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

Abstract

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.

1. 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 [1–3]). 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 laws𝜌_𝑡+(𝑓(𝜇𝜇,𝜌))_𝑥=0,(𝑡,𝑥)∈Ω_𝑇=(0,𝑇)×ℝ,_𝑡𝜌(0,𝑥)=−𝜌₀𝜇_𝑥=𝑢(𝑡,𝑥),(𝑡,𝑥)∈Ω_𝑇,(𝑥),𝜇(0,𝑥)=𝜇₀(𝑥),𝑥∈ℝ.(1) Throughout this paper we suppose that the structure of the source term 𝑢(𝑡,𝑥) is prescribed, namely,𝑢=𝑢(𝑡,𝑥)=^𝑁_𝑖=1with−∞<𝑎<𝑢𝜏_𝑖1(𝑡)𝛿_𝜏𝑖(𝑥),<⋯<𝜏_𝑁<𝑏<+∞,(2) where the functions {𝑢_𝑖∈𝐿²(0,𝑇)}^𝑁𝑖=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 problem𝐼_𝜀(𝑣^𝜀,𝜌^𝜀)=‖𝑣^𝜀‖^2𝐿_{2(0,𝑇;ℳ(𝒪))}+𝜀⁻¹‖‖𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀‖‖_{𝐿2(0,𝑇;𝐻−1(𝒪))}⟶inf(3) subject to the constraints𝜌𝜇^𝜀𝑡_𝜀𝑡(𝑡,𝑥)−𝜀𝜌^𝜀𝑥𝑥(𝑡,𝑥)+𝑓₁(𝜌^𝜀(𝑡,𝑥))_𝑥=𝑣(𝑡,𝑥),(𝑡,𝑥)∈Ω_𝑇,𝜌_𝜀(𝑡,𝑥)−𝜀𝜇^𝜀𝑥𝑥(𝑡,𝑥)−𝜇^𝜀𝑥(𝑡,𝑥)=𝑢(𝑡,𝑥),(𝑡,𝑥)∈Ω_𝑇,𝑢(𝑡(,0𝑥,)𝑥)==𝑢₁𝜌₀(𝑥),𝜇^𝜀(0,𝑥)=𝜇₀(𝑥),𝑥∈ℝ,𝑢(𝑡)𝛿_𝜏1(𝑥)+𝑢₂(𝑡)𝛿_𝜏2(𝑥)+⋯+𝑢_𝑁(𝑡)𝛿_𝜏𝑁(𝑥),_𝑖∈𝑣𝐿∈²𝐿(0₂,𝑇),∀𝑖={1,…,𝑁},𝜌(0,𝑇;ℳ(ℝ)),_𝜀(𝑡,𝑥)=0,𝜇^𝜀(𝑡,𝑥)=0,on(0,𝑇)×𝜕𝒪,(4) 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^𝜀0,𝜌^𝜀0)∈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).

2. 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^𝑝loc(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 {𝜇_𝑘}_𝑘∈ℕ converges weakly-∗ to a measure 𝜇∈ℳ(ℝ) (in symbols 𝜇⇀_𝑘∗𝜇) iflim_𝑘→∞_ℝ𝜑𝑑𝜇_𝑘=_ℝ𝜑𝑑𝜇,∀𝜑∈𝐶₀(ℝ).(5)

A subset 𝔐 of ℳ(ℝ) is called to be bounded if for every compact set 𝐾⊂ℝ we havesup_𝜇∈𝔐||𝜇||(𝐾)<+∞,(6) where |𝜇| denotes the total variation of μ. The following compactness result for measures is well known.

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.

According to the Riesz theory, every Radon measure μ on ℝ can be identified with an element of the dual space (𝐶₀(ℝ))^, that is, μ is a linear form on 𝐶₀(ℝ) and for every compact set 𝐾⊂ℝ there exists a constant 𝐶>0 depending only on 𝐾 and μ such that||⟨𝜇,𝑓⟩||≤𝐶‖𝑓‖_𝐶(𝐾),∀𝑓∈𝐶₀(ℝ)withsupp𝑓⊆𝐾.(7) As an example of a Radon measure on ℝ, we consider the following one. Let {𝑎_𝑘}_𝑘∈ℕ and {𝑏_𝑘}_𝑘∈ℕ be two sequences in ℝ such that ∑^∞𝑘=1|𝑎_𝑘|≤𝐂<+∞. Let 𝛿_𝑐 be the Dirac measure located at the point 𝑐∈ℝ, that is, this measure is defined as follows⟨𝛿_𝑐,𝜑⟩=_ℝ𝛿_𝑐(𝑥)𝜑(𝑥)𝑑𝑥∶=𝜑(𝑐),∀𝜑∈𝐶₀(ℝ).(8) Since|||||_∞_𝑘=1𝑎_𝑘𝜑𝑏_𝑘|||||≤^∞_𝑘=1||𝑎_𝑘||‖𝜑‖_𝐶(ℝ)≤𝐂‖𝜑‖_𝐶(ℝ)(9) for every continuous function with compact support 𝜑∈𝐶₀(ℝ), it follows that the linear form𝜇^∗=^∞_𝑘=1𝑎_𝑘𝛿_𝑏𝑘(10) is continuous on 𝐶₀(ℝ). Hence 𝜇^∗ is an element of the space of Radon measures ℳ(ℝ).

Let 𝒪 be a bounded open subset of ℝ. Let 𝑓∶𝒪→ℝ be an element of 𝐿¹(𝒪). Define _𝒪=s||𝐷up𝑓||_𝒪𝑓𝜑^𝑑𝑥∶𝜑∈𝐶¹⁰(𝒪),||𝜑(𝑥)||≤1for𝑥∈𝒪.(11) According to the Radon-Nikodym theorem, if ∫_𝒪|𝐷𝑓|<+∞ then the distribution 𝐷𝑓 is a measure and there exist a function 𝑓^∈𝐿¹(𝒪) and a measure 𝐷_𝑠𝑓, singular with respect to the one-dimensional Lebesgue measure ℒ⌊𝒪 restricted to 𝒪, such that 𝐷𝑓=𝑓^ℒ⌊𝒪+𝐷_𝑠𝑓.(12)

Definition 2. A function 𝑓∈𝐿¹(𝒪) is said to have a bounded variation in 𝒪 if the derivative 𝐷𝑓 exists in the sense of distributions and belongs to the class of Radon measures with bounded total variation, that is, ∫_𝒪|𝐷𝑓|<+∞. By BV(𝒪) we denote the space of all functions in 𝐿¹(𝒪) with bounded variation.

Under the norm ‖𝑓‖_BV(𝒪)=‖𝑓‖_𝐿¹_(𝒪)+_𝒪||𝐷𝑓||,(13)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 𝐿¹(𝒪), that is, if {𝑓_𝑘}^∞𝑘=1⊂BV(𝒪) and sup_𝑘∈ℕ‖𝑓_𝑘‖_BV(𝒪)<+∞, then there exists a subsequence of {𝑓_𝑘}^∞𝑘=1 strongly converging in 𝐿¹(𝒪) to some 𝑓∈BV(𝒪).

Definition 4. A sequence {𝑓_𝑘}^∞𝑘=1⊂BV(𝒪) weakly converges to some 𝑓∈BV(𝒪), and we write 𝑓_𝑘⇀𝑓 if and only if the two following conditions hold: 𝑓_𝑘→𝑓 strongly in 𝐿¹(𝒪), and 𝐷𝑓_𝑘⇀𝐷𝑓 weakly* in ℳ(𝒪).

In the following proposition we give a compactness result related to this convergence, together with the lower semicontinuity property (see [5]).

Proposition 5. Let {𝑓_𝑘}^∞𝑘=1 be a sequence in BV(𝒪) strongly converging to some 𝑓 in 𝐿¹(𝒪) and satisfying sup_𝑘∈ℕ∫_𝒪|𝐷𝑓_𝑘|<+∞. Then (i)𝑓∈BV(𝒪) and ∫_𝒪|𝐷𝑓|≤liminf_𝑘→∞∫_𝒪|𝐷𝑓_𝑘|; (ii)𝑓_𝑘⇀𝑓 in BV(𝒪).

3. Statement of Problem and Main Motivation

Let {𝜏_𝑘}^𝑁𝑘=1⊂ℝ be a given finite family of points such that 𝑎<𝜏₁<⋯<𝜏_𝑁<𝑏. We focus on the following fluid dynamic model, expressed by the nonlinear inhomogeneous hyperbolic conservation laws:𝜌𝜇_𝑡+(𝑓(𝜇,𝜌))_𝑥=0,(𝑡,𝑥)∈Ω_𝑇,(14)_𝑡𝜌(0,𝑥)=−𝜌₀𝜇_𝑥=𝑢(𝑡,𝑥),(𝑡,𝑥)∈Ω_𝑇,(15)(𝑥),𝜇(0,𝑥)=𝜇₀(𝑥),𝑥∈ℝ,(16) where the source term is subjected to the following constraints:𝑢(𝑡,𝑥)=𝑢₁𝑢(𝑡)𝛿_𝜏1(𝑥)+𝑢₂(𝑡)𝛿_𝜏2(𝑥)+⋯+𝑢_𝑁(𝑡)𝛿_𝜏𝑁(𝑥),(17)_𝑖∈𝐿²(0,𝑇),∀𝑖={1,…,𝑁}.(18) Here 𝑢_𝑖∈𝐿²(0,𝑇) are some external distributed sources located at the corresponding points 𝜏_𝑖∈(𝑎,𝑏), 𝜌₀,𝜇₀∈BV(ℝ)∩𝐿^∞(ℝ) are data functions, and 𝑓=𝑓(𝜇,𝜌)=𝑓₁(𝜌)+𝑓₂(𝜇) is a flux function.

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^𝑇0‖=𝑢(𝑡,⋅)‖^2ℳ(ℝ)𝑑𝑡_𝑇0⎛⎜⎜⎜⎝sup_{‖𝜑‖𝐶(ℝ)=1𝜑∈𝐶0(𝑅)}⟨𝑢(𝑡,⋅),𝜑(⋅)⟩_{ℳ(ℝ),𝐶0(ℝ)}⎞⎟⎟⎟⎠²=𝑑𝑡_𝑇0⎛⎜⎜⎜⎝sup_{‖𝜑‖𝐶(ℝ)=1𝜑∈𝐶0𝑁(𝑅)}_𝑘=1≤𝑢_𝑘(𝑡)_ℝ𝛿_𝜏𝑘(𝑥)𝜑(𝑥)𝑑𝑥⎞⎟⎟⎟⎠²𝑑𝑡_𝑇0^𝑁_𝑘=1≤𝑁_𝑁𝑢_𝑘(𝑡)²𝑑𝑡_𝑘=1‖‖𝑢_𝑘‖‖^2𝐿_2(0,𝑇).(19)

Hence, it is natural to define the following class:𝒰_ad=𝑢sa∈tis𝐿fy²((02,5𝑇);-ℳ(26(ℝ))f)o∣r𝑢so=m𝑢e(𝑎𝑡,<𝑥)𝜏₁<…<𝜏_𝑁<𝑏.(20)

Definition 6. Let 𝑢∈𝒰_ad be a fixed source term. We say that a vector value function 𝑌=^𝜌𝜇∈[𝐿²(0,𝑇;𝐿^2loc(ℝ))]² is a weak solution to (14)–(16) if the identities ^𝑇0_ℝ+𝑌⊙𝜕𝜕𝜑𝑡+𝐹(𝑌)⊙𝜕𝜕𝜑𝑥𝑑𝑥𝑑𝑡_𝑇0𝑁_𝑘=1lim_𝑡→0+𝑈_𝑘(𝑡)⊙𝜑𝜏_𝑘𝑑𝑡=0,1𝑡^𝑡0||||_ℝ𝑌(𝑡,𝑥)⊙𝜓(𝑥)𝑑𝑥−_ℝ𝑌₀⊙𝜓(𝑥)𝑑𝑥||||_ℝ2𝑑𝑡=0(21) hold true for all 𝐶^∞0-functions 𝜑∶[0,𝑇]×ℝ→ℝ² and 𝜓∶ℝ→ℝ² with compact supports in (0,𝑇)×ℝ and ℝ, respectively. Here 𝑌₀=𝜌⁰𝜇₀,𝑈^𝑘(𝑡)=𝑢_𝑘0(𝑡),𝐹(𝑌)=𝑓(−𝜇𝜇,𝜌),(22) and the symbol ⊙ denotes the tensor product ^𝑎1𝑎₂⊙^𝑏1_𝑏2=^{𝑎1𝑏1𝑎}₂^𝑏₂.

The characteristic feature of the initial value problem (14)–(16) is that even for arbitrary smooth functions 𝜌₀, 𝜇₀, 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 𝐶¹-function 𝜂∶ℝ²→ℝ is an entropy for the system (14)-(15), if it is convex and there exists a 𝐶¹-function 𝑞∶ℝ²→ℝ such that 𝐷𝜂(𝑣)⋅𝐷𝐹(𝑣)=𝐷𝑞(𝑣),∀𝑣∈ℝ².(23) The function 𝑞∶ℝ²→ℝ is said an entropy flux for 𝜂. The pair (𝜂,𝑞) is said an entropy flux pair for the system (14)-(15).

Remark 8. Note that the 𝐶¹-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 𝐶⁰-function 𝜂 an entropy, if there exists a sequence of 𝐶¹-entropies {𝜂^𝜈∶ℝ²→ℝ}^∞𝜈=1 converging to 𝜂 locally uniformly as 𝜈→∞. Moreover a 𝐶⁰-function 𝑞 is a corresponding entropy flux, if there exists a sequence {𝑞^𝜈}^∞𝜈=1 of 𝐶¹-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<𝜏₁<⋯<𝜏_𝑁<𝑏. A weak solution 𝑌=^𝜌𝜇∶[0,𝑇]×ℝ→ℝ² to the Cauchy problem (14)–(16) is said entropy admissible if for any constants 𝑘,𝑙∈ℝ the entropy inequalities ^𝑇0_ℝ𝜈_𝑙−(𝜌)𝜓_𝑡+𝑔_𝑙(𝜌)𝜓_𝑥𝑑𝑥𝑑𝑡_𝑇0_ℝsign(𝜌−𝑙)𝑓₂(𝜇)_𝑥𝜓𝑑𝑥𝑑𝑡≥0,_𝑇0_ℝ𝜈_𝑘+(𝜇)𝜑_𝑡−𝑞_𝑘(𝜇)𝜑_𝑥𝑑𝑥𝑑𝑡_𝑇_𝑖=1^𝑇0sign𝜇𝑡,𝜏_𝑖−𝑘𝑢_𝑖(𝑡)𝜑𝑡,𝜏_𝑖𝑑𝑡≥0(24) hold true for all positive functions 𝜑,𝜓∈𝐶^∞0(Ω_𝑇), provided that 𝜈_𝑘(𝜇)∶𝑔=||𝜇−𝑘||,𝑞_𝑘(𝜇)∶=(𝜇−𝑘)sign(𝜇−𝑘),_𝑙(𝜌)∶=𝑓₁(𝜌)−𝑓₁(𝑙)sign(𝜌−𝑙).(25)

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 𝐶([0,𝑇];𝐿¹(𝑎,𝑏))∩𝐿^∞(Ω)∩𝐿^∞(0,𝑇;BV(𝑎,𝑏))²(26) 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 𝐿¹(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 𝑌=^𝜌𝜇∈[𝐿²(0,𝑇;𝐿^2loc(ℝ))] is an approximately entropy solution to the Cauchy problem (14)–(16) in a domain (0,𝑇)×𝒪, if 𝑌=^𝜌𝜇∶[0,𝑇]×ℝ→ℝ² is a weak solution in the sense of Definition 6 and there exists a sequence {𝑌^𝜀=^𝜌𝜀_𝜇𝜀}_𝜀>0⊂[𝐿²(0,𝑇;𝐿²(𝒪))]² such that (B1)𝜌^𝜀⇀𝜌 and 𝜇^𝜀⇀𝜇 in 𝐿²(0,𝑇;𝐿²(𝒪)) as 𝜀→0; (B2)for any constants 𝑘,𝑙∈ℝ and for all positive concave functions 𝜑∈𝐶^∞0((0,𝑇)×𝒪) the entropy inequalities ^𝑇0_𝒪𝜈_𝑙+(𝜌^𝜀)𝜑_𝑡+𝑔_𝑙(𝜌^𝜀)𝜑_𝑥𝑑𝑥𝑑𝑡_𝑇0×sig𝑓n(𝜌^𝜀(𝑡,⋅)−𝑙)₂(𝜇^𝜀(𝑡,⋅)_𝑥,𝜑(𝑡,⋅)_{ℳ(𝒪),𝐶0(𝒪)}𝑑𝑡≥0,(27)_𝑇0_𝒪𝜈_𝑘+(𝜇^𝜀)𝜑_𝑡−𝑞_𝑘(𝜇^𝜀)𝜑_𝑥𝑑𝑥𝑑𝑡_𝑁_𝑖=1^𝑇0sign𝜇^𝜀𝑡,𝜏_𝑖−𝑘𝑢_𝑖(𝑡)𝜑𝑡,𝜏_𝑖𝑑𝑡≥0(28) it hold true for every 𝜀>0 with 𝜈_𝑙(𝜌)∶=|𝜌−𝑙|, 𝑔_𝑙(𝜌)∶=(𝑓₁(𝜌)−𝑓₁(𝑙))sign(𝜌−𝑙), 𝜈_𝑘(𝜇)∶=|𝜇−𝑘|, and 𝑞_𝑘(𝜇)∶=(𝜇−𝑘)sign(𝜇−𝑘).

4. 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∈𝐿²(0,𝑇) for all 𝑖=1,…,𝑁, and with initial distributions 𝜌₀,𝜇₀∈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: 𝑓(𝜇,𝜌)=𝑓₁(𝜌)+𝑓₂(𝜇),(29)

(A1) the function 𝑓₁∶ℝ→ℝ is locally Lipschitz, that is, ||𝑓₁𝜌₁−𝑓₁𝜌₂||≤𝐿_𝜌||𝜌₁−𝜌₂||,∀𝜌₁,𝜌₂∈−𝑀_𝜌,𝑀_𝜌,(30)𝑓₁(0)=0, and 𝑓^2∶ℝ↦ℝ is a piecewise linear mapping.

Remark 12. As was shown in recent works [4, 9–11], a flux function of the fluid dynamic model for supply chains is the following: 𝑓(𝜇,𝜌)=𝑓₁(𝜌)+𝑓₂(𝜇)=𝜌0,,iiff𝜌𝜌<≥𝜇𝜇+𝜇0,,iiff𝜌𝜌<≥𝜇𝜇.(31) 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:𝜌^𝜀𝑡(𝜇𝑡,𝑥)−𝜀𝜌^𝜀𝑥𝑥(𝑡,𝑥)+𝑓₁(𝜌^𝜀(𝑡,𝑥))_𝑥=𝑣(𝑡,𝑥),(𝑡,𝑥)∈Ω_𝑇,(32)_𝜀𝑡(𝑡𝜌,𝑥)−𝜀𝜇^𝜀𝑥𝑥(𝑡,𝑥)−𝜇^𝜀𝑥(𝑡,𝑥)=𝑢(𝑡,𝑥),(𝑡,𝑥)∈Ω_𝑇,(33)_𝜀(0,𝑥)=𝜌₀(𝑥),𝜇^𝜀(0,𝑥)=𝜇₀(𝑥),𝑥∈ℝ(34) subjected to the constraints𝑢(𝑡,𝑥)=𝑢₁𝑢(𝑡)𝛿_𝜏1(𝑥)+𝑢₂(𝑡)𝛿_𝜏2(𝑥)+⋯+𝑢_𝑁(𝑡)𝛿_𝜏𝑁(𝑥),(35)_𝑖∈𝐿𝑣²(∈0,𝐿𝑇₂),∀𝑖={1,…,𝑁},(36)(0,𝑇;ℳ(ℝ)),(37) where 𝑣=𝑣(𝑡,𝑥) is a fictitious control. By 𝒱_ad we denote the set of all fictitious controls satisfying conditions (37).

Since 𝜌₀,𝜇₀∈𝐿¹(ℝ^𝑁)∩𝐿^∞(ℝ^𝑁), it is natural to assume that there is a compact interval 𝐼⊂ℝ such that 𝜌₀=0 and 𝜇₀=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):𝜌^𝜀(𝑡,𝑥)=0,𝜇^𝜀(𝑡,𝑥)=0,on(0,𝑇)×𝜕𝒪.(38)

Since by the initial assumptions the influx-rate 𝑢=∑^𝑁𝑘=1𝑢_𝑘(𝑡)𝛿_𝑘 and the fictitious control 𝑣 belong to the space of measure data 𝐿²(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: 𝒲=𝑦∶𝑦∈𝐿²0,𝑇;𝐻¹⁰(𝒪),𝜕𝑦𝜕𝑡∈𝐿²0,𝑇;𝐻⁻¹(𝒪),(39)𝒲₁=𝑦∶𝑦∈𝐿²0,𝑇;𝐿²(𝒪),𝜕𝑦𝜕𝑡∈𝐿²0,𝑇;𝐻⁻¹(𝒪),(40) equipped with the norm of the graph. Then, the following properties hold true: (1)the embeddings 𝒲↪𝐿²(0,𝑇;𝐿²(𝒪)), 𝒲₁↪𝐿²(0,𝑇;𝐻⁻¹(𝒪)) are compact; (2)one has the embedding 𝒲↪𝐶[0,𝑇];𝐿²(𝒪),𝒲₁↪[0,𝑇];𝐻⁻¹(𝒪),(41) where, for 𝐗=𝐿²(𝒪) or 𝐗=𝐻⁻¹(𝒪), 𝐶([0,𝑇];𝐗) 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; (3) for any 𝑢,𝑣∈𝒲𝑑𝑑𝑡_𝒪=𝑢𝑢(𝑡,𝑥)𝑣(𝑡,𝑥)𝑑𝑥_(𝑡,⋅),𝑣(𝑡,⋅)_{𝐻−1(𝒪),𝐻10(𝒪)}+𝑣^(𝑡,⋅),𝑢(𝑡,⋅)_{𝐻−1(𝒪),𝐻10(𝒪)};(42)(4)let 𝑦∈𝐿²(0,𝑇;𝐻¹⁰(𝒪))∩𝐶([0,𝑇];𝐿²(𝒪)). Then the following density result holds: for any 𝛿>0 there exists Φ∈𝐶^∞([0,𝑇];𝐶^∞0(𝒪)), such that‖𝑦−Φ‖_{𝐶([0,𝑇];𝐿}²_(𝒪))≤𝛿,‖∇𝑦−∇Φ‖_𝐿²_{((0,𝑇)×𝒪)}≤𝛿.(43)

Further we note that by the Friedrichs inequality, we have ||||_𝕆𝑦𝜑^𝑑𝑥||||≤≤𝑐𝑐₁_𝕆𝑦²𝑑𝑥_𝕆||𝜑^||²𝑑𝑥_𝕆||𝑦^||²𝑑𝑥_𝕆||𝜑^||²𝑑𝑥,∀𝑦,𝜑∈𝐻¹⁰(𝒪).(44) Hence the bilinear form ∫_𝕆𝑦𝜑^𝑑𝑥 is bounded on 𝐻¹⁰(𝒪). Moreover, this form is skew-symmetric by the identity_𝕆𝑦𝜑^𝑑𝑥==−_𝕆(_𝕆𝑦𝜑)^𝑑𝑥−_𝕆𝑦^𝜑𝑑𝑥𝑦^𝜑𝑑𝑥,∀𝑦,𝜑∈𝐶^∞0(𝒪),(45) which remains valid for all 𝑦,𝜑∈𝐻¹⁰(𝒪) by continuity. Then, we come to the following classical result (see [12, 13]).

Theorem 14. Assume that 𝜇₀∈BV(𝒪)∩𝐿^∞(𝒪) and Hypothesis (A1) holds true. Then for every 𝜀>0 the initial-boundary value problem (32)–(38) admits a unique solution (𝜌^𝜀,𝜇^𝜀)∈𝒲×𝒲 satisfying the integral identities: ^𝑇0_𝒪𝜕𝜕𝜌𝑡^𝜀𝜓+𝜀𝜕𝜕𝜌𝑥^𝜀𝜕𝜕𝜓𝑥−𝑓¹(𝜌^𝜀)𝜕𝜓=𝜕𝑥𝑑𝑥𝑑𝑡_𝑇0⟨𝑣(𝑡,∀⋅)𝜓,𝜓∈(𝐿𝑡,₂⋅)⟩_𝐻⁻¹_(𝒪),𝐻¹⁰_(𝒪)𝑑𝑡,0,𝑇;𝐻¹⁰(𝒪),(46)_𝑇0_𝒪𝜕𝜕𝜇𝑡^𝜀𝜑+𝜀𝜕𝜕𝜇𝑥^𝜀𝜕𝜕𝜑𝑥+𝜇^𝜀𝜕𝜑=𝜕𝑥𝑑𝑥𝑑𝑡_𝑇0⟨𝑢(𝑡,∀⋅)𝜑,𝜑∈(𝐿𝑡,₂⋅)⟩_𝐻⁻¹_(𝒪),𝐻¹⁰_(𝒪)𝑑𝑡,0,𝑇;𝐻¹⁰(𝒪),(47) with a priori estimates ^𝑇0_𝒪||𝜌^𝜀≤𝐶(_𝑇0𝑡,𝑥)||²+𝜀||𝜌^𝜀𝑥(𝑡,𝑥)||²+||𝜌^𝜀𝑡(𝑡,𝑥)||²𝑑𝑥𝑑𝑡_𝒪||𝑔̂^𝑡+𝐶‖𝑣‖(𝑡,𝑥)||²_{2𝐿20,𝑇;𝐻−1(𝒪)}+||𝑔̂_𝑥(𝑡,𝑥)||²𝑑𝑥𝑑𝑡,(48)_𝑇0_𝒪||𝜇^𝜀≤𝐶_𝑇0(𝑡,𝑥)||²+𝜀||𝜇^𝜀𝑥(𝑡,𝑥)||²+||𝜇^𝜀𝑡(𝑡,𝑥)||²𝑑𝑥𝑑𝑡_𝒪||𝑔^𝑡+𝐶‖𝑢‖(𝑡,𝑥)||²_{2𝐿20,𝑇;𝐻−1(𝒪)}+||𝑔_𝑥(𝑡,𝑥)||²𝑑𝑥𝑑𝑡,(49)_𝑇0‖‖≤𝜌^𝜀𝑡𝐶(𝑡‖,⋅𝑔)̂‖‖‖^2𝐻_−1(𝒪)𝑑𝑡_{2𝑊−1,2((0,𝑇)×𝒪)}+𝐶‖𝑣‖^2𝐿_{20,𝑇;𝐻−1(𝒪)},(50)_𝑇0‖‖≤𝜇^𝜀𝑡𝐶(𝑡‖,𝑔⋅)‖‖‖^2𝐻_−1(𝒪)𝑑𝑡_{2𝑊−1,2((0,𝑇)×𝒪)}+𝐶‖𝑢‖^2𝐿_{2(0,𝑇;𝐻−1(𝒪))},(51) where 𝐶>0 is a constant independent of 𝜀 and 𝑔,𝑔̂∈𝑊^1,2((0,𝑇)×𝒪) are such that 𝑔|_𝜕𝒪=0, 𝑔̂|_𝜕𝒪=0, 𝑔̂(0,⋅)=𝜌₀, and 𝑔(0,⋅)=𝜇₀ in 𝒪 (the so-called compatibility condition).

Note that in this case 𝜌^𝜀∈𝐶[0,𝑇];𝐿²(𝒪),𝜇^𝜀∈𝐶[0,𝑇];𝐿²(𝒪)(52) by the embedding (41), and the terms in the right-hand sides of (46)-(47) are well defined, because 𝐻¹⁰(𝒪)⊂𝐶₀(𝒪) by the classical Sobolev Embedding Theorem. Moreover, in the one-dimensional case every Radon measure 𝜈∈ℳ(𝒪) can be identified with an element of 𝐻⁻¹(𝒪), that is, ℳ(𝒪)⊂𝐻⁻¹(𝒪). As a result, the integral identity (47) with a source term 𝑢(𝑡,𝑥)=^𝑁_𝑘=1𝑢_𝑘(𝑡)𝛿_𝑘(𝑥)(53) can be rewritten as follows:^𝑇0_𝒪𝜕𝜕𝜇𝑡^𝜀𝜑+𝜀𝜕𝜕𝜇𝑥^𝜀𝜕𝜕𝜑𝑥+𝜇^𝜀𝜕𝜑=𝜕𝑥𝑑𝑥𝑑𝑡_𝑁_𝑘=1^𝑇0𝑢_𝑘(𝑡)𝜑𝑡,𝜏_𝑘𝑑𝑡,∀𝜑∈𝐿²0,𝑇;𝐻¹⁰(𝒪).(54)

In conclusion of this section we state the following entropy property of the weak solutions to the initial-boundary value problem (32)–(38).

Lemma 15. Let 𝑢^∗=∑^𝑁𝑘=1𝑢^∗𝑘(𝑡)𝛿_𝜏𝑘∈𝒰_ad be a given source term with prescribed location 𝑎<𝜏₁<⋯<𝜏_𝑁<𝑏. Let {(𝜌^𝜀,𝜇^𝜀)}_𝜀>0 be 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∈𝐿²(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: ^𝑇0_𝒪𝜈_𝑙+(𝜌^𝜀)𝜑_𝑡+𝑔_𝑙(𝜌^𝜀)𝜑_𝑥𝑑𝑥𝑑𝑡_𝑇0⟨sign(𝜌^𝜀(𝑡,⋅)−𝑙)𝑣^𝜀(𝑡,⋅),𝜑(𝑡,⋅)⟩_{ℳ(𝒪),𝐶0(𝒪)}𝑑𝑡≥,(55)_𝑇0_𝒪𝜈_𝑘+(𝜇^𝜀)𝜑_𝑡−𝑞_𝑘(𝜇^𝜀)𝜑_𝑥𝑑𝑥𝑑𝑡_𝑁_𝑖=1^𝑇0sign𝜇^𝜀𝑡,𝜏_𝑖−𝑘𝑢^∗𝑖(𝑡)𝜑𝑡,𝜏_𝑖𝑑𝑡≥0(56) with 𝜈_𝑙(𝜌)∶=|𝜌−𝑙|, 𝑔_𝑙(𝜌)∶=(𝑓₁(𝜌)−𝑓₁(𝑙))sign(𝜌−𝑙), 𝜈_𝑘(𝜇)∶=|𝜇−𝑘|, and 𝑞_𝑘(𝜇)∶=(𝜇−𝑘)sign(𝜇−𝑘).

Proof. Let 𝐸=𝐸(𝜌)∈𝐶²(ℝ) be any convex function. We multiply (32) by 𝐸^(𝜌). Then the equalities 𝐸^(𝜌)𝜌_𝑡=𝜕𝐸(𝜌(𝑡,𝑥))𝑓𝜕𝑡_1,(𝜌)𝐸^(𝜌)𝜌_𝑥=𝜕𝜕𝑥^{𝑘𝜌(𝑡,𝑥)}𝐸𝑓^1(𝜉)𝐸^(𝜉)𝑑𝜉_,(𝜌)𝜌_𝑥𝑥=(𝐸(𝜌))_𝑥𝑥−𝐸^(𝜌)𝜌^2𝑥,(57) imply the following relation (𝐸(𝜌^𝜀))_𝑡+^𝜌𝜀_𝑘𝑓^1(𝜉)𝐸^(𝜉)𝑑𝜉_𝑥=𝜀(𝐸(𝜌^𝜀))_𝑥𝑥−𝜀𝐸^(𝜌^𝜀)(𝜌^𝜀)^2𝑥+𝐸^(𝜌^𝜀)𝑣^𝜀,in𝒟^((0,𝑇)×𝒪).(58) 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: −^𝑇0_ℝ𝐸(𝜌^𝜀)𝜑_𝑡+^𝜌𝜀_𝑘𝑓^1(𝜉)𝐸^(𝜉)𝑑𝜉𝜑_𝑥=𝜀_𝑇0𝑑𝑥𝑑𝑡_ℝ−𝜀_𝑇0(𝐸(𝜌^𝜀))𝜑_𝑥𝑥𝑑𝑥𝑑𝑡_ℝ𝐸^(𝜌^𝜀)(𝜌^𝜀)^2𝑥𝜑𝑑𝑥𝑑𝑡+^𝑇0_ℝ≤𝜀_𝑇0𝐸^(𝜌^𝜀)𝑣^𝜀𝜑𝑑𝑥𝑑𝑡_ℝ𝐸(𝜌^𝜀)𝜑_𝑥𝑥𝑑𝑥𝑑𝑡+^𝑇0_ℝ𝐸^(𝜌^𝜀)𝑣^𝜀𝜑𝑑𝑥𝑑𝑡.(59) Since 𝜌^𝜀∈𝐿²(0,𝑇;𝐻¹⁰(𝒪)) for all 𝜀>0 and 𝐻¹⁰(𝒪)↪𝐶(𝒪) by the classical Sobolev Embedding Theorem, it follows that the following term is well defined: ^𝑇0_ℝ𝐸^(𝜌^𝜀)𝑣^𝜀𝜑𝑑𝑥𝑑𝑡=^𝑇0𝐸^(𝜌^𝜀)𝑣^𝜀,𝜑_{ℳ(ℝ),𝐶0(ℝ)}𝑑𝑡.(60) Further we use the well-known trick. Let {𝐸_𝑚}_𝑚∈ℕ be a sequence of 𝐶²-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 𝐸^𝑚(𝜉)→sign(𝜉−𝑘) for all 𝜉∈ℝ, 𝜉≠𝑘. Since 𝜑_𝑥𝑥≤0 in Ω_𝑇 and ^𝜌𝜀_𝑘=s𝑓ig^1n(𝜉()𝜌𝐸^𝑚_𝜀(𝜉)𝑑𝜉⟶^𝜌𝜀_𝑘𝑓^1(𝜉)sign(𝜉−𝑘)𝑑𝜉−𝑘)^𝜌𝜀_𝑘𝑓^1(𝜉)𝑑𝜉=sign(𝜌^𝜀−𝑘)𝑓₁(𝜌^𝜀)−𝑓₁(𝑘),(61) it immediately leads us to the entropy inequality (55) from (59). The verification of inequality (56) can be done by similar arguments.

5. The Penalized Optimization Problem

In this section, for every 𝜀>0 and a given influx-rate 𝑢^𝜀∈𝒰_ad, we analyze the following penalized optimization problem associated with the singular perturbed initial-boundary value problem (32)–(38):𝐼_𝜀(𝑣^𝜀,𝜌^𝜀)=‖𝑣^𝜀‖^2𝐿_{2(0,𝑇;ℳ(𝒪))}+𝜀⁻¹‖‖𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀‖‖_{𝐿2(0,𝑇;𝐻−1(𝒪))}⟶inf(62)subjecttotheconstraints(33)-(39).(63)

Definition 16. We say that a pair (𝑣^𝜀,𝜌^𝜀) is admissible to the optimization problem (62)-(63) if 𝑣^𝜀∈𝐿²(0,𝑇;ℳ(ℝ)) and 𝜌^𝜀=𝜌^𝜀(𝑣^𝜀)∈𝒲 is the corresponding weak solution to the initial boundary value problem (32), (34)1, and (38)1.

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:𝒴=𝐿²(0,𝑇;ℳ(𝒪))×𝐿²((0,𝑇)×𝒪).(64)

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 𝜇^𝜀∈𝒲. Hence (𝑓^2(𝜇^𝜀))𝜇^𝜀𝑥 is in 𝐿²((0,𝑇)×𝒪), and 𝑣^𝜀∈𝐿²(0,𝑇;ℳ(𝒪)) by the definition of the class 𝒱_ad. Since 𝐿²((0,𝑇)×𝒪)⊂𝐿²(0,𝑇;ℳ(𝒪)), we come to the required conclusion.

We define the 𝜏-topology on 𝒴 as follows: 𝜏 is the product of the weak-∗ topology of 𝐿²(0,𝑇;ℳ(𝒪)) and the topology of norm in 𝐿²((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).

Lemma 18. Assume that supposition (A1) holds true. Then the set Ξ_𝜀 is nonempty and sequentially 𝜏-closed for every 𝜀>0.

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 𝜇∶^𝜀=,𝜌^𝜀∈𝑦∶𝒲𝑦∈𝐿₂0,𝑇;𝐻¹⁰(𝒪),𝜕𝜕𝑦𝑡∈𝐿²0,𝑇;𝐻⁻¹(𝒪)(65) and 𝒲↪𝐿²((0,𝑇)×𝒪), we conclude that (𝑣^𝜀,𝜌^𝜀)∈Ξ_𝜀 and hence Ξ_𝜀≠∅.
To establish the 𝜏-closedness of Ξ_𝜀, we fix an arbitrary 𝜏-converging sequence of admissible solutions to the perturbed problem (32)–(38) and (62) {(𝑣^𝜀𝑘,𝜌^𝜀𝑘)∈Ξ_𝜀}^∞𝑘=1 and show that (𝑣^𝜀∗,𝜌^𝜀∗)∈Ξ_𝜀, where (𝑣^𝜀∗,𝜌^𝜀∗) is its 𝜏-limit.
We have that 𝑣⇀^𝜀𝑘∗𝑣^𝜀∗ in 𝐿²(0,𝑇;ℳ(𝒪)) and 𝜌^𝜀𝑘→𝜌^𝜀∗ in 𝐿²((0,𝑇)×𝒪). Hence 𝑣^𝜀∗∈𝒱_ad and it remains to show that 𝜌^𝜀∗ is the corresponding weak solution of the initial-boundary value problem (32), (34)1, and (38)1. Indeed, in view of the a priori estimate (48), it is easy to see that the 𝐿²((0,𝑇)×𝒪-limit function 𝜌^𝜀∗ belongs to the space 𝒲 and satisfies conditions: 𝑓₁𝜌𝜌^𝜀𝑘⟶𝑓₁𝜌^𝜀∗,in𝐿²0,𝑇;𝐿²(𝒪)as𝑘⟶∞,_𝜀𝑘𝜌_𝑥⇀𝜌^𝜀∗_𝑥,in𝐿²0,𝑇;𝐿²(𝒪)as𝑘⟶∞,_𝜀𝑘𝜌_𝑡⇀𝜌^𝜀∗_𝑡,in𝐿²0,𝑇;𝐻⁻¹(𝒪)as𝑘⟶∞,_𝜀𝑘,𝜌^𝜀∗∈𝐶[0,𝑇];𝐿²(𝒪),𝜌^𝜀𝑘(0,𝑥)=∀𝑘𝜌₀∈(𝑥ℕ).in𝒪,(66) This enables us to pass to the limit in the integral identity (46) as 𝑘→∞ with 𝜌^𝜀=𝜌^𝜀𝑘 and 𝑣=𝑣^𝜀𝑘, and eo ipso to show that the limit function 𝜌^𝜀∗ is a weak solution to the parabolic problem (32), (34)1, (38)1.
Thus, the pair (𝑣^𝜀∗,𝜌^𝜀∗) is an admissible solution to the perturbed optimization problem (32)–(38), (62). The proof is complete.

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 (𝑣^𝜀0,𝜌^𝜀0)∈Ξ_𝜀 such that 𝐼_𝜀𝑣^𝜀0,𝜌^𝜀0=inf_{(𝑣𝜀,𝜌𝜀)∈Ξ𝜀}𝐼_𝜀(𝑣^𝜀,𝜌^𝜀),(67) 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 {(𝑣^𝜀𝑘,𝜌^𝜀𝑘)}_𝑘∈ℕ⊂Ξ_𝜀 such that 𝐼_𝜀𝑣^𝜀𝑘,𝜌^𝜀𝑘⟶_𝑘→∞𝐼^𝜀min≡inf_{(𝑣𝜀,𝜌𝜀)∈Ξ𝜀}𝐼_𝜀(𝑣^𝜀,𝜌^𝜀)≥0,(68) that is, {(𝑣^𝜀𝑘,𝜌^𝜀𝑘)}_𝑘∈ℕ⊂Ξ_𝜀 is a minimizing sequence for the problem (32)–(38), (62).
To begin with, we show that for any 𝜆>0 the set Ξ^𝜆𝜀=(𝑣^𝜀,𝜌^𝜀)∈Ξ_𝜀∶𝐼_𝜀(𝑣^𝜀,𝜌^𝜀)≤𝜆(69) is bounded in 𝐿²(0,𝑇;ℳ(𝒪))×𝒲. Indeed, as follows from inequality (68), the sequence of fictitious controls {𝑣^𝜀𝑘}_𝑘∈ℕ is bounded in 𝐿²(0,𝑇;ℳ(𝒪)). Hence, we may assume that there exists an element 𝑣^𝜀0∈𝒱_ad such that 𝑣⇀^𝜀𝑘∗𝑣^𝜀0 in 𝐿²(0,𝑇;ℳ(𝒪)) as 𝑘→∞, that is, lim_𝑘→∞^𝑇0𝑣^𝜀𝑘=(𝑡,⋅),𝜑_{𝑀(𝒪),𝐶0(𝒪)}𝜓(𝑡)𝑑𝑡_𝑇0∀𝜑∈𝑣𝐶^𝜀0₀(𝑡,⋅),𝜑_{𝑀(𝒪),𝐶0(𝒪)}𝜓(𝑡)𝑑𝑡,(𝒪),∀𝜓∈𝐶^∞0(0,𝑇).(70) Then having used the a priori estimate (48), we see that {𝜌^𝜀𝑘=𝜌^𝜀(𝑣^𝜀𝑘)}_𝑘∈ℕ form a uniformly bounded sequence in 𝒲. Hence, we may again assume that, up to a subsequence, there exists an element 𝜌^𝜀0∈𝒲 such that 𝜌^𝜀𝑘⇀𝜌^𝜀0 weakly in 𝒲 and strongly in 𝐿²((0,𝑇)×𝒪). As a result, (𝑣^𝜀0,𝜌^𝜀0)∈Ξ_𝜀 by Lemma 18.
Let us show that the 𝜏-limit pair (𝑣^𝜀0,𝜌^𝜀0) is an optimal solution to the penalized problem (32)–(38), (62). Indeed, taking into account supposition (A1) and Theorem 14, we have 𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀𝑘⇀^∗𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀0,in𝐿²(0,𝑇;ℳ(𝒪)).(71) Using the property of lower semi-continuity for 𝐼_𝜀 with respect to the 𝜏-topology, we get 0≤𝐼_𝜀𝑣^𝜀0,𝜌^𝜀0≤lim_𝑘→∞𝐼(𝑣^𝜀,𝜌^𝜀)=𝐼^𝜀min.(72) Thus the pair (𝑢^𝜀0,𝜌^𝜀0) is optimal for the problem (32)–(38), (62).

6. 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 (𝒴₁,𝜎), where𝒴₁=𝐿²(0,𝑇;ℳ(𝒪))×𝐿²0,𝑇;𝐻⁻¹(𝒪),(73) and 𝜎 is the product of the weak-∗ topology of 𝐿²(0,𝑇;ℳ(𝒪)) and the strong topology of 𝐿²(0,𝑇;𝐻⁻¹(𝒪)). So, we can take 𝜎 as the main topology for the asymptotic analysis.

Lemma 20. Let 𝑢^∗=∑^𝑁𝑘=1𝑢^∗𝑘(𝑡)𝛿_𝜏𝑘∈𝒰_ad a given source term with prescribed location 𝑎<𝜏₁<…<𝜏_𝑁<𝑏. Let {𝜇^𝜀}_𝜀>0 be a sequence of corresponding weak solutions to the initial boundary value problem (33), (34)2, (38)2 when the small parameter 𝜀>0 varies in a strictly decreasing sequence of positive numbers converging to 0. Let {(𝑣^𝜀0,𝜌^𝜀0)∈Ξ_𝜀}_𝜀>0 be a sequence of optimal solutions to the penalized problem (32)–(38), (62). Assume that the fictitious controls {𝑣^𝜀0}_𝜀>0 are bounded in 𝐿²(0,𝑇;ℳ(𝒪)) and supposition (A1) holds true. Then subsequences of {𝜇^𝜀}_𝜀>0 and of {(𝑣^𝜀0,𝜌^𝜀0)}_𝜀>0, still denoted by the suffix 𝜀, can be extracted such that (a)𝑣⇀^𝜀0∗𝑣^∗ in 𝐿²(0,𝑇;ℳ(𝒪)); (b)𝜌^𝜀0→𝜌^∗ and 𝜇^𝜀→𝜇^∗ weakly in 𝐿²((0,𝑇)×𝒪) and strongly in 𝐿²(0,𝑇;𝐻⁻¹(𝒪)); (c)(𝜌^∗,𝜇^∗) is a weak solution in [𝐿²((0,𝑇)×𝒪)]² of the Cauchy problem:𝜌^∗𝑡+𝜇𝑓₁(𝜌^∗)_𝑥=𝑣^∗,𝜌^∗(0,⋅)=𝜌₀,(74)_∗𝑡−𝜇^∗𝑥=𝑢^∗,𝜇^∗(0,⋅)=𝜇₀.(75)

Proof. As follows from the a priori estimates (50)-(51), the sequences {𝜌^𝜀0}_𝜀>0 and {𝜇^𝜀}_𝜀>0 are bounded in 𝒲₁=𝑦∶𝑦∈𝐿²0,𝑇;𝐿²(𝒪),𝜕𝑦𝜕𝑡∈𝐿²0,𝑇;𝐻⁻¹(𝒪).(76) Hence the compactness properties (a)-(b) of the sequences {𝑣^𝜀0}_𝜀>0, {𝜌^𝜀0}_𝜀>0, and {𝜇^𝜀}_𝜀>0 are a direct consequence of the initial suppositions, the Banach-Alaoglu Theorem, and the compactness embedding 𝒲₁↪𝐿²(0,𝑇;𝐻⁻¹(𝒪)). Moreover, as follows from estimates (48)-(49), the sequence {(𝜌^𝜀0,𝜇^𝜀)}_𝜀>0 is bounded in 𝐿²(0,𝑇;𝐿²(𝒪)). So, we can suppose that 𝜌^𝜀0⟶𝜌^∗,𝜇^𝜀⟶𝜇^∗,as𝜀⟶0(77) strongly in 𝐿²(0,𝑇;𝐻⁻¹(𝒪)) and weakly in 𝐿²(0,𝑇;𝐿²(𝒪)). In view of estimates (48)-(49), there are elements 𝜂,𝜂̂∈𝐿²((0,𝑇)×𝒪) such that, up to subsequences, we have √𝜀𝜌^𝜀0_𝑥⇀𝜂,√𝜀(𝜇^𝜀)_𝑥⇀𝜂âsi𝜀n⟶𝐿²((00.,𝑇)×𝒪),(78) In order to verify the item (c), we note that the integral identity (46) leads us to the following relation: ^𝑇0_𝒪−𝜌^𝜀0𝜕𝜑𝜕𝑡+√𝜀√𝜀𝜕𝜌^𝜀0=𝜕𝑥𝜕𝜕𝜑𝑥−𝑓¹𝜌^𝜀0𝜕𝜕𝜑𝑥𝑑𝑥𝑑𝑡_𝑇0𝑣^𝜀0(𝑡,⋅),𝜑(𝑡,⋅)_{ℳ(𝒪),𝐶0(𝒪)}𝑑𝑡,(79) which holds true for every 𝜀>0 and any test function 𝜑∈𝐶^∞0((0,𝑇)×𝒪). Since 𝑣⇀^𝜀0∗𝑣^∗ in 𝐿²(0,𝑇;ℳ(𝒪)) as 𝜀→0, we can pass to the limit in (79) using the property (77)1-(78)1. As a result, we come to the relation ^𝑇0_𝒪−𝜌^∗𝜕𝜑=𝜕𝑡−𝑓¹𝜌^∗𝜕𝜕𝜑𝑥𝑑𝑥𝑑𝑡_𝑇0⟨𝑣^∗(𝑡,⋅),𝜑(𝑡,⋅)⟩_{ℳ(𝒪),𝐶0(𝒪)}𝑑𝑡,(80) which gives us the weak formulation of the hyperbolic conservation law (75)1. As for the initial condition (75)1, we note that by continuity property (41) the following identity lim_𝑡→0+1𝑡^𝑡0||||_𝒪𝜌^𝜀0(𝑠,⋅)−𝜌₀𝜓𝑑𝑥||||𝑑𝑠=0,∀𝜓∈𝐶^∞0(𝒪)(81) is valid for every 𝜀>0. So, we can pass to the limit in (81) as 𝜀→0 using the weak convergence of 𝜌^𝜀0→𝜌^∗ in 𝐿²((0,𝑇)×𝒪). As a result, the initial condition for the limit function 𝜌^∗ is satisfied in the following sense: lim_𝑡→0+1𝑡^𝑡0||||_𝒪𝜌^∗(𝑠,⋅)−𝜌₀𝜓𝑑𝑥||||𝑑𝑠=0,∀𝜓∈𝐶^∞0(𝒪).(82) Thus, 𝜌^∗∈𝐿²((0,𝑇)×𝒪) is a weak solution to the Cauchy problem (74). By analogy, similar properties for the limit function 𝜇^∗ can be proved. This concludes the proof.

The next result is crucial in this paper. We show that approximately entropy weak solutions to the system of nonlinear conservation laws with impulse controls can be constructed by optimal solutions to the penalized problem (32)–(38) and (62).

Theorem 21. Let 𝑢=∑^𝑁𝑘=1𝑢_𝑘(𝑡)𝛿_𝜏𝑘∈𝒰_ad be a given source term with prescribed location 𝑎<𝜏₁<⋯<𝜏_𝑁<𝑏. Assume that there exists a sequence of pairs {(𝑣^𝜀̂,𝜌̂^𝜀)∈Ξ_𝜀}_𝜀>0 satisfying the following relation: limsup_𝜀→0𝐼_𝜀𝑣^𝜀̂,𝜌̂^𝜀<+∞.(83) Let {(𝑣^𝜀0,𝜌^𝜀0)∈Ξ_𝜀}_𝜀>0 be a sequence of optimal solutions to the penalized problem (32)–(38) and (62). Then, under supposition (A1), for every 𝜎-cluster point (𝑣^∗,𝜌^∗)∈𝒴₁ of the sequence {(𝑣^𝜀0,𝜌^𝜀0)∈Ξ_𝜀}_𝜀>0 we has that the triplet (𝑢^∗,𝜌^∗,𝜇^∗) is an approximately entropy solution to the Cauchy problem (14)–(16) in the domain (0,𝑇)×𝒪 and the equality 𝑣^∗=(𝑓₂(𝜇^∗))_𝑥 is valid almost everywhere in (0,𝑇)×𝒪. Here the distribution 𝜇^∗ is defined by (75).

Remark 22. It is worth to notice that the existence of a sequence {(𝑣^𝜀̂,𝜌̂^𝜀)∈Ξ_𝜀}_𝜀>0 satisfying relation (83) is rather important for our further analysis and this assumption is coming from the regularity property of the original Cauchy problem (14)–(18). Here by the regularity of Cauchy problem (14)–(18) we mean that this problem admits at least one entropy solution. Since the existence of such solutions is unknown in general, we must assume it. Only in this case it has a sense to construct an approximation of entropy solutions. So, for the regular Cauchy problem (14)–(18), the sequence {(𝑣^𝜀̂,𝜌̂^𝜀)∈Ξ_𝜀}_𝜀>0 can be constructed as follows: 𝑣^𝜀̂=(𝑓₂(𝜇^𝜀))_𝑥 for all 𝑒>0, and (𝜌_𝜀,𝜇_𝜀) is the corresponding solution of the perturbed problem (32)–(37). As for the general case, we demand the fulfilment of the condition (83).

Proof. As Lemma 20 indicates, the sequence {(𝜇^𝜀)∈𝒲₁}_𝜀>0 is relatively compact with respect to the strong convergence in 𝐿²(0,𝑇;𝐻⁻¹(𝒪)) and the weak convergence in 𝐿²(0,𝑇;𝐿²(𝒪)). So, passing to a subsequence, when the occasion requires, we get 𝜇𝜇^𝜀⟶𝜇^∗,in𝐿²0,𝑇;𝐻⁻¹(𝒪)_𝜀,⇀𝜇^∗,in𝐿²0,𝑇;𝐿²(𝒪),(84) where 𝜇^∗∈𝐿²((0,𝑇)×𝒪) is a weak solution to the Cauchy problem (75). For our further analysis we have to show that 𝑓₂(𝜇^𝜀)_𝑥⇀𝑓₂(𝜇^∗)_𝑥,in𝐿²0,𝑇;𝐻⁻¹(𝒪).(85) Indeed, let 𝜑∈𝐶^∞0((0,𝑇)×𝒪) be a fixed test function. Then the following estimate holds: ||||^𝑇0_𝒪=||||𝜇^𝜀𝑥𝜑𝑑𝑥𝑑𝑡_𝑇0||||_𝒪𝜇^𝜀𝑡≤𝜑+𝜇^𝜀𝑥𝜑_𝑥−𝑢𝜑𝑑𝑥𝑑𝑡_𝑇0|||||||𝜇^𝜀𝑡,𝜑_{𝐻−1(𝒪),𝐻10(𝒪)}|||+√𝜀_𝒪|||√𝜀𝜇^𝜀𝑥𝜑_𝑥+|||⟨𝑢,𝜑⟩_{𝐻−1(𝒪),𝐻10(𝒪)}|||𝑑𝑥|||𝑑𝑡_by(59),(61)≤𝐶+‖𝑢‖_𝐿²_(0,𝑇;𝐻⁻¹_(𝒪))‖𝜑‖_𝐿²_(0,𝑇;𝐻¹⁰_(𝒪)).(86) Hence the sequence {𝜇^𝜀𝑥}_𝜀>0 is uniformly bounded in 𝐿²(0,𝑇;𝐻⁻¹(𝒪). Therefore, in view of (84), we can suppose that 𝜇^∗𝑥∈𝐿²(0,𝑇;𝐻⁻¹(𝒪)) and 𝜇⇀^𝜀𝑥∗𝜇^∗𝑥,in𝐿²0,𝑇;𝐻⁻¹(𝒪).(87) As a result, applying the arguments of Remark 17, we come to the required conclusion (85).
Let {(𝑣^𝜀̂,𝜌̂^𝜀)∈Ξ_𝜀}_𝜀>0 be a sequence with property (83). Then there exist a value 𝜀₀>0 and a constant 𝑐>0 independent of 𝜀 such that the following inequality holds true: ‖‖𝑣^𝜀0‖‖^2𝐿_{2(0,𝑇;ℳ(𝒪))}+𝜀⁻¹‖‖𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀0‖‖_{𝐿2(0,𝑇;𝐻−1(𝒪))}≤𝐼_𝜀𝑣^𝜀̂,𝜌̂^𝜀≤𝑐,∀𝜀∈0,𝜀₀.(88) Hence the sequence of optimal fictitious controls {𝑣^𝜀0}_𝜀>0 is bounded in 𝐿²(0,𝑇;ℳ(𝒪)). Therefore, by Lemma 20 the sequence of optimal pairs {(𝑣^𝜀0,𝜌^𝜀0)∈Ξ_𝜀}_𝜀>0 is relatively compact with respect to the 𝜎-topology of 𝐿²(0,𝑇;ℳ(𝒪))×𝐿²(0,𝑇;𝐻⁻¹(𝒪)). Moreover, every 𝜎-cluster point (𝑣^∗,𝜌^∗)∈𝒴₁ possesses the properties (a)–(c) of Lemma 20.
Further we note that the inequality (88) leads to the estimate 0≤‖‖𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀0‖‖_{𝐿2(0,𝑇;𝐻−1(𝒪))}≤𝜀𝑐,∀𝜀∈0,𝜀₀.(89) Since (𝑓₂(𝜇^𝜀))_𝑥−𝑣^𝜀0⇀(𝑓₂(𝜇^∗))_𝑥−𝑣^∗ in 𝐿²(0,𝑇;𝐻⁻¹(𝒪)) (see (85)), 𝑣⇀^𝜀0∗𝑣^∗ in 𝐿²(0,𝑇;ℳ(𝒪)), and ℳ(𝒪)⊂𝐻⁻¹(𝒪), we can pass to the limit in (89) as 𝜀→0. Then, in view of the lower semicontinuity property, we obtain 0≤‖‖𝑓₂(𝜇^∗)_𝑥−𝑣^∗‖‖_{𝐿2(0,𝑇;𝐻−1(𝒪))}≤liminf_𝜀→0‖‖𝑓₂(𝜇^𝜀)_𝑥−𝑣^𝜀0‖‖_{𝐿2(0,𝑇;𝐻−1(𝒪))}≤0.(90) Since this is equivalent to the equality 𝑣^∗=(𝑓₂(𝜇^∗))_𝑥 almost every where in (0,𝑇)×Ω, by Lemma 15 it follows that the pair (𝜌^∗,𝜇^∗) is an approximately entropy solution to the initial-boundary value problem (32)–(38). This concludes the proof.

7. 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 𝜎-cluster pair (𝑣^∗,𝜌^∗)∈𝒴₁ of the sequence {(𝑣^𝜀0,𝜌^𝜀0)∈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).