On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms

We study one class of nonlinear fluid dynamic models with impulse source terms. The model consists of a system of two hyperbolic conservation laws: a nonlinear conservation law for the goods density and a linear evolution equation for the processing rate. We consider the case when influx-rates in the second equation take the form of impulse functions. Using the vanishing viscosity method and the so-called principle of fictitious controls, we show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems.


Introduction
The main goal of this paper is to approximate entropy solutions to a Cauchy problem for the system of nonlinear balance laws with an impulse source term.Conservation laws, taking the form of hyperbolic partial differential equations, appear in a variety of applications that offer control or identification of parameters, including the control of traffic and water flows, the modeling of supply chains, gas pipelines, blood flows, and so forth.The analysis of conservation laws is a very active research area.The main difficulty in dealing with them is the fact that the solution of such systems may develop discontinuities (after a finite time), that propagate in time even for smooth initial and boundary conditions (see [1][2][3]).Usually such solutions can be formed by the so-called rarefaction or shock waves.Therefore, it makes a sense to consider a more flexible notion of solutions, which are physically meaningful and whose admissibility issue is related to the notions of entropy and energy.
We analyze the following initial value problem for the system of nonlinear conservation laws (1) Throughout this paper we suppose that the structure of the source term u(t, x) is prescribed, namely, where the functions {u i ∈ L 2 (0, T)} N i=1 can play the role of control factors, and δ τi denote the Dirac measures located at the points τ i .
In the recent applications of the model (1) to the supply chain problem [4], ρ = ρ(t, x) represents the density of objects or the concentration of a physical quantity processed by the supply chain (modeled by a real line R), and μ = μ(t, x) is the processing rate.However, to the best knowledge of authors, the existence and uniqueness of entropy solutions to the problems of conservation laws with impulse controls is an open problem even for the simplest situation.Thus our prime interest is to discuss the approximation approach to the construction of entropy solutions for the above problem.To this end, we apply the vanishing viscosity method and the so-called principle of fictitious controls.We prove that entropy solutions to the Cauchy problem (1)-( 2) can be approximated by optimal solutions of special optimization problems.Namely, we introduce the following penalized optimization problem subject to the constraints where v = v(t, x) is a fictitious control.We carry out the analysis of this problem and show that under some additional assumptions every cluster pair (v * , ρ * ) (in an appropriate topology) of the sequence {(v ε 0 , ρ ε 0 ) ∈ Ξ ε } ε>0 of optimal solutions to the penalized problem (3)-( 4) is an entropy solution (u * , ρ * , μ * ) to the Cauchy problem (1).

Notation and Preliminaries
Let a and b be two fixed constants such that −∞ < a < b < +∞.For a given T > 0 we set Ω T = (0, T) × R and Ω = (0, T) × (a, b).Let L p loc (Ω T ), with 1 ≤ p ≤ ∞, be the locally convex space of all measurable functions g : Ω T → R such that g| (0,T)×K ∈ L p ((0, T) × K) for all compact sets K ⊂ R.
Let M(R) be the set of all Radon measures on R, that is, μ ∈ M(R) if μ is a countably additive set function defined on the Borel subsets of R such that μ is finite on every compact subset of R. We say that a sequence of Radon measures A subset M of M(R) is called to be bounded if for every compact set K ⊂ R we have where |μ| denotes the total variation of μ.The following compactness result for measures is well known.According to the Riesz theory, every Radon measure μ on R can be identified with an element of the dual space (C 0 (R)) , that is, μ is a linear form on C 0 (R) and for every compact set K ⊂ R there exists a constant C > 0 depending only on K and μ such that As an example of a Radon measure on R, we consider the following one.Let Let δ c be the Dirac measure located at the point c ∈ R, that is, this measure is defined as follows Since (9) for every continuous function with compact support ϕ ∈ C 0 (R), it follows that the linear form According to the Radon-Nikodym theorem, if O |D f | < +∞ then the distribution D f is a measure and there exist a function f ∈ L 1 (O) and a measure D s f , singular with respect to the one-dimensional Lebesgue measure L O restricted to O, such that Under the norm BV(O) is a Banach space.The following compactness result for BV-functions is well known.
Proposition 3. The uniformly bounded sets in BV-norm are relatively compact in weakly converges to some f ∈ BV(O), and we write f k f if and only if the two following conditions hold: In the following proposition we give a compactness result related to this convergence, together with the lower semicontinuity property (see [5]).
k=1 be a sequence in BV(O) strongly converging to some f in L 1 (O) and satisfying

Statement of Problem and Main Motivation
Let {τ k } N k=1 ⊂ R be a given finite family of points such that a < τ 1 < • • • < τ N < b.We focus on the following fluid dynamic model, expressed by the nonlinear inhomogeneous hyperbolic conservation laws: where the source term is subjected to the following constraints: Here u i ∈ L 2 (0, T) are some external distributed sources located at the corresponding points τ i ∈ (a, b), ρ 0 , μ 0 ∈ BV(R) ∩ L ∞ (R) are data functions, and We note that a particular case of the initial value problem (14)-( 16) is a perturbed model for the supply chain (represented by a real line), where ρ = ρ(t, x) represents the density of objects or the concentration of a physical quantity processed by the supply chain (modeled by a real line R), μ = μ(t, x) is the processing rate, and u = u(t, x) is a source term associated with an influx-rate.
In order to give a precise description of the set of admissible source terms to the Cauchy problem ( 14)-( 18), we note that for any function u(t, x) of type (17), we have Hence, it is natural to define the following class: Definition 6.Let u ∈ U ad be a fixed source term.We say that a vector value function Here and the symbol denotes the tensor product The characteristic feature of the initial value problem ( 14)-( 16) is that even for arbitrary smooth functions ρ 0 , μ 0 , and smooth external sources u k , k = 1, . . ., N, a weak solution Y (t, x) = (ρ(t, x), μ(t, x)) to ( 14)-( 16), is, in general, not unique (see [2,3]).Hence, in order to select the "physically" relevant solution, some additional conditions must be imposed.Following [2,3,6], we can introduce the entropy-admissibility condition, coming from physical considerations.Definition 7. A C 1 -function η : R 2 → R is an entropy for the system ( 14)-( 15), if it is convex and there exists a C 1 -function q : R 2 → R such that The function q : R 2 → R is said an entropy flux for η.
Remark 8.Note that the C 1 -functions η, q in Definition 7 form a special family of convex entropy pairs.However, any convex function η defined on an open set is locally Lipschitz, and therefore Dη is defined almost everywhere.This allows us to call a C 0 -function η an entropy, if there exists a sequence of C 1 -entropies {η ν : R 2 → R} ∞ ν=1 converging to η locally uniformly as ν → ∞.Moreover a C 0 -function q is a corresponding entropy flux, if there exists a sequence {q ν } ∞ ν=1 of C 1 -entropy fluxes of η ν converging to q locally uniformly.
As a result, an entropy solution of ( 14)-( 16) for a given u ∈ U ad can be defined as follows.
Definition 9. Let u ∈ U ad be a given source term with prescribed location a < τ 16) is said entropy admissible if for any constants k, l ∈ R the entropy inequalities hold true for all positive functions ϕ, ψ ∈ C ∞ 0 (Ω T ), provided that Remark 10.Note that the existence and differential properties of entropy solutions to the Cauchy problem ( 14)-( 16) with impulse influx-rate (17) in the sense of Definition 9 are unknown in general.To the best knowledge of authors, the problems ( 14)-( 17) with measure data in the right hand side are not covered by the classical theory of nonlinear hyperbolic conservation laws.Moreover, we cannot assert that entropy admissible solutions (ρ(u), μ(u)) to the above problem are elements of the class which is a natural functional space for the scalar hyperbolic conservation laws (see [2,7,8]).Usually these properties essentially depend not only on the flux function f (μ, ρ), but also on the properties of the admissible source terms u(t, x), which typically, in contrast to our case, are supposed to be bounded in L ∞ (Ω T ) and closed in L 1 (Ω T ) (see [8]).
Taking this motivation into account, it is reasonable to introduce the following concept.Definition 11.Let u ∈ U ad be a given source term.We say that a vector value function ] is an approximately entropy solution to the Cauchy problem ( 14)-( 16) in a domain (0, a weak solution in the sense of Definition 6 and there exists a sequence (B2) for any constants k, l ∈ R and for all positive concave functions ϕ ∈ C ∞ 0 ((0, T)×O) the entropy inequalities it hold true for every ε > 0 with ν l (ρ) :

A Perturbation Framework
As was mentioned above the existence and uniqueness of entropy solutions for nonlinear hyperbolic conservation laws ( 14)-( 16) with source terms (17), where u i ∈ L 2 (0, T) for all i = 1, . . ., N, and with initial distributions ρ 0 , μ 0 ∈ BV(R) ∩ L ∞ (R), is not covered by the classical theory.In view of this, we apply in this section the scheme of "vanishing viscosity" method and the principle of fictitious controls.
To begin with, we impose the following assumptions on the flux function: (A1) the function f 1 : R → R is locally Lipschitz, that is, f 1 (0) = 0, and f 2 : R → R is a piecewise linear mapping.
Remark 12.As was shown in recent works [4,[9][10][11], a flux function of the fluid dynamic model for supply chains is the following: Hence, the fulfilment of Hypothesis (A1) is obvious in this case.
Let ε be a small positive parameter associated with a viscosity coefficient.Then instead of the fluid dynamic system ( 14)-( 16), we focus on the following singular perturbed system of nonlinear PDEs: subjected to the constraints where v = v(t, x) is a fictitious control.By V ad we denote the set of all fictitious controls satisfying conditions (37).Since ρ 0 , , it is natural to assume that there is a compact interval I ⊂ R such that ρ 0 = 0 and μ 0 = 0 almost everywhere in R \ I. Then taking a sufficiently big open bounded interval O ⊂ R including the interval I, we can suppose that the rate processing μ ε and the density ρ ε vanish at the ends of O.As a result, we can introduce the following boundary conditions into the model (32)-(37): Since by the initial assumptions the influx-rate u = N k=1 u k (t)δ k and the fictitious control v belong to the space of measure data L 2 (0, T; M(O)), we make the notion of solution for the problem (32)-(38) precise.To this end, we give the following theorem which plays an important role in the study of partial differential equations (see [12]).
Theorem 13.Let one defines the Banach spaces: equipped with the norm of the graph.Then, the following properties hold true: (1) the embeddings where, for denotes the space of measurable functions on [0, T]×O such that y(t, •) ∈ X for any t ∈ [0, T] and such that the map t ∈ [0, T] → y(t, •) ∈ X is continuous; Then the following density result holds: for any δ > 0 there exists Further we note that by the Friedrichs inequality, we have Hence the bilinear form O yϕ dx is bounded on H 1 0 (O).Moreover, this form is skew-symmetric by the identity Journal of Control Science and Engineering which remains valid for all y, ϕ ∈ H 1 0 (O) by continuity.Then, we come to the following classical result (see [12,13]).Theorem 14. Assume that μ 0 ∈ BV(O) ∩ L ∞ (O) and Hypothesis (A1) holds true.Then for every ε > 0 the initialboundary value problem (32)-(38) admits a unique solution (ρ ε , μ ε ) ∈ W × W satisfying the integral identities: with a priori estimates where C > 0 is a constant independent of ε and g, g ∈ W 1,2 ((0, T) × O) are such that g| ∂O = 0, g| ∂O = 0, g(0, •) = ρ 0 , and g(0, •) = μ 0 in O (the so-called compatibility condition).

Note that in this case
by the embedding (41), and the terms in the right-hand sides of ( 46)-( 47) are well defined, because the classical Sobolev Embedding Theorem.Moreover, in the one-dimensional case every Radon measure ν ∈ M(O) can be identified with an element of As a result, the integral identity (47) with a source term can be rewritten as follows: In conclusion of this section we state the following entropy property of the weak solutions to the initialboundary value problem (32)-(38).

a sequence of corresponding weak solutions to the initial boundary value problem (32)-(38)
where the small parameter ε > 0 varies in a strictly decreasing sequence of positive numbers converging to 0. Let {v ε ∈ L 2 (0, T; M(R))} ε>0 be a bounded sequence of fictitious controls.Assume that supposition (A1) holds true.Then for every ε > 0, k, l ∈ R, and for all positive concave functions ϕ ∈ C ∞ 0 ((0, T) × O), each of the pairs ((ρ ε , μ ε )) satisfies the following integral inequalities: Proof.Let E = E(ρ) ∈ C 2 (R) be any convex function.We multiply (32) by E (ρ).Then the equalities imply the following relation By the initial assumptions, for every ε > 0 the functions ρ ε ∈ W can be zero-extended to the domain Ω T = (0, T)×R.Now let us multiply equality (58) by a test function ϕ ∈ C ∞ 0 (Ω T ) and integrate it over Ω T .Using the integration by parts and the fact that ε > 0 and E (ρ ε ) ≥ 0 a.e. in Ω T , we transfer all derivatives to the test function ϕ: Since ρ ε ∈ L 2 (0, T; H 1 0 (O)) for all ε > 0 and H 1 0 (O) → C(O) by the classical Sobolev Embedding Theorem, it follows that the following term is well defined: (60) Further we use the well-known trick.Let {E m } m∈N be a sequence of C 2 -functions approximating the function ξ → |ξ−k| uniformly on R. Substitute E = E m (ρ) in the inequality (59) and pass to the limit as m → ∞.Note that we can choose E m in such way that E m is bounded and it immediately leads us to the entropy inequality (55) from (59).The verification of inequality (56) can be done by similar arguments.
Let Ξ ε be the set of all admissible solutions to the perturbed problem (62)-(63).As follows from Theorem 13, for every ε > 0, Ξ ε is a nonempty subset of the space: Remark 17.We note that the cost functional (63) is well defined on Ξ ε for every ε > 0. Indeed, let (v ε , ρ ε ) be any representative of Ξ ε .By supposition (A1), we have that f 2 : R → R is a piecewise linear mapping and , we come to the required conclusion.
We define the τ-topology on Y as follows: τ is the product of the weak- * topology of L 2 (0, T; M(O)) and the topology of norm in L 2 ((0, T) × O).Then we have the following topological properties of the set Ξ ε of admissible solutions to the perturbed optimization problem (62)-(63).
Proof.For a fixed ε > 0 let (u ε , v ε ) ∈ U ad ×V ad be an arbitrary pair of source terms.Then Theorem 14 implies the existence of a unique pair (ρ ε , μ ε ) such that ρ ε = ρ ε (v ε ) and μ ε = μ ε (u ε ) are the corresponding weak solutions to the initial boundary value problem (32)-( 34), (38).Since and To establish the τ-closedness of Ξ ε , we fix an arbitrary τconverging sequence of admissible solutions to the perturbed problem (32)-( 38) and ( 62) Journal of Control Science and Engineering in view of the a priori estimate (48), it is easy to see that the L 2 ((0, T) × O-limit function ρ ε * belongs to the space W and satisfies conditions: This enables us to pass to the limit in the integral identity (46) as k → ∞ with ρ ε = ρ ε k and v = v ε k , and eo ipso to show that the limit function ρ ε * is a weak solution to the parabolic problem (32), (34) 1 , (38) 1 .
In conclusion of this section, we prove that the penalized problem (32)-( 38), (62) has a nonempty set of optimal solutions.Theorem 19.Assume that supposition (A1) holds true.Then for every ε > 0 and u ε ∈ U ad there exists at least one pair that is, the problem (32)-( 38), (62) is solvable.
Proof.Since Ξ ε / = ∅ and the cost functional I ε is bounded below on Ξ ε , it follows that there exists a sequence is a minimizing sequence for the problem (32)-( 38), (62).
To begin with, we show that for any λ > 0 the set is bounded in L 2 (0, T; M(O)) × W . Indeed, as follows from inequality (68), the sequence of fictitious controls {v ε k } k∈N is bounded in L 2 (0, T; M(O)).Hence, we may assume that there exists an element Then having used the a priori estimate (48), we see that } k∈N form a uniformly bounded sequence in W . Hence, we may again assume that, up to a subsequence, there exists an element ρ ε 0 ∈ W such that ρ ε k ρ ε 0 weakly in W and strongly in L 2 ((0, T) × O).As a result, (v ε 0 , ρ ε 0 ) ∈ Ξ ε by Lemma 18.

Approximation Properties of the Perturbed Optimization Problem
The aim of this section is to study the asymptotic behavior of the optimal solutions to the penalized optimization problem (32)-( 38), (62) as the small parameter ε tends to zero.To begin with, we note that for every ε > 0 the set of admissible solutions Ξ ε is embedded in the topological space (Y 1 , σ), where and σ is the product of the weak- * topology of L 2 (0, T; M(O)) and the strong topology of L 2 (0, T; H −1 (O)).So, we can take σ as the main topology for the asymptotic analysis.

Conclusion
In this article, we have proposed the approximation of entropy solutions for the system of two hyperbolic conservation laws ( 14)-( 16) with impulse source terms.We have considered the case when influx-rates in the second equation (15) take the form of impulse functions (17)-(18).Since the existence of entropy solutions for Cauchy problem ( 14)-( 18) is not covered by the classical theory, we combine the vanishing viscosity method and the so-called principle of fictitious controls in order to show that entropy solutions to the original Cauchy problem can be approximated by optimal solutions of special optimization problems.The main result is given by Theorem 21, where we conclude that every σcluster pair (v * , ρ * ) ∈ Y 1 of the sequence {(v ε 0 , ρ ε 0 ) ∈ Ξ ε } ε>0 of optimal solutions to the penalized problem (32)-( 38), ( 62) is an approximately entropy solution (u * , ρ * , μ * ) to the Cauchy problem ( 14)-( 16).

Proposition 1 .
Let {μ k } k∈N be a bounded sequence of Radon measures on R. Then there exist a subsequence {μ kj } j∈N and a Radon measure μ ∈ M(R) such that μ kj * μ.
Definition 2. A function f ∈ L 1 (O) is said to have a bounded variation in O if the derivative D f exists in the sense of distributions and belongs to the class of Radon measures with bounded total variation, that is, O |D f | < +∞.By BV(O) we denote the space of all functions in L 1 (O) with bounded variation.