bounded. This article only considers the problem of perishable demand and the minimization in the L 2 -sense. Partial differential equations models for such manufacturing systems are motivated by the very high volume (number of parts manufactured per unit time) and the very large number of consecutive production steps which typically number in the many hundreds. They are popular due to their superior analytic properties and the availability of efficient numerical tools for simulation. For more detailed discussions see e.g. [5,2,3,6,16,17,19]. In many aspects these models are very similar to those of traffic flows, compare e.g. [12].
The study of hyperbolic conservation laws, and especially of control systems governed by such laws, have a rich history. A modern introduction to the subject is the text [8]. From a mathematical perspective, the choice of spaces in which to consider the conservations laws (and their data) provides for distinct levels of challenges. Fundamental are question of wellposedness, regularity properties of solutions, controllability, existence, uniqueness and regularity of optimal controls. Existence of solutions, regularity and well-posedness of nonlinear conservation laws have been widely studied under diverse sets of hypotheses, commonly in the context of vector values systems of conservation laws, see e.g. [4,7,9]. Further results on uniqueness may be found in [11], while [10] introduced an a distinct notion of differentiability of the solution of hyperbolic systems. For the controllability of linear hyperbolic systems, see, in particular, the important survey [22]. The attainable sets of nonlinear conservation laws are studied in [1,15,18,20,21], while [14] provides a comprehensive survey of controllability that also includes nonlinear conservation laws.
This article is, in particular, motivated by the recent work [19] which, among others, considered the optimal control problem of minimizing y − y d L 2 (0,T ) (the L 2 norm of the difference between a demand forecast and the actual outflux). That work derived necessary conditions and used these to numerically compute optimal controls corresponding to piecewise constant desired outputs y d .
The organization of the following sections is as follows: First we rigorously prove the existence of weak solutions of the Cauchy problem for the conservation law (1) for the case when the initial data and boundary condition (2) lie in L 1 (0, 1) and L 1 (0, T ), respectively. Next we establish the existence and uniqueness of solutions for the optimal control problem of minimizing the L 2 -norm of the difference between any desired L 2 -demand forecast y d and actual outflux y(t) = λ(W (t)) · ρ(t, 1). Finally, in the classical special case where we prove that the natural candidate control for transferring the system from one equilibrium state to another one is indeed time-optimal. While preparing the final version of this article, the authors received a copy of the related manuscript [13] which is also motivated in part by [5,2,19] and which addresses wellposedness for systems of hyperbolic conservation laws with a nonlocal speed on all of R n . It also includes a study of the solutions with respect to the initial datum and a necessary condition for the optimality of integral functionals. There are substantial differences between [13] and our paper, especially the treatment of the boundary conditions and the method of proof.
2. Existence, uniqueness, and regularity of solutions in L 1 .
We can estimate the first two terms on the right hand side of (14) as where K is a constant independent of ε. While for the last term on the right hand side of (14), we get from (11) that In view of (15)- (17), letting ε → 0 in (14) one gets (9). Proof. We first prove the existence of weak solution for small time: there exists a small δ ∈ (0, T ] such that the Cauchy problem (1) and (2) has a weak solution ρ ∈ C 0 ([0, δ]; L 1 (0, 1)). The idea is to find first the characteristic curve ξ = ξ(t) passing through (0, 0), then construct a solution to the Cauchy problem. Let where λ, λ are defined by (5) and We point out here that the case d(M ) = 0 (by (5), λ is a constant in [0, M ]) is trivial. We only prove Theorem 2.3 for the case d(M ) > 0. We define a map F : Now we prove that, if δ is small enough, F is a contraction mapping on Ω δ,M with respect to the C 0 norm defined by Let ξ 1 , ξ 2 ∈ Ω δ,M . We define ξ 1 ∈ C 0 ([0, δ]) and ξ 2 ∈ C 0 ([0, δ]) by ξ 1 (t) := max{ξ 1 (t), ξ 2 (t)} and ξ 2 (t) := min{ξ 1 (t), ξ 2 (t)}. By (5) and changing the order of the integrations (see Figure 1), we have Using the definitions of ξ 1 , ξ 2 and of Ω δ,M , we obtain that, for every y ∈ [0, ξ 2 (t)] (see Figure 2), Therefore,  Since ρ 0 ∈ L 1 (0, 1), we can choose δ ∈ (0, 1) small enough such that Then By means of the contraction mapping principle, there exists a unique fixed point ξ = F (ξ) in Ω δ,M . By (20), the fix point ξ is an increasing function in C 1 ([0, δ]), and one has Then we define a function ρ by which is obviously nonnegative almost everywhere. Direct computations give that, Using (5), (27) and (29), we obtain the following estimates of ξ ′ from above and below: We now prove that As for the first term on the right hand side of (31), we choose {u n } ∞ n=1 ⊂ C 1 ([0, T ]) which converges to u in L 1 (0, T ), then we have By (30), where C n is a constant independent of s and t but depending on u n . By changing the order of integrations, we obtain furthermore (see Figure 3) and (see Figure 4)  Figure  4. Change order of integration on y and x in (35) As for the second term on the right hand side of (31), it is easy to get that As for the last term on the right hand side of (31), we choose {ρ n 0 } ∞ n=1 ⊂ C 1 ([0, 1]) which converges to ρ 0 in L 1 (0, 1), then we have where D n is a constant independent of s and t but depending on ρ n 0 . Using (19) together with the estimates (31) to (37), we obtain for any s, t ∈ [0, δ] with s ≥ t, We can choose u n and ρ n 0 such that T 0 |u(σ) − u n (σ)|dσ and 1 0 |ρ 0 (y) − ρ n 0 (y)|dy are small as we want. Then according to (30) and the fact that u ∈ L 1 (0, T ) and ρ 0 ∈ L 1 (0, 1), the right hand side of (38) is sufficiently small if s and t are close enough to each other. This proves that the function ρ defined by (28) belongs to C 0 ([0, δ]; L 1 (0, 1)).
Moreover, by (48) and ρ 0 ∈ L 1 (0, 1), u ∈ L 1 (0, T ), one can find a suitably small δ 0 > 0 independent of τ such that Step by step, we finally have a unique global weak solution ρ ∈ C 0 ([0, T ]; L 1 (0, 1)). This concludes the proof of Theorem 2.3. Then, it follows from our proof of Theorem 2.3 that Moreover, W (t) can be expressed as which implies that and Finally, W is absolutely continuous: and 0 ≤ we can expect ρ ∈ L 1 (0, 1; L 1 (0, T )). However, the weak solution is more regular than expected. In fact, under the assumptions of Theorem 2.3, we have the hidden regularity that ρ ∈ C 0 ([0, 1]; L 1 (0, T )) so that the function t → ρ(t, x) ∈ L 1 (0, T ) is well defined for any fixed x ∈ [0, 1]. The proof of the hidden regularity is quite similar to our proof of ρ ∈ C 0 ([0, 1]; L 1 (0, T )) by means of the explicit expression of ρ (see also (54)-(56) that we use when T is large).
For any fixed demand signal y d ∈ L 2 (0, T ) and initial data ρ 0 , define a functional on L 2 + (0, T ) by where y(t) := ρ(t, 1)λ(W (t)) (59) is the out-flux corresponding to the in-flux u ∈ L 2 + (0, T ) and initial data ρ 0 . Theorem 3.1. The infimum of the functional J in L 2 Then we have u n L 2 (0,T ) + y n L 2 (0,T ) ≤ C, ∀n ∈ Z + . (62) In (62) and hereafter, we denote by C various constants which do not depend on n.
The uniform boundedness of u n in L 2 (0, T ) shows that there exists u ∞ ∈ L 2 + (0, T ) and a subsequence of {u n k } ∞ k=1 such that u n k ⇀ u ∞ in L 2 + (0, T ). For simplicity, we still denote the subsequence as {u n } ∞ n=1 . Let ρ n be the weak solution to the Cauchy problem of equation (1) with the initial and boundary conditions ρ(t, 0)λ(W (t)) = u n (t), Thus by (51), we have In view of (62) and (64), we can derive from (51) that which in turn gives with (64) that Moreover, let us point out that ξ ′ n is uniformly bounded from above and below: where λ, λ are defined by (5) with Then it follows from Arzelà-Ascoli Theorem that there exists ξ ∞ ∈ C 0 ([0, T ]) and a subsequence {ξ n l } ∞ l=1 such that ξ n l → ξ ∞ in C 0 ([0, T ]). Now we choose the corresponding subsequence {u n l } ∞ l=1 and again, denote it as {u n } ∞ n=1 . Thus we have u n ⇀ u ∞ in L 2 (0, T ), as n → ∞ (70) and ξ n → ξ ∞ in C 0 ([0, T ]), as n → ∞.

4.
Time-optimal transition between equilibria. In this section, we focus on the specific model that relates the nonlocal speed to the total mass according to the assumption (4).
It is immediate that constant boundary data ρ(·, 0) = ρ in ≥ 0 eventually drive the state to the equilibrium ρ ≡ ρ in . Together with the symmetry (t, x, ρ(t, x)) −→ (T − t, 1 − x, ρ(T − t, 1 − x)) of the conservation law (1) this establishes (long-time state) controllability. Of particular interest is the question of how long it takes to drive the system from one equilibrium state ρ 0 to another equilibrium state ρ 1 , compare also the numerical studies of transfers between equilibria in [19].
We first explicitly calculate all quantities for the corresponding piecewise constant boundary data ρ(·, 0), and subsequently prove that this boundary control is indeed time-optimal.
A natural choice for the boundary values is ρ(t, 0) = ρ 1 for t ≥ 0. This determines for 0 ≤ t ≤ T the control influx and the outflux via u(t) = ρ 1 λ(W (t)) and y(t) = ρ 0 λ(W (t)), where W is a solution of the initial value problem This can be integrated in closed form, yielding and similar expressions for the fluxes and the speed. All characteristic curves are translations of the solution of the initial value problem which has the explicit solution The time T to achieve this transition between equilibria is uniquely determined by ξ(T ) = 1 and evaluates to In the sequel we prove that this time is indeed minimal.
Note that W is a continuous function, and, in particular W (T ) = ρ 1 . It is convenient to extend ρ, u, v, and W to negative times by setting ρ(t, x) = W (t) = ρ 0 and u(t) = y(t) = ρ0 1+ρ0 = u 0 = y 0 for all t < 0. Then u is continuous except for a jump at t = 0, and y is continuous except for a jump at T . Note that the height of the jump of u at t = 0 is larger than the corresponding jump of y at T .
(102) While it may seem intuitive that this control is time-optimal, we need to rigorously prove that it is indeed not possible to improve on this time by e.g. temporarily increasing the speed via smaller influxes.
The primary interest is the case of t 0 < t 1 . For t 0 ≤ t ≤ t 1 estimate