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Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves.


Introduction
The goal of this paper is to study the reachable sets, optimal controllability and exact controllability of the following scalar conservation laws with discontinuous flux u t + F (x, u) x = 0, if x ∈ R, t > 0, u(x, 0) = u 0 (x), if x ∈ R, (1.1) where the flux F is given by, F (x, u) = H(x)f (u) + (1 − H(x))g(u), H is the Heaviside function. Through out this article we assume the fluxes f, g to be C 1 (R), strictly convex with superlinear growth (i.e., lim |p|→∞ f (p) |p| , g(p) |p| = (∞, ∞)) and initial data u 0 ∈ L ∞ (R). We denote by θ f , θ g the unique minima of the fluxes f, g respectively. In this article, by entropy solution we mean a weak solution to (1.1) satisfying interface entropy condition as in [5].
Here we explore three aspects of control theory in conservation laws with discontinuous flux: (i) characterization of reachable set, (ii) exact controlability and (iii) optimal controlability. Above three problems are classical and they are answered for f = g case in [2,3]. It is an open question for f = g case. One may think of clubbing states obtained from two boundary control problems which are separately known from [2,3]. Unfortunately, this does not work since the equation (1.1) is completely different from solving two different boundary value problems. Furthermore, adopting the method of backward construction [2,3] to characterize the reachable set is a big challenge due to the following facts: (1) Unlike the scalar conservation laws, for (1.1), L 1 -contraction is still unknown in general setting even if f, g are convex.
(3) Reflected characteristic curves (see definition 2.4) from the boundary can occur in the structure of entropy solution to (1.1).
We resolve the above difficulties by introducing a new backward construction to characterize the reachable sets. Then we adopt this to obtain the optimal control result. The scalar conservation laws with discontinuous flux of type (1.1) has a huge variety of applications in several fields, namely traffic flow modeling, modeling gravity, modeling continuous sedimentation in clarifier-thickener units, ion etching in the semiconductor industry and many more. In the past two decades the first order model of type (1.1) has been extensively studied from both the theoretical and numerical point of view. Concerning the uniqueness it is worth to mention that the following Kružkov type entropy inequalities, in both the two upper quarter-planes are not sufficient to guarantee the uniqueness,  Here (φ 1 , ψ 1 ) denote the entropy pair corresponding to the flux f , (φ 2 , ψ 2 ) denote the entropy pair corresponding to the flux g, and s ∈ C 1 0 (R × R + ), a non-negative test function. Consequently one need an extra criteria on the interface called "interface entropy condition" (see [5]) given by meas{t : f ′ (u(0+, t)) > 0, g ′ (u(0−, t)) < 0} = 0. (1.3) Using this extra entropy along with the above Kružkov type inequalities the uniqueness result has been obtained in [5]. On the other hand, the existence result has been proved in several ways, namely via Hamilton-Jacobi, convergence of numerical schemes, vanishing viscosity method, for further details we refer the reader to [5,6,7,8,11,16,17,24,26,32] and the references therein. The present article uses the explicit formula obtained in [5], via the Hamilton-Jacobi Cauchy problem. By using this formula it can be shown that if the initial data v 0 is uniformly Lipschitz then the viscosity solution v(·, t) is also uniformly Lipschitz, for all t > 0. Let u := ∂v ∂x , then u is the unique weak solution (see [5], Theorem 2.2) of (1.1), enjoys (1.3) near interface and satisfies the following Rankine-Hugoniot condition on the interface.
meas t : f (u(0+, t)) = g(u(0−, t)) = 0. (1.4) Note that in general TV of entropy solution to (1.1) can blow up [1,25] at finite time even for BV initial data which makes the current article more technical while obtaining the compactness. Regarding the well-posedness theory to f = g case, we refer the reader to [23] for Cauchy problem and for the initial boundary value problem to [31].
Concerning the exact controllability for the scalar convex conservation laws the first work has been done in [13], where they considered the initial boundary value problem in a quarter plane with u 0 = 0 and by using one boundary control they investigated the reachable set. As in [2], they considered u 0 ∈ L ∞ and three possible cases, namely pure initial value problem with initial data control outside any domain, initial boundary value problem in a quarter plane with one boundary control and initial boundary problem in a strip with two boundary controls to get the reachable sets in a complete generality. In both the articles the Lax-Oleinik type formulas has been exploited. An alternative approach has been provided in [30] by using the return method (see [20,21]). For the viscous Burgers equation any non-zero state can be reached in finite time by two boundary controls [29], recently, it has been proved [12] that there exist many pairs (C, T ) so that the state C is not reachable from zero state at time T for the viscosity 1. Control theory for the system of conservation laws is still largely open. We refer to [10,14,15,22,27,28] and references therein for controllability results on system of hyperbolic conservation laws.
Let us briefly discuss the optimal controllability results for the case f = g. Assume the target function k ∈ L 2 loc (R), support of f ′ (k) is compact and T > 0. We denote by J {f =g} , a cost functional, defined in the following way where u 0 ∈ L ∞ (R), u 0 ≡ θ f outside a compact set, θ f being the only critical point of the flux f . Here u(·, T ) denotes the unique weak solution at t = T to the Cauchy problem (1.1), in the case f = g with initial datum u 0 . Then in this case, the optimal control reads like: find a w 0 such that J {f =g} (w 0 ) = min u 0 J {f =g} (u 0 ). In [18,19], they considered the above optimal control problem for the Burgers' equation and proved such minimizer exists and proposed a numerical scheme called "alternating decent algorithm", although the convergence of these scheme still remains open. Whereas in [3], they made use of the Lax-Oleinik formula and derived a numerical backward construction which converges to a solution of the above problem. The latter method can be applied also to general convex fluxes as long as a Lax-Oleinik type formula is available. It has to be noticed that even for the case f = g, due to the occurrence of the shocks in the solution of (1.1), one may have several minimizers of the optimal control problem (1.5).
One if the main results in this paper is to characterize the reachable set (see subsection 1.1) and then we prove the exact and optimal controllability (subsections 1.2 and 1.3 respectively) for (1.1). In order to do so, we divide R × (0, T ) into three sub domains: D 1 , D 2 and D 3 (see subsection 4.1). These three domains correspond to the solution (a) with reflected characteristics, (b) having interface interactions and (c) solving pure initial value problem (i.e. f = g case) respectively. Now we define a reachable set at t = T in such a way that a given solution corresponds to an element in the reachable set. This imposes a constraint on the elements of the reachable set. Then for any element in the reachable set, using this constraint, we first construct a data inD 1 ∩ {(x, t) : t = 0} and the solution. Using the 'no forward interface rarefaction' from the t-axis (see lemma 3.2), we construct another initial data and a solution in D 2 . Construction of solution in D 3 is as in the f = g case. Using the R-H condition, we glue all the three solutions to form a single solution which corresponds to the given element in the reachable set. In the construction, we use finer analysis of characteristic curves from [2,3,4] and explicit formula from [5]. Similar construction is valid (see section 6) for (A, B)-connection [8].
The paper is organized as follows. In the next three subsections we state our main results. In section 2, we recall some known results from [5]. Section 3 deals with the non existence of forward rarefaction from the interface, backward construction for shock and continuous solution. Also the construction of (τ 0 , ξ 0 ) which is used to define the reachable set R(T ). Subsections 3.3 and section 4 deals with the backward construction when the reflected characteristics exist. Sections 4 and 5 proves the main results of the paper. In section 6 we indicate how to extend the above results for the (A, B) entropy solution. In section 7, we prove a stability lemma which is used to prove the main results.
Through out this article we assume that f (θ f ) ≤ g(θ g ). The other case f (θ f ) > g(θ g ) follows in a similar way.
Authors declaration: It is to be noted most of the ideas and technical details was in the arxiv version but that there was a gap in the proof in [9] because the reflected characteristics was not being considered in the proof and hence the definition of reachable set was incomplete. In this article we fill this gap in subsection 3.3 and present a modified version of [9].
if and only if there exists a u 0 ∈ L ∞ (R) and the solution u of (1.1) such that for i = 1, 2, where R i and y(x, T ) defined in theorem 2.1.

Optimal control
Let u 0 ∈ L ∞ (R) and u be the solution of (1.1) with initial data u 0 . Let T > 0. Let k ∈ L ∞ (R) and c > 0 such that Define Note that K ∈ L ∞ (R) and support of K ⊂ [−c, c]. Denote g −1 + to be the inverse of g on [g(θ g ), ∞). Let u 0 ∈ L ∞ (R) and u be the corresponding solution of (1.1). Define the cost functional J : (1.12) Then we have the following result on optimal control problem. Theorem 1.3. Let A be the admissible class of functions defined by Then there exists a u 0 ∈ A such that We prove the Theorem 1.3 via an explicit construction and hence can be adopted to numerical computation.
Remark 1.1. We can obtain the similar results, when one of the flux is concave and another one is convex in the equation (1.1). One can use the explicit formulas as in [8] and similar analysis in the present paper.

Preliminaries
In order to make the paper self contained we recall some results, definitions and notations from [5]. 1. γ is piece-wise affine and it can have at most 3 affine segments such that each affine part lies

If γ has three affine segments {γ
Let 0 < t, x ∈ R and let c(x, t) be the set of all control curves such that γ(t) = x. The set c(x, t) can be partitioned into three categories: 1. c 0 (x, t) ⊂ c(x, t) consists of control curves γ which have only one affine segment and satisfies xγ(s) ≥ 0 for s ∈ [0, t].
2. c r (x, t) ⊂ c(x, t) consists of control curves γ which have exactly 3 affine segments and satisfies xγ(s) ≥ 0 for s ∈ [0, t]. Here we say c r (x, t) to be the set of all reflected control curves. Observe that (f * ) ′ = (f ′ ) −1 .

Definition 2.3.
Cost function: Let f * , g * be the respective convex duals of the fluxes f and g. Let us assume that v 0 : R → R be an uniformly Lipschitz continuous function. Let (x, t) ∈ R × R + , γ ∈ c(x, t).
The cost functional Γ associated to v 0 is defined by Then we define the value function v : We say an element of the set ch(x, t) ∩ c r (x, t) to be a reflected characteristics curves.
Let t > 0, define (see figure 2 for further illustration) R 1 (t) if the above set is empty.
L 1 (t) if the above set is empty.
Definition 2.5. Let (X, d) be a metric space and A k , A are subsets of X for each k ≥ 1. We say that lim k→∞ A k ⊂ A if for every sequence {x k } with x k ∈ A k , there exists a subsequence {x k i } converges to some Let f, g be in C 1 (R) and strictly convex function with F (x, u) = H(x)f (u) + (1 − H(x))g(u). Let {f k }, {g k }in C 1 (R) be sequences of strictly convex functions and F k (x, u) = H(x)f k (u) + (1 − H(x))g k (u) such that lim k→∞ (f k (u), g k (u)) = (f (u), g(u)) in C 1 loc (R), Let u 0 ∈ L ∞ (R) and {u 0,k } ⊂ L ∞ (R) be such that Let v 0 (x) = x 0 u 0 (θ)dθ and v 0,k (x) = x 0 u 0,k (θ)dθ be the associated primitives of u 0 and u 0,k . Notice that in [5], it was assumed that u 0 ∈ C(R). Now it is easy to show that results in [5] continue to hold for u 0 ∈ L ∞ (R). In order to prove this result, we need the following stability lemma.
Lemma 2.1 (Stability lemma). With the data as above, let v, v k be the corresponding value functions associated to the fluxes F and F k and initial data v 0 and v 0,k respectively. Let ch(x, t) and ch k (x, t) be the respective characteristic sets. Let u = ∂v ∂x and u k = ∂v k ∂x , then Proof of this lemma is given in the appendix (section 7). Then u(x, t) = ∂v ∂x (x, t) exists for t > 0, a.e. x ∈ R and is a solution to (1.1). Furthermore there exist Lipschitz continuous curves R i (t), L i (t), for i = 1, 2 such that for each t > 0 a.e. x ∈ R, we have

2.
Let T > 0, then one of the following holds: 3. The following properties are true: (ii) x → y(x, t) is a non decreasing function and x → t + (x, t) is a non increasing function on the domain of definitions.
(viii) u(0+, t), u(0−, t) exist and RH condition holds, i.e., f (u(0+, t)) = g(u(0−, t)) for a.e. t > 0, Interface entropy condition: Let L 1 -denotes the one dimensional Lebesgue measure, then The entropy solution u is explicitly given by the following Lax-Oleinik type formula, for t > 0, a.e., x ∈ R, (xii) L 1 -Contractivity: Let u 0 , v 0 ∈ L ∞ (R) and u, v be the solution of (1.1) with corresponding initial data u 0 , v 0 respectively. Assume that the set of discontinuity of u and v are discrete set of Lipschitz curves.
We remark that there is a change in the notation used here and in [5] and is as follows: See equations (4.13), (4.10), page 51 in [5]: See equation (4.25), page 54 in [5] Now comes to the identification of (i) to (xii) in [5].

Some technical lemmas
First observe that η ∈ ch(x, t), then η is a curve consists of atmost three line segments and denotė Proof. Let 0 < x 1 < x 2 < R 2 (t) and Then by dynamic programming principle, If x 2 < 0, then θ 1 ≤ θ 2 and γ 1 (0) ≤ γ 2 (0). This prove the lemma. Figure 3: Rarefaction from the interface cannot occur, hence the above figure is not possible.

No rarefaction from the interface
One of the key factor of this article is that there exists no rarefaction from the interface for the solution of (1.1). This is useful for backward construction (section 4).

Definition 3.2 (Forward rarefaction from the interface). We say that the solution u admits a forward rarefaction on the interface if
Lemma 3.2. There does not exists forward rarefaction from the interface (see figure 3).
Proof. Suppose not, without loss of generality we can assume that there exist 0 < x 1 < x 2 < R 2 (t), , and from lemma 3.1, we have Letting k → ∞ to obtain .
which is contradiction. This proves the Lemma.
As an immediate consequence of this lemma, we have the following: is a strictly decreasing function and is strictly increasing function.
Proof. We prove this for t + (x, t) and similarly proof holds for Hence u admits a forward rarefaction from the interface and from lemma 3.2 we get a contradiction. This proves the lemma.
contradicting the non decreasing hypothesis on ρ. This proves (i) and (ii). If α > 0, then T − α f ′ (θ g ) < T and hence (i) and (ii) together imply (iv). Let α = 0 and t 0 = lim which is a contradiction. Hence t 0 = T and Proof of (2): As . Thus, a 2 < a 1 and b 2 < b 1 . By the choice of a 1 and a 2 , . This proves the lemma.

Building blocks: Construction of shock solution and continuous solution
then the solution u of (1.1) in R × [0, T ] with initial data u 0 is given by (see figure 4) Proof. From lemma 3.5, , hence u is a weak solution satisfying the interior and interface entropy condition with initial data u 0 . This proves the lemma. In lemma 3.6, under suitable hypothesis on x 0 , ρ i , i = 1, 2, we constructed a solution which admits shocks. Next we consider the case where 0 ≤ x 1 < x 2 and ρ 0 < 0. Under a suitable hypothesis, we construct a continuous solution to (1.1).
Then we have following: Let u(x, t) be the solution of (1.1) with initial data u 0 as above. Then (3.14) Proof. Define the regions in R × (0, T ) by Let x > 0, 0 < t ≤ T and w ∈ c b (x, t) ∩ ch(x, t). Then w = (w 1 , ∅, w 3 ) is given by τ , then from the convexity of f and g and the above relation gives That is f (p 1 ) = g(q 1 ). Observe that (x, t) ∈ Ω 3 if and only if x > γ 2 (t) = f ′ (a 2 )(t − t 2 ). If t − t 2 ≤ 0, then τ < t ≤ t 2 . Nowγ(t) = g ′ (b 2 )(t − τ ) is the only characteristic of u in x < 0 andγ(τ ) = 0. Since w 3 is a characteristic in x < 0 and w 3 (τ ) = 0, hence w 3 (t) = g ′ (b 2 )(t − τ ). This implies ρ 0 < w 3 (0) < 0. If t > t 2 , As y → t + (y, t) is an increasing continuous function, constant on the line γ 2 (t) and hence for x > γ 2 (t), it follows that t + (x, t) < t 2 . Due to t > t 2 , we therefore have and therefore w 3 (0) < ρ 0 . Since no two characteristic intersects, hence τ > t 1 and − , and subsequently, we obtain and therefore b 1 > b 2 . But b 1 < b 2 which is a contradiction. Hence τ < t 2 and from the non intersecting of characteristics, it follows that ρ 0 < w 3 (0) < 0 and w As a consequence of this if max{γ 1 (t), 0} < x < γ 2 (t) and w = ( is an increasing function, hence at the point of differentiability of t(·, t), we have u( . This proves the lemma. Previous construction (as in lemma 3.6 and 3.7) of solutions corresponds to the case when no reflected characteristic occurs. Now we deal with the case when reflected characteristics occur and the definition of (τ 0 , ξ 0 ) (see subsection 1.1) which is needed to define the reachable set R(T ). We do it in the region x > 0. Similar construction follows for x < 0.
(2) t → s ξ (t) is a non decreasing convex function with Hence s ξ is unique.
From the convexity, we prove this by induction on i. Asᾱ ≤ β <ᾱ, by the convexity of f , we have Hence integrating from θ to T to obtain Now choose t k−1 by Since R 1 − ξ = T f ′ (β) and u k−1 < u k = β to obtain . Now by induction on i (a 1 ), (a 2 ) and (a 3 ) holds.
Let v k (x, t) be as in (viii), then for t / ∈ {t p , t 1 , · · · , t k }, Hence From Arzela-Ascoli, we can find a subsequence still denoted by {s ξ k } such that s ξ k → s ξ uniformly and t p (ξ k ) → t 0 (ξ) as k → ∞. Since s k is convex for each k, hence s ξ is convex and satisfies (1) and (2).

Solution with reflected characteristics
Earlier we build two solutions via backward construction, namely one has shock and other is a continuous solution. Now we need to construct another solution by backward construction for the reflected case and is as follows: Let (T, R 1 , R 2 , y(·)) ∈ R(T ). Assume that there are constants y − ≤ 0 ≤ y + such that Since (T, R 1 , R 2 , y(·)) ∈ R(T ), we have to consider three cases. In each case we construct a u 1,0 ∈ L ∞ (R) and the description of the corresponding solution u such that for i = 1, 2, R i (T ) = R i , y(·, T ) = y(·).
Since (T 1 , R 1 , R 2 , y(·)) ∈ R(T ) hence y + must satisfy In this case define the following quantities: let where t 1 be such that γ 6 (t 1 ) = 0. Then by the definition of h + , t 0 and convexity of f, g, it follows Figure 7: Solution with reflected characteristics as in Case 1.
Since ξ 0 ≤ y + , hence from convexity of f , we have In this case define the initial data u 1,0 by It is easy to verify that the solution u 1 (·, ·) of (1.1) with initial data u 1,0 is given by In this case define the domain D 1 for t ≤ T by Figure 8: The figure illustrate as in Subcase 1.
Let t 1 be such that then the corresponding solution u 1 of (1.1) is given by Define then u 1 satisfies Case 3: Figure 11: Solution with reflected characteristics as in Case 3. Define Due to ξ 0 ≤ y + = γ 1 (0), we have u + ≤θ g . Hence by convexity of f , and the corresponding solution u 1 is given by Let

Backward construction
Lemma 4.1. Let 0 < R 2 and y : [0, R 2 ] → (−∞, 0] be a non decreasing function. Define Then there exists a u 2,0 ∈ L ∞ (R) and the corresponding solution u 2 of (1.1) such that Proof. Without loss of generality by approximation, we assume that y is a strictly increasing continuous function and N > 1, let k > 1 and define a discreatization by where δ i is defined by α i (δ i ) = 0. Then from the convexity of f and g, we have τ 2i−1 < δ i < τ 2i , Then the solution u N 2 of (1.1) with initial data u N 2,0 in R × (0, T ) is given by (see figure 12) Next we show that the above sequences converges: Convergence Analysis: First we show that ||u N 2,0 || ∞ is uniformly bounded. Let i 0 = sup{i : Since {τ 2i } is a decreasing sequence and {x i } is an increasing sequence, we have Therefore, by Helly's Theorem, there exists a subsequence still denoting by {g ′ (u N 2,0 )} converges pointwise to g ′ (u 2,0 ). Hence ∀y ∈ R, lim N →∞ u N 2,0 (y) = u 2,0 (y), and u 2,0 ∈ L ∞ (R) with Then then for a.e. x ∈ (0, R 2 ), we have, t N is a strictly increasing function, t 1 ≤ t N (x) ≤ T and for a.e., x ∈ (0, R 2 ), we have Subsequently, we have Hence for a subsequence still denoted by {u N 2 } converges to u 2 , a solution of (1.1) with initial data u 2,0 . From Helly's Theorem, again for a subsequence, Then from (4.4), letting N → ∞ to obtain for a.e.
and u 2 satisfies the conclusion of the lemma. If f and g are not uniformly convex (and just strictly convex), then approximate f and g by f ǫ and g ǫ respectively which are uniformly convex and by stability lemma 2.1, the lemma follows as ǫ → 0.

Optimal control
Let K be given, the associated cost functional J and admissible set A are as in (1.12). Then we have the following: Proof. Because of finite speed of propagation, it is immediate.

Proof of
Step 1: Let Then w is the solution to (1.1) with data w 0 , where This proves Step 1.
Since K(x) = 0 for x > c, we get which is a contradiction. Therefore, there exists M 2 > 0 such that R 1 ≤ M 2 .
As earlier we can decompose the domain R × (0, T ) into three disjoints regions D 1 , D 2 and D 3 . Here we only sketch the proof of backward construction and the rest follows as earlier.
I. Backward construction (continuous and shock solutions): Define Then h ± are isomorphisms and by R-H condition across the interface, using h ± it follows as in earlier case (i) There are no forward rarefaction from the interface.
(ii) Continuous and shock solutions are constructed.
(iii) Using this and L 1 -contractivity for solutions with discrete set of discontinuities of the solution, one can glue them to obtain a solution in D 2 ∪ D 3 , where D 2 and D 3 are described earlier.

Appendix
Proof of lemma 2.1. From the hypothesis on {f k } and {g k }, it follows that lim k→∞ (f * k , g * k ) → (f * , g * ) in C 1 loc (R). Since {u 0,k } is uniformly bounded in L ∞ (R) and converges to u 0 in weak * − L ∞ (R). Hence {v 0,k } converges to v 0 uniformly on compact subsets of R and having uniformly Lipschitz constant. Hence {v k } are having uniformly Lipschitz constant. Hence by Arzela-Ascoli theorem, there exists a subsequence still denoted by {v k } converges to w in C 0 loc (R × [0, ∞)). Claim: lim k→∞ ch k (x, t) ⊂ ch(x, t), v = w.
For γ k ∈ ch k (x, t), then from lemma 4.2 of [5] (page 38), dγ k dθ is uniformly bounded and hence for subsequence {γ k i } converges toγ ∈ ch(x, t). In order to prove the claim we need to show thatγ ∈ ch(x, t).
If γ ∈ c(x, t) then v k (x, t) = Γ v 0,k ,γ k (x, t) ≤ Γ v 0,k,γ (x, t). Letting k = k i and k i → ∞ to obtain w(x, t) = lim Henceγ ∈ ch(x, t) and w(x, t) = inf γ∈c(x,t) This proves the claim. Hence by uniqueness of the limit, it follows that lim k→∞ v k = v in C 0 loc (R) and lim k→∞ ch k (x, t) ⊂ ch(x, t). Since Lipschitz constant of {v k } are uniformly bounded, hence for any ϕ ∈