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Journal of Control Science and Engineering
Volume 2010 (2010), Article ID 982369, 10 pages
http://dx.doi.org/10.1155/2010/982369
Research Article

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

1Dipartimento di Ingegneria Elettronica e Ingegneria Informatica, Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano, Italy
2Department of Differential Equations, Dnipropetrovsk National University, Gagarin Avenue, 72, 49010 Dnipropetrovsk, Ukraine

Received 9 September 2010; Accepted 18 December 2010

Academic Editor: Yoshito Ohta

Copyright © 2010 Ciro D'Apice et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [13]). 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 laws𝜌𝑡+(𝑓(𝜇,𝜌))𝑥=0,(𝑡,𝑥)Ω𝑇𝜇=(0,𝑇)×,𝑡𝜇𝑥=𝑢(𝑡,𝑥),(𝑡,𝑥)Ω𝑇,𝜌(0,𝑥)=𝜌0(𝑥),𝜇(0,𝑥)=𝜇0(𝑥),𝑥.(1) Throughout this paper we suppose that the structure of the source term 𝑢(𝑡,𝑥) is prescribed, namely,𝑢=𝑢(𝑡,𝑥)=𝑁𝑖=1𝑢𝑖(𝑡)𝛿𝜏𝑖(𝑥),with<𝑎<𝜏1<<𝜏𝑁<𝑏<+,(2) where the functions {𝑢𝑖𝐿2(0,𝑇)}𝑁𝑖=1 can play the role of control factors, and 𝛿𝜏𝑖 denote the Dirac measures located at the points 𝜏𝑖.

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 ), 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 problem𝐼𝜀(𝑣𝜀,𝜌𝜀)=𝑣𝜀2𝐿2(0,𝑇;(𝒪))+𝜀1𝑓2(𝜇𝜀)𝑥𝑣𝜀𝐿2(0,𝑇;𝐻1(𝒪))inf(3) subject to the constraints𝜌𝜀𝑡(𝑡,𝑥)𝜀𝜌𝜀𝑥𝑥𝑓(𝑡,𝑥)+1(𝜌𝜀(𝑡,𝑥))𝑥=𝑣(𝑡,𝑥),(𝑡,𝑥)Ω𝑇,𝜇𝜀𝑡(𝑡,𝑥)𝜀𝜇𝜀𝑥𝑥(𝑡,𝑥)𝜇𝜀𝑥(𝑡,𝑥)=𝑢(𝑡,𝑥),(𝑡,𝑥)Ω𝑇,𝜌𝜀(0,𝑥)=𝜌0(𝑥),𝜇𝜀(0,𝑥)=𝜇0(𝑥),𝑥,𝑢(𝑡,𝑥)=𝑢1(𝑡)𝛿𝜏1(𝑥)+𝑢2(𝑡)𝛿𝜏2(𝑥)++𝑢𝑁(𝑡)𝛿𝜏𝑁𝑢(𝑥),𝑖𝐿2,(0,𝑇),𝑖={1,,𝑁}𝑣𝐿2𝜌(0,𝑇;()),𝜀(𝑡,𝑥)=0,𝜇𝜀(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 {(𝑣𝜀0,𝜌𝜀0)Ξ𝜀}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 Ω𝑇=(0,𝑇)× and Ω=(0,𝑇)×(𝑎,𝑏). Let 𝐿𝑝loc(Ω𝑇), with 1𝑝, be the locally convex space of all measurable functions 𝑔Ω𝑇 such that 𝑔|(0,𝑇)×𝐾𝐿𝑝((0,𝑇)×𝐾) for all compact sets 𝐾.

Let () be the set of all Radon measures on , that is, 𝜇() if μ is a countably additive set function defined on the Borel subsets of such that μ is finite on every compact subset of . We say that a sequence of Radon measures {𝜇𝑘}𝑘 converges weakly- to a measure 𝜇() (in symbols 𝜇𝑘𝜇) iflim𝑘𝜑𝑑𝜇𝑘=𝜑𝑑𝜇,𝜑𝐶0().(5)

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

Proposition 1. Let {𝜇𝑘}𝑘 be a bounded sequence of Radon measures on . Then there exist a subsequence {𝜇𝑘𝑗}𝑗 and a Radon measure 𝜇() such that 𝜇𝑘𝑗𝜇.

According to the Riesz theory, every Radon measure μ on can be identified with an element of the dual space (𝐶0()), that is, μ is a linear form on 𝐶0() and for every compact set 𝐾 there exists a constant 𝐶>0 depending only on 𝐾 and μ such that||||𝜇,𝑓𝐶𝑓𝐶(𝐾),𝑓𝐶0()withsupp𝑓𝐾.(7) As an example of a Radon measure on , we consider the following one. Let {𝑎𝑘}𝑘 and {𝑏𝑘}𝑘 be two sequences in such that 𝑘=1|𝑎𝑘|𝐂<+. Let 𝛿𝑐 be the Dirac measure located at the point 𝑐, that is, this measure is defined as follows𝛿𝑐,𝜑=𝛿𝑐(𝑥)𝜑(𝑥)𝑑𝑥=𝜑(𝑐),𝜑𝐶0().(8) Since|||||𝑘=1𝑎𝑘𝜑𝑏𝑘|||||𝑘=1||𝑎𝑘||𝜑𝐶()𝐂𝜑𝐶()(9) for every continuous function with compact support 𝜑𝐶0(), it follows that the linear form𝜇=𝑘=1𝑎𝑘𝛿𝑏𝑘(10) is continuous on 𝐶0(). Hence 𝜇 is an element of the space of Radon measures ().

Let 𝒪 be a bounded open subset of . Let 𝑓𝒪 be an element of 𝐿1(𝒪). Define 𝒪||||𝐷𝑓=sup𝒪𝑓𝜑𝑑𝑥𝜑𝐶10||||.(𝒪),𝜑(𝑥)1for𝑥𝒪(11) According to the Radon-Nikodym theorem, if 𝒪|𝐷𝑓|<+ then the distribution 𝐷𝑓 is a measure and there exist a function 𝑓𝐿1(𝒪) and a measure 𝐷𝑠𝑓, singular with respect to the one-dimensional Lebesgue measure 𝒪 restricted to 𝒪, such that 𝐷𝑓=𝑓𝒪+𝐷𝑠𝑓.(12)

Definition 2. A function 𝑓𝐿1(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 𝑓BV(𝒪)=𝑓𝐿1(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 {𝑓𝑘}𝑘=1BV(𝒪) and sup𝑘𝑓𝑘BV(𝒪)<+, then there exists a subsequence of {𝑓𝑘}𝑘=1 strongly converging in 𝐿1(𝒪) to some 𝑓BV(𝒪).

Definition 4. A sequence {𝑓𝑘}𝑘=1BV(𝒪) weakly converges to some 𝑓BV(𝒪), and we write 𝑓𝑘𝑓 if and only if the two following conditions hold: 𝑓𝑘𝑓 strongly in 𝐿1(𝒪), and 𝐷𝑓𝑘𝐷𝑓 weakly* in (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 {𝑓𝑘}𝑘=1 be a sequence in BV(𝒪) strongly converging to some 𝑓 in 𝐿1(𝒪) and satisfying sup𝑘𝒪|𝐷𝑓𝑘|<+. Then (i)𝑓BV(𝒪) and 𝒪|𝐷𝑓|liminf𝑘𝒪|𝐷𝑓𝑘|; (ii)𝑓𝑘𝑓 in BV(𝒪).

3. Statement of Problem and Main Motivation

Let {𝜏𝑘}𝑁𝑘=1 be a given finite family of points such that 𝑎<𝜏1<<𝜏𝑁<𝑏. We focus on the following fluid dynamic model, expressed by the nonlinear inhomogeneous hyperbolic conservation laws:𝜌𝑡+(𝑓(𝜇,𝜌))𝑥=0,(𝑡,𝑥)Ω𝑇,𝜇(14)𝑡𝜇𝑥=𝑢(𝑡,𝑥),(𝑡,𝑥)Ω𝑇,(15)𝜌(0,𝑥)=𝜌0(𝑥),𝜇(0,𝑥)=𝜇0(𝑥),𝑥,(16) where the source term is subjected to the following constraints:𝑢(𝑡,𝑥)=𝑢1(𝑡)𝛿𝜏1(𝑥)+𝑢2(𝑡)𝛿𝜏2(𝑥)++𝑢𝑁(𝑡)𝛿𝜏𝑁𝑢(𝑥),(17)𝑖𝐿2(0,𝑇),𝑖={1,,𝑁}.(18) Here 𝑢𝑖𝐿2(0,𝑇) are some external distributed sources located at the corresponding points 𝜏𝑖(𝑎,𝑏), 𝜌0,𝜇0BV()𝐿() 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 ), 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 have𝑇0𝑢(𝑡,)2()=𝑑𝑡𝑇0sup𝜑𝐶()=1𝜑𝐶0(𝑅)𝑢(𝑡,),𝜑()(),𝐶0()2=𝑑𝑡𝑇0sup𝜑𝐶()=1𝜑𝐶0𝑁(𝑅)𝑘=1𝑢𝑘(𝑡)𝛿𝜏𝑘(𝑥)𝜑(𝑥)𝑑𝑥2𝑑𝑡𝑇0𝑁𝑘=1𝑢𝑘(𝑡)2𝑑𝑡𝑁𝑁𝑘=1𝑢𝑘2𝐿2(0,𝑇).(19)

Hence, it is natural to define the following class:𝒰ad=𝑢𝐿2(0,𝑇;())𝑢=𝑢(𝑡,𝑥)satisfy(25)-(26)forsome𝑎<𝜏1<<𝜏𝑁.<𝑏(20)

Definition 6. Let 𝑢𝒰ad be a fixed source term. We say that a vector value function 𝑌=𝜌𝜇[𝐿2(0,𝑇;𝐿2loc())]2 is a weak solution to (14)–(16) if the identities 𝑇0𝑌𝜕𝜑𝜕𝑡+𝐹(𝑌)𝜕𝜑+𝜕𝑥𝑑𝑥𝑑𝑡𝑇0𝑁𝑘=1𝑈𝑘𝜏(𝑡)𝜑𝑘𝑑𝑡=0,lim𝑡0+1𝑡𝑡0||||𝑌(𝑡,𝑥)𝜓(𝑥)𝑑𝑥𝑌0||||𝜓(𝑥)𝑑𝑥2𝑑𝑡=0(21) hold true for all 𝐶0-functions 𝜑[0,𝑇]×2 and 𝜓2 with compact supports in (0,𝑇)× and , respectively. Here 𝑌0=𝜌0𝜇0,𝑈𝑘0𝑢(𝑡)=𝑘(𝑡),𝐹(𝑌)=𝑓(𝜇,𝜌)𝜇,(22) and the symbol denotes the tensor product 𝑎1𝑎2𝑏1𝑏2=𝑎1𝑏1𝑎2𝑏2.

The characteristic feature of the initial value problem (14)–(16) is that even for arbitrary smooth functions 𝜌0, 𝜇0, and smooth external sources 𝑢𝑘, 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 𝜂2 is an entropy for the system (14)-(15), if it is convex and there exists a 𝐶1-function 𝑞2 such that 𝐷𝜂(𝑣)𝐷𝐹(𝑣)=𝐷𝑞(𝑣),𝑣2.(23) The function 𝑞2 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 {𝜂𝜈2}𝜈=1 converging to 𝜂 locally uniformly as 𝜈. Moreover a 𝐶0-function 𝑞 is a corresponding entropy flux, if there exists a sequence {𝑞𝜈}𝜈=1 of 𝐶1-entropy fluxes of 𝜂𝜈 converging to 𝑞 locally uniformly.

As a result, an entropy solution of (14)–(16) for a given 𝑢𝒰ad can be defined as follows.

Definition 9. Let 𝑢𝒰ad be a given source term with prescribed location 𝑎<𝜏1<<𝜏𝑁<𝑏. A weak solution 𝑌=𝜌𝜇[0,𝑇]×2 to the Cauchy problem (14)–(16) is said entropy admissible if for any constants 𝑘,𝑙 the entropy inequalities 𝑇0𝜈𝑙(𝜌)𝜓𝑡+𝑔𝑙(𝜌)𝜓𝑥𝑑𝑥𝑑𝑡𝑇0𝑓sign(𝜌𝑙)2(𝜇)𝑥𝜓𝑑𝑥𝑑𝑡0,𝑇0𝜈𝑘(𝜇)𝜑𝑡𝑞𝑘(𝜇)𝜑𝑥+𝑑𝑥𝑑𝑡𝑇𝑖=1𝑇0𝜇sign𝑡,𝜏𝑖𝑢𝑘𝑖(𝑡)𝜑𝑡,𝜏𝑖𝑑𝑡0(24) hold true for all positive functions 𝜑,𝜓𝐶0(Ω𝑇), provided that 𝜈𝑘||||(𝜇)=𝜇𝑘,𝑞𝑘𝑔(𝜇)=(𝜇𝑘)sign(𝜇𝑘),𝑙𝑓(𝜌)=1(𝜌)𝑓1(𝑙)sign(𝜌𝑙).(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(Ω𝑇) and closed in 𝐿1(Ω𝑇) (see [8]).

Taking this motivation into account, it is reasonable to introduce the following concept.

Definition 11. Let 𝑢𝒰ad be a given source term. We say that a vector value function 𝑌=𝜌𝜇[𝐿2(0,𝑇;𝐿2loc())] is an approximately entropy solution to the Cauchy problem (14)–(16) in a domain (0,𝑇)×𝒪, if 𝑌=𝜌𝜇[0,𝑇]×2 is a weak solution in the sense of Definition 6 and there exists a sequence {𝑌𝜀=𝜌𝜀𝜇𝜀}𝜀>0[𝐿2(0,𝑇;𝐿2(𝒪))]2 such that (B1)𝜌𝜀𝜌 and 𝜇𝜀𝜇 in 𝐿2(0,𝑇;𝐿2(𝒪)) as 𝜀0; (B2)for any constants 𝑘,𝑙 and for all positive concave functions 𝜑𝐶0((0,𝑇)×𝒪) the entropy inequalities 𝑇0𝒪𝜈𝑙(𝜌𝜀)𝜑𝑡+𝑔𝑙(𝜌𝜀)𝜑𝑥+𝑑𝑥𝑑𝑡𝑇0sign(𝜌𝜀×𝑓(𝑡,)𝑙)2(𝜇𝜀(𝑡,)𝑥,𝜑(𝑡,)(𝒪),𝐶0(𝒪)𝑑𝑡0,(27)𝑇0𝒪𝜈𝑘(𝜇𝜀)𝜑𝑡𝑞𝑘(𝜇𝜀)𝜑𝑥+𝑑𝑥𝑑𝑡𝑁𝑖=1𝑇0𝜇sign𝜀𝑡,𝜏𝑖𝑢𝑘𝑖(𝑡)𝜑𝑡,𝜏𝑖𝑑𝑡0(28) it hold true for every 𝜀>0 with 𝜈𝑙(𝜌)=|𝜌𝑙|, 𝑔𝑙(𝜌)=(𝑓1(𝜌)𝑓1(𝑙))sign(𝜌𝑙), 𝜈𝑘(𝜇)=|𝜇𝑘|, and 𝑞𝑘(𝜇)=(𝜇𝑘)sign(𝜇𝑘).

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 𝑢𝑖𝐿2(0,𝑇) for all 𝑖=1,,𝑁, and with initial distributions 𝜌0,𝜇0BV()𝐿(), 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 is locally Lipschitz, that is, ||𝑓1𝜌1𝑓1𝜌2||𝐿𝜌||𝜌1𝜌2||,𝜌1,𝜌2𝑀𝜌,𝑀𝜌,(30)𝑓1(0)=0, and 𝑓2 is a piecewise linear mapping.

Remark 12. As was shown in recent works [4, 911], a flux function of the fluid dynamic model for supply chains is the following: 𝑓(𝜇,𝜌)=𝑓1(𝜌)+𝑓2+(𝜇)=𝜌,if𝜌<𝜇0,if𝜌𝜇0,if𝜌<𝜇𝜇,if𝜌𝜇.(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:𝜌𝜀𝑡(𝑡,𝑥)𝜀𝜌𝜀𝑥𝑥𝑓(𝑡,𝑥)+1(𝜌𝜀(𝑡,𝑥))𝑥=𝑣(𝑡,𝑥),(𝑡,𝑥)Ω𝑇,𝜇(32)𝜀𝑡(𝑡,𝑥)𝜀𝜇𝜀𝑥𝑥(𝑡,𝑥)𝜇𝜀𝑥(𝑡,𝑥)=𝑢(𝑡,𝑥),(𝑡,𝑥)Ω𝑇𝜌,(33)𝜀(0,𝑥)=𝜌0(𝑥),𝜇𝜀(0,𝑥)=𝜇0(𝑥),𝑥(34) subjected to the constraints𝑢(𝑡,𝑥)=𝑢1(𝑡)𝛿𝜏1(𝑥)+𝑢2(𝑡)𝛿𝜏2(𝑥)++𝑢𝑁(𝑡)𝛿𝜏𝑁𝑢(𝑥),(35)𝑖𝐿2(0,𝑇),𝑖={1,,𝑁},(36)𝑣𝐿2(0,𝑇;()),(37) where 𝑣=𝑣(𝑡,𝑥) is a fictitious control. By 𝒱ad we denote the set of all fictitious controls satisfying conditions (37).

Since 𝜌0,𝜇0𝐿1(𝑁)𝐿(𝑁), it is natural to assume that there is a compact interval 𝐼 such that 𝜌0=0 and 𝜇0=0 almost everywhere in 𝐼. Then taking a sufficiently big open bounded interval 𝒪 including the interval 𝐼, we can suppose that the rate processing 𝜇𝜀 and the density 𝜌𝜀 vanish at the ends of 𝒪. As a result, we can introduce the following boundary conditions into the model (32)–(37):𝜌𝜀(𝑡,𝑥)=0,𝜇𝜀(𝑡,𝑥)=0,on(0,𝑇)×𝜕𝒪.(38)

Since by the initial assumptions the influx-rate 𝑢=𝑁𝑘=1𝑢𝑘(𝑡)𝛿𝑘 and the fictitious control 𝑣 belong to the space of measure data 𝐿2(0,𝑇;(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=𝑦𝑦𝐿20,𝑇;𝐻10,(𝒪)𝜕𝑦𝜕𝑡𝐿20,𝑇;𝐻1(𝒪),(39)𝒲1=𝑦𝑦𝐿20,𝑇;𝐿2,(𝒪)𝜕𝑦𝜕𝑡𝐿20,𝑇;𝐻1(𝒪),(40) equipped with the norm of the graph. Then, the following properties hold true: (1)the embeddings 𝒲𝐿2(0,𝑇;𝐿2(𝒪)), 𝒲1𝐿2(0,𝑇;𝐻1(𝒪)) are compact; (2)one has the embedding []𝒲𝐶0,𝑇;𝐿2(𝒪),𝒲1[]0,𝑇;𝐻1,(𝒪)(41) where, for 𝐗=𝐿2(𝒪) or 𝐗=𝐻1(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,𝑣𝒲𝑑𝑑𝑡𝒪=𝑢𝑢(𝑡,𝑥)𝑣(𝑡,𝑥)𝑑𝑥(𝑡,),𝑣(𝑡,)𝐻1(𝒪),𝐻10(𝒪)+𝑣(𝑡,),𝑢(𝑡,)𝐻1(𝒪),𝐻10(𝒪);(42)(4)let 𝑦𝐿2(0,𝑇;𝐻10(𝒪))𝐶([0,𝑇];𝐿2(𝒪)). Then the following density result holds: for any 𝛿>0 there exists Φ𝐶([0,𝑇];𝐶0(𝒪)), such that𝑦Φ𝐶([0,𝑇];𝐿2(𝒪))𝛿,𝑦Φ𝐿2((0,𝑇)×𝒪)𝛿.(43)

Further we note that by the Friedrichs inequality, we have ||||𝕆𝑦𝜑||||𝑑𝑥𝑐𝕆𝑦2𝑑𝑥𝕆||𝜑||2𝑑𝑥𝑐1𝕆||𝑦||2𝑑𝑥𝕆||𝜑||2𝑑𝑥,𝑦,𝜑𝐻10(𝒪).(44) Hence the bilinear form 𝕆𝑦𝜑𝑑𝑥 is bounded on 𝐻10(𝒪). Moreover, this form is skew-symmetric by the identity𝕆𝑦𝜑𝑑𝑥=𝕆(𝑦𝜑)𝑑𝑥𝕆𝑦𝜑𝑑𝑥=𝕆𝑦𝜑𝑑𝑥,𝑦,𝜑𝐶0(𝒪),(45) which remains valid for all 𝑦,𝜑𝐻10(𝒪) by continuity. Then, we come to the following classical result (see [12, 13]).

Theorem 14. Assume that 𝜇0BV(𝒪)𝐿(𝒪) and Hypothesis (A1) holds true. Then for every 𝜀>0 the initial-boundary value problem (32)–(38) admits a unique solution (𝜌𝜀,𝜇𝜀)𝒲×𝒲 satisfying the integral identities: 𝑇0𝒪𝜕𝜌𝜀𝜕𝑡𝜓+𝜀𝜕𝜌𝜀𝜕𝑥𝜕𝜓𝜕𝑥𝑓1(𝜌𝜀)𝜕𝜓=𝜕𝑥𝑑𝑥𝑑𝑡𝑇0𝑣(𝑡,),𝜓(𝑡,)𝐻1(𝒪),𝐻10(𝒪)𝑑𝑡,𝜓𝐿20,𝑇;𝐻10,(𝒪)(46)𝑇0𝒪𝜕𝜇𝜀𝜕𝑡𝜑+𝜀𝜕𝜇𝜀𝜕𝑥𝜕𝜑𝜕𝑥+𝜇𝜀𝜕𝜑=𝜕𝑥𝑑𝑥𝑑𝑡𝑇0𝑢(𝑡,),𝜑(𝑡,)𝐻1(𝒪),𝐻10(𝒪)𝑑𝑡,𝜑𝐿20,𝑇;𝐻10,(𝒪)(47) with a priori estimates 𝑇0𝒪||𝜌𝜀||(𝑡,𝑥)2||𝜌+𝜀𝜀𝑥||(𝑡,𝑥)2+||𝜌𝜀𝑡||(𝑡,𝑥)2𝑑𝑥𝑑𝑡𝐶𝑇0𝒪||̂𝑔𝑡||(𝑡,𝑥)2+||̂𝑔𝑥||(𝑡,𝑥)2𝑑𝑥𝑑𝑡+𝐶𝑣2𝐿20,𝑇;𝐻1(𝒪),(48)𝑇0𝒪||𝜇𝜀||(𝑡,𝑥)2||𝜇+𝜀𝜀𝑥||(𝑡,𝑥)2+||𝜇𝜀𝑡||(𝑡,𝑥)2𝑑𝑥𝑑𝑡𝐶𝑇0𝒪||𝑔𝑡||(𝑡,𝑥)2+||𝑔𝑥||(𝑡,𝑥)2𝑑𝑥𝑑𝑡+𝐶𝑢2𝐿20,𝑇;𝐻1(𝒪),(49)𝑇0𝜌𝜀𝑡(𝑡,)2𝐻1(𝒪)𝑑𝑡𝐶̂𝑔2𝑊1,2((0,𝑇)×𝒪)+𝐶𝑣2𝐿20,𝑇;𝐻1(𝒪),(50)𝑇0𝜇𝜀𝑡(𝑡,)2𝐻1(𝒪)𝑑𝑡𝐶𝑔2𝑊1,2((0,𝑇)×𝒪)+𝐶𝑢2𝐿2(0,𝑇;𝐻1(𝒪)),(51) where 𝐶>0 is a constant independent of 𝜀 and 𝑔,̂𝑔𝑊1,2((0,𝑇)×𝒪) are such that 𝑔|𝜕𝒪=0, ̂𝑔|𝜕𝒪=0, ̂𝑔(0,)=𝜌0, and 𝑔(0,)=𝜇0 in 𝒪 (the so-called compatibility condition).

Note that in this case 𝜌𝜀[]𝐶0,𝑇;𝐿2(𝒪),𝜇𝜀[]𝐶0,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(𝒪) can be identified with an element of 𝐻1(𝒪), that is, (𝒪)𝐻1(𝒪). As a result, the integral identity (47) with a source term 𝑢(𝑡,𝑥)=𝑁𝑘=1𝑢𝑘(𝑡)𝛿𝑘(𝑥)(53) can be rewritten as follows:𝑇0𝒪𝜕𝜇𝜀𝜕𝑡𝜑+𝜀𝜕𝜇𝜀𝜕𝑥𝜕𝜑𝜕𝑥+𝜇𝜀𝜕𝜑=𝜕𝑥𝑑𝑥𝑑𝑡𝑁𝑘=1𝑇0𝑢𝑘(𝑡)𝜑𝑡,𝜏𝑘𝑑𝑡,𝜑𝐿20,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=1𝑢𝑘(𝑡)𝛿𝜏𝑘𝒰ad be a given source term with prescribed location 𝑎<𝜏1<<𝜏𝑁<𝑏. Let {(𝜌𝜀,𝜇𝜀)}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 {𝑣𝜀𝐿2(0,𝑇;())}𝜀>0 be a bounded sequence of fictitious controls. Assume that supposition (A1) holds true. Then for every 𝜀>0, 𝑘,𝑙, and for all positive concave functions 𝜑𝐶0((0,𝑇)×𝒪), each of the pairs ((𝜌𝜀,𝜇𝜀)) satisfies the following integral inequalities: 𝑇0𝒪𝜈𝑙(𝜌𝜀)𝜑𝑡+𝑔𝑙(𝜌𝜀)𝜑𝑥+𝑑𝑥𝑑𝑡𝑇0sign(𝜌𝜀(𝑡,)𝑙)𝑣𝜀(𝑡,),𝜑(𝑡,)(𝒪),𝐶0(𝒪)𝑑𝑡,(55)𝑇0𝒪𝜈𝑘(𝜇𝜀)𝜑𝑡𝑞𝑘(𝜇𝜀)𝜑𝑥+𝑑𝑥𝑑𝑡𝑁𝑖=1𝑇0𝜇sign𝜀𝑡,𝜏𝑖𝑢𝑘𝑖(𝑡)𝜑𝑡,𝜏𝑖𝑑𝑡0(56) with 𝜈𝑙(𝜌)=|𝜌𝑙|, 𝑔𝑙(𝜌)=(𝑓1(𝜌)𝑓1(𝑙))sign(𝜌𝑙), 𝜈𝑘(𝜇)=|𝜇𝑘|, and 𝑞𝑘(𝜇)=(𝜇𝑘)sign(𝜇𝑘).

Proof. Let 𝐸=𝐸(𝜌)𝐶2() be any convex function. We multiply (32) by 𝐸(𝜌). Then the equalities 𝐸(𝜌)𝜌𝑡=𝜕𝐸(𝜌(𝑡,𝑥)),𝑓𝜕𝑡1(𝜌)𝐸(𝜌)𝜌𝑥=𝜕𝜕𝑥𝑘𝜌(𝑡,𝑥)𝑓1(𝜉)𝐸,𝐸(𝜉)𝑑𝜉(𝜌)𝜌𝑥𝑥=(𝐸(𝜌))𝑥𝑥𝐸(𝜌)𝜌2𝑥,(57) imply the following relation (𝐸(𝜌𝜀))𝑡+𝜌𝜀𝑘𝑓1(𝜉)𝐸(𝜉)𝑑𝜉𝑥=𝜀(𝐸(𝜌𝜀))𝑥𝑥𝜀𝐸(𝜌𝜀)(𝜌𝜀)2𝑥+𝐸(𝜌𝜀)𝑣𝜀,in𝒟((0,𝑇)×𝒪).(58) By the initial assumptions, for every 𝜀>0 the functions 𝜌𝜀𝒲 can be zero-extended to the domain Ω𝑇=(0,𝑇)×. Now let us multiply equality (58) by a test function 𝜑𝐶0(Ω𝑇) and integrate it over Ω𝑇. Using the integration by parts and the fact that 𝜀>0 and 𝐸(𝜌𝜀)0 a.e. in Ω𝑇, we transfer all derivatives to the test function 𝜑: 𝑇0𝐸(𝜌𝜀)𝜑𝑡+𝜌𝜀𝑘𝑓1(𝜉)𝐸(𝜉)𝑑𝜉𝜑𝑥𝑑𝑥𝑑𝑡=𝜀𝑇0(𝐸(𝜌𝜀))𝜑𝑥𝑥𝑑𝑥𝑑𝑡𝜀𝑇0𝐸(𝜌𝜀)(𝜌𝜀)2𝑥𝜑𝑑𝑥𝑑𝑡+𝑇0𝐸(𝜌𝜀)𝑣𝜀𝜑𝑑𝑥𝑑𝑡𝜀𝑇0𝐸(𝜌𝜀)𝜑𝑥𝑥𝑑𝑥𝑑𝑡+𝑇0𝐸(𝜌𝜀)𝑣𝜀𝜑𝑑𝑥𝑑𝑡.(59) Since 𝜌𝜀𝐿2(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: 𝑇0𝐸(𝜌𝜀)𝑣𝜀𝜑𝑑𝑥𝑑𝑡=𝑇0𝐸(𝜌𝜀)𝑣𝜀,𝜑(),𝐶0()𝑑𝑡.(60) Further we use the well-known trick. Let {𝐸𝑚}𝑚 be a sequence of 𝐶2-functions approximating the function 𝜉|𝜉𝑘| uniformly on . Substitute 𝐸=𝐸𝑚(𝜌) in the inequality (59) and pass to the limit as 𝑚. Note that we can choose 𝐸𝑚 in such way that 𝐸𝑚 is bounded and 𝐸𝑚(𝜉)sign(𝜉𝑘) for all 𝜉, 𝜉𝑘. Since 𝜑𝑥𝑥0 in Ω𝑇 and 𝜌𝜀𝑘𝑓1(𝜉)𝐸𝑚(𝜉)𝑑𝜉𝜌𝜀𝑘𝑓1(𝜉)sign(𝜉𝑘)𝑑𝜉=sign(𝜌𝜀𝑘)𝜌𝜀𝑘𝑓1(𝜉)𝑑𝜉=sign(𝜌𝜀𝑓𝑘)1(𝜌𝜀)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 𝑢𝜀𝒰ad, we analyze the following penalized optimization problem associated with the singular perturbed initial-boundary value problem (32)–(38):𝐼𝜀(𝑣𝜀,𝜌𝜀)=𝑣𝜀2𝐿2(0,𝑇;(𝒪))+𝜀1𝑓2(𝜇𝜀)𝑥𝑣𝜀𝐿2(0,𝑇;𝐻1(𝒪))inf(62)subjecttotheconstraints(33)-(39).(63)

Definition 16. We say that a pair (𝑣𝜀,𝜌𝜀) is admissible to the optimization problem (62)-(63) if 𝑣𝜀𝐿2(0,𝑇;()) and 𝜌𝜀=𝜌𝜀(𝑣𝜀)𝒲 is the corresponding weak solution to the initial boundary value problem (32), (34)1, and (38)1.

Let Ξ𝜀 be the set of all admissible solutions to the perturbed problem (62)-(63). As follows from Theorem 13, for every 𝜀>0, Ξ𝜀 is a nonempty subset of the space:𝒴=𝐿2(0,𝑇;(𝒪))×𝐿2((0,𝑇)×𝒪).(64)

Remark 17. We note that the cost functional (63) is well defined on Ξ𝜀 for every 𝜀>0. Indeed, let (𝑣𝜀,𝜌𝜀) be any representative of Ξ𝜀. By supposition (A1), we have that 𝑓2 is a piecewise linear mapping and 𝜇𝜀𝒲. Hence (𝑓2(𝜇𝜀))𝜇𝜀𝑥 is in 𝐿2((0,𝑇)×𝒪), and 𝑣𝜀𝐿2(0,𝑇;(𝒪)) by the definition of the class 𝒱ad. Since 𝐿2((0,𝑇)×𝒪)𝐿2(0,𝑇;(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;(𝒪)) and the topology of norm in 𝐿2((0,𝑇)×𝒪). Then we have the following topological properties of the set Ξ𝜀 of admissible solutions to the perturbed optimization problem (62)-(63).

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

Proof. For a fixed 𝜀>0 let (𝑢𝜀,𝑣𝜀)𝒰ad×𝒱ad be an arbitrary pair of source terms. Then Theorem 14 implies the existence of a unique pair (𝜌𝜀,𝜇𝜀) such that 𝜌𝜀=𝜌𝜀(𝑣𝜀) and 𝜇𝜀=𝜇𝜀(𝑢𝜀) are the corresponding weak solutions to the initial boundary value problem (32)–(34), (38). Since 𝜇𝜀,𝜌𝜀𝒲=𝑦𝑦𝐿20,𝑇;𝐻10,(𝒪)𝜕𝑦𝜕𝑡𝐿20,𝑇;𝐻1(𝒪)(65) and 𝒲𝐿2((0,𝑇)×𝒪), we conclude that (𝑣𝜀,𝜌𝜀)Ξ𝜀 and hence Ξ𝜀.
To establish the 𝜏-closedness of Ξ𝜀, we fix an arbitrary 𝜏-converging sequence of admissible solutions to the perturbed problem (32)–(38) and (62) {(𝑣𝜀𝑘,𝜌𝜀𝑘)Ξ𝜀}𝑘=1 and show that (𝑣𝜀,𝜌𝜀)Ξ𝜀, where (𝑣𝜀,𝜌𝜀) is its 𝜏-limit.
We have that 𝑣𝜀𝑘𝑣𝜀 in 𝐿2(0,𝑇;(𝒪)) and 𝜌𝜀𝑘𝜌𝜀 in 𝐿2((0,𝑇)×𝒪). Hence 𝑣𝜀𝒱ad and it remains to show that 𝜌𝜀 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 𝜌𝜀 belongs to the space 𝒲 and satisfies conditions: 𝑓1𝜌𝜀𝑘𝑓1𝜌𝜀,in𝐿20,𝑇;𝐿2𝜌(𝒪)as𝑘,𝜀𝑘𝑥𝜌𝜀𝑥,in𝐿20,𝑇;𝐿2𝜌(𝒪)as𝑘,𝜀𝑘𝑡𝜌𝜀𝑡,in𝐿20,𝑇;𝐻1𝜌(𝒪)as𝑘,𝜀𝑘,𝜌𝜀[]𝐶0,𝑇;𝐿2(𝒪),𝜌𝜀𝑘(0,𝑥)=𝜌0(𝑥)in𝒪,𝑘.(66) This enables us to pass to the limit in the integral identity (46) as 𝑘 with 𝜌𝜀=𝜌𝜀𝑘 and 𝑣=𝑣𝜀𝑘, and eo ipso to show that the limit function 𝜌𝜀 is a weak solution to the parabolic problem (32), (34)1, (38)1.
Thus, the pair (𝑣𝜀,𝜌𝜀) 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 𝑢𝜀𝒰ad there exists at least one pair (𝑣𝜀0,𝜌𝜀0)Ξ𝜀 such that 𝐼𝜀𝑣𝜀0,𝜌𝜀0=inf(𝑣𝜀,𝜌𝜀)Ξ𝜀𝐼𝜀(𝑣𝜀,𝜌𝜀),(67) that is, the problem (32)–(38), (62) is solvable.

Proof. Since Ξ𝜀 and the cost functional 𝐼𝜀 is bounded below on Ξ𝜀, it follows that there exists a sequence {(𝑣𝜀𝑘,𝜌𝜀𝑘)}𝑘Ξ𝜀 such that 𝐼𝜀𝑣𝜀𝑘,𝜌𝜀𝑘𝑘𝐼𝜀mininf(𝑣𝜀,𝜌𝜀)Ξ𝜀𝐼𝜀(𝑣𝜀,𝜌𝜀)0,(68) that is, {(𝑣𝜀𝑘,𝜌𝜀𝑘)}𝑘Ξ𝜀 is a minimizing sequence for the problem (32)–(38), (62).
To begin with, we show that for any 𝜆>0 the set Ξ𝜆𝜀=(𝑣𝜀,𝜌𝜀)Ξ𝜀𝐼𝜀(𝑣𝜀,𝜌𝜀)𝜆(69) is bounded in 𝐿2(0,𝑇;(𝒪))×𝒲. Indeed, as follows from inequality (68), the sequence of fictitious controls {𝑣𝜀𝑘}𝑘 is bounded in 𝐿2(0,𝑇;(𝒪)). Hence, we may assume that there exists an element 𝑣𝜀0𝒱ad such that 𝑣𝜀𝑘𝑣𝜀0 in 𝐿2(0,𝑇;(𝒪)) as 𝑘, that is, lim𝑘𝑇0𝑣𝜀𝑘(𝑡,),𝜑𝑀(𝒪),𝐶0(𝒪)=𝜓(𝑡)𝑑𝑡𝑇0𝑣𝜀0(𝑡,),𝜑𝑀(𝒪),𝐶0(𝒪)𝜓(𝑡)𝑑𝑡,𝜑𝐶0(𝒪),𝜓𝐶0(0,𝑇).(70) Then having used the a priori estimate (48), we see that {𝜌𝜀𝑘=𝜌𝜀(𝑣𝜀𝑘)}𝑘 form a uniformly bounded sequence in 𝒲. Hence, we may again assume that, up to a subsequence, there exists an element 𝜌𝜀0𝒲 such that 𝜌𝜀𝑘𝜌𝜀0 weakly in 𝒲 and strongly in 𝐿2((0,𝑇)×𝒪). As a result, (𝑣𝜀0,𝜌𝜀0)Ξ𝜀 by Lemma 18.
Let us show that the 𝜏-limit pair (𝑣𝜀0,𝜌𝜀0) is an optimal solution to the penalized problem (32)–(38), (62). Indeed, taking into account supposition (A1) and Theorem 14, we have 𝑓2(𝜇𝜀)𝑥𝑣𝜀𝑘𝑓2(𝜇𝜀)𝑥𝑣𝜀0,in𝐿2(0,𝑇;(𝒪)).(71) Using the property of lower semi-continuity for 𝐼𝜀 with respect to the 𝜏-topology, we get 0𝐼𝜀𝑣𝜀0,𝜌𝜀0lim𝑘𝐼(𝑣𝜀,𝜌𝜀)=𝐼𝜀min.(72) Thus the pair (𝑢𝜀0,𝜌𝜀0) 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 Ξ𝜀 is embedded in the topological space (𝒴1,𝜎), where𝒴1=𝐿2(0,𝑇;(𝒪))×𝐿20,𝑇;𝐻1,(𝒪)(73) and 𝜎 is the product of the weak- topology of 𝐿2(0,𝑇;(𝒪)) and the strong topology of 𝐿2(0,𝑇;𝐻1(𝒪)). So, we can take 𝜎 as the main topology for the asymptotic analysis.

Lemma 20. Let 𝑢=𝑁𝑘=1𝑢𝑘(𝑡)𝛿𝜏𝑘𝒰ad a given source term with prescribed location 𝑎<𝜏1<<𝜏𝑁<𝑏. Let {𝜇𝜀}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 {(𝑣𝜀0,𝜌𝜀0)Ξ𝜀}𝜀>0 be a sequence of optimal solutions to the penalized problem (32)–(38), (62). Assume that the fictitious controls {𝑣𝜀0}𝜀>0 are bounded in 𝐿2(0,𝑇;(𝒪)) and supposition (A1) holds true. Then subsequences of {𝜇𝜀}𝜀>0 and of {(𝑣𝜀0,𝜌𝜀0)}𝜀>0, still denoted by the suffix 𝜀, can be extracted such that (a)𝑣𝜀0𝑣 in 𝐿2(0,𝑇;(𝒪)); (b)𝜌𝜀0𝜌 and 𝜇𝜀𝜇 weakly in 𝐿2((0,𝑇)×𝒪) and strongly in 𝐿2(0,𝑇;𝐻1(𝒪)); (c)(𝜌,𝜇) is a weak solution in [𝐿2((0,𝑇)×𝒪)]2 of the Cauchy problem:𝜌𝑡+𝑓1(𝜌)𝑥=𝑣,𝜌(0,)=𝜌0,𝜇(74)𝑡𝜇𝑥=𝑢,𝜇(0,)=𝜇0.(75)

Proof. As follows from the a priori estimates (50)-(51), the sequences {𝜌𝜀0}𝜀>0 and {𝜇𝜀}𝜀>0 are bounded in 𝒲1=𝑦𝑦𝐿20,𝑇;𝐿2,(𝒪)𝜕𝑦𝜕𝑡𝐿20,𝑇;𝐻1(𝒪).(76) Hence the compactness properties (a)-(b) of the sequences {𝑣𝜀0}𝜀>0, {𝜌𝜀0}𝜀>0, and {𝜇𝜀}𝜀>0 are a direct consequence of the initial suppositions, the Banach-Alaoglu Theorem, and the compactness embedding 𝒲1𝐿2(0,𝑇;𝐻1(𝒪)). Moreover, as follows from estimates (48)-(49), the sequence {(𝜌𝜀0,𝜇𝜀)}𝜀>0 is bounded in 𝐿2(0,𝑇;𝐿2(𝒪)). So, we can suppose that 𝜌𝜀0𝜌,𝜇𝜀𝜇,as𝜀0(77) strongly in 𝐿2(0,𝑇;𝐻1(𝒪)) and weakly in 𝐿2(0,𝑇;𝐿2(𝒪)). In view of estimates (48)-(49), there are elements 𝜂,̂𝜂𝐿2((0,𝑇)×𝒪) such that, up to subsequences, we have 𝜀𝜌𝜀0𝑥𝜂,𝜀(𝜇𝜀)𝑥̂𝜂in𝐿2((0,𝑇)×𝒪),as𝜀0.(78) In order to verify the item (c), we note that the integral identity (46) leads us to the following relation: 𝑇0𝒪𝜌𝜀0𝜕𝜑+𝜕𝑡𝜀𝜀𝜕𝜌𝜀0𝜕𝑥𝜕𝜑𝜕𝑥𝑓1𝜌𝜀0𝜕𝜑=𝜕𝑥𝑑𝑥𝑑𝑡𝑇0𝑣𝜀0(𝑡,),𝜑(𝑡,)(𝒪),𝐶0(𝒪)𝑑𝑡,(79) which holds true for every 𝜀>0 and any test function 𝜑𝐶0((0,𝑇)×𝒪). Since 𝑣𝜀0𝑣 in 𝐿2(0,𝑇;(𝒪)) as 𝜀0, we can pass to the limit in (79) using the property (77)1-(78)1. As a result, we come to the relation 𝑇0𝒪𝜌𝜕𝜑𝜕𝑡𝑓1𝜌𝜕𝜑=𝜕𝑥𝑑𝑥𝑑𝑡𝑇0𝑣(𝑡,),𝜑(𝑡,)(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 lim𝑡0+1𝑡𝑡0||||𝒪𝜌𝜀0(𝑠,)𝜌0||||𝜓𝑑𝑥𝑑𝑠=0,𝜓𝐶0(𝒪)(81) is valid for every 𝜀>0. So, we can pass to the limit in (81) as 𝜀0 using the weak convergence of 𝜌𝜀0𝜌 in 𝐿2((0,𝑇)×𝒪). As a result, the initial condition for the limit function 𝜌 is satisfied in the following sense: lim𝑡0+1𝑡𝑡0||||𝒪𝜌(𝑠,)𝜌0||||𝜓𝑑𝑥𝑑𝑠=0,𝜓𝐶0(𝒪).(82) Thus, 𝜌𝐿2((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=1𝑢𝑘(𝑡)𝛿𝜏𝑘𝒰ad be a given source term with prescribed location 𝑎<𝜏1<<𝜏𝑁<𝑏. Assume that there exists a sequence of pairs ̂𝑣{(𝜀,̂𝜌𝜀)Ξ𝜀}𝜀>0 satisfying the following relation: limsup𝜀0𝐼𝜀̂𝑣𝜀,̂𝜌𝜀<+.(83) Let {(𝑣𝜀0,𝜌𝜀0)Ξ𝜀}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 {(𝑣𝜀0,𝜌𝜀0)Ξ𝜀}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{(𝜀,̂𝜌𝜀)Ξ𝜀}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{(𝜀,̂𝜌𝜀)Ξ𝜀}𝜀>0 can be constructed as follows: ̂𝑣𝜀=(𝑓2(𝜇𝜀))𝑥 for all 𝑒>0, and (𝜌𝜀,𝜇𝜀) 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 {(𝜇𝜀)𝒲1}𝜀>0 is relatively compact with respect to the strong convergence in 𝐿2(0,𝑇;𝐻1(𝒪)) and the weak convergence in 𝐿2(0,𝑇;𝐿2(𝒪)). So, passing to a subsequence, when the occasion requires, we get 𝜇𝜀𝜇,in𝐿20,𝑇;𝐻1,𝜇(𝒪)𝜀𝜇,in𝐿20,𝑇;𝐿2,(𝒪)(84) where 𝜇𝐿2((0,𝑇)×𝒪) is a weak solution to the Cauchy problem (75). For our further analysis we have to show that 𝑓2(𝜇𝜀)𝑥𝑓2(𝜇)𝑥,in𝐿20,𝑇;𝐻1.(𝒪)(85) Indeed, let 𝜑𝐶0((0,𝑇)×𝒪) be a fixed test function. Then the following estimate holds: ||||𝑇0𝒪𝜇𝜀𝑥||||=||||𝜑𝑑𝑥𝑑𝑡𝑇0𝒪𝜇𝜀𝑡𝜑+𝜇𝜀𝑥𝜑𝑥||||𝑢𝜑𝑑𝑥𝑑𝑡𝑇0|||𝜇𝜀𝑡,𝜑𝐻1(𝒪),𝐻10(𝒪)|||+𝜀𝒪|||𝜀𝜇𝜀𝑥𝜑𝑥|||+|||𝑑𝑥𝑢,𝜑𝐻1(𝒪),𝐻10(𝒪)|||𝑑𝑡by(59),(61)𝐶+𝑢𝐿2(0,𝑇;𝐻1(𝒪))𝜑𝐿2(0,𝑇;𝐻10(𝒪)).(86) Hence the sequence {𝜇𝜀𝑥}𝜀>0 is uniformly bounded in 𝐿2(0,𝑇;𝐻1(𝒪). Therefore, in view of (84), we can suppose that 𝜇𝑥𝐿2(0,𝑇;𝐻1(𝒪)) and 𝜇𝜀𝑥𝜇𝑥,in𝐿20,𝑇;𝐻1.(𝒪)(87) As a result, applying the arguments of Remark 17, we come to the required conclusion (85).
Let ̂𝑣{(𝜀,̂𝜌𝜀)Ξ𝜀}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: 𝑣𝜀02𝐿2(0,𝑇;(𝒪))+𝜀1𝑓2(𝜇𝜀)𝑥𝑣𝜀0𝐿2(0,𝑇;𝐻1(𝒪))𝐼𝜀̂𝑣𝜀,̂𝜌𝜀𝑐,𝜀0,𝜀0.(88) Hence the sequence of optimal fictitious controls {𝑣𝜀0}𝜀>0 is bounded in 𝐿2(0,𝑇;(𝒪)). Therefore, by Lemma 20 the sequence of optimal pairs {(𝑣𝜀0,𝜌𝜀0)Ξ𝜀}𝜀>0 is relatively compact with respect to the 𝜎-topology of 𝐿2(0,𝑇;(𝒪))×𝐿2(0,𝑇;𝐻1(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 𝑓02(𝜇𝜀)𝑥𝑣𝜀0𝐿2(0,𝑇;𝐻1(𝒪))𝜀𝑐,𝜀0,𝜀0.(89) Since (𝑓2(𝜇𝜀))𝑥𝑣𝜀0(𝑓2(𝜇))𝑥𝑣 in 𝐿2(0,𝑇;𝐻1(𝒪)) (see (85)), 𝑣𝜀0𝑣 in 𝐿2(0,𝑇;(𝒪)), and (𝒪)𝐻1(𝒪), we can pass to the limit in (89) as 𝜀0. Then, in view of the lower semicontinuity property, we obtain 𝑓02(𝜇)𝑥𝑣𝐿2(0,𝑇;𝐻1(𝒪))liminf𝜀0𝑓2(𝜇𝜀)𝑥𝑣𝜀0𝐿2(0,𝑇;𝐻1(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 {(𝑣𝜀0,𝜌𝜀0)Ξ𝜀}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).

References

  1. A. Bressan, Hyperbolic Systems of Conservation Laws—The One-dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000.
  2. S. Kruzhkov, “First-order quasilinear equations in several independent variables,” Mathematics of the USSR-Sbornik, vol. 10, pp. 217–243, 1970. View at Google Scholar
  3. P. D. Lax, Hyperbolic System of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, Pa, USA, 1973.
  4. C. D'apice and R. Manzo, “A fluid dynamic model for supply chains,” Networks and Heterogeneous Media, vol. 1, no. 3, pp. 379–398, 2006. View at Google Scholar
  5. E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, Mass, USA, 1984.
  6. M. Garavello and B. Piccoli, Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, 2006.
  7. G. A. Chechkin and A. Yu. Goritsky, “S. N. Kruzhkov’s lectures on first-order quasilinear PDEs,” Analytical and Numerical Aspects of PDEs, pp. 1–67, 2010. View at Google Scholar
  8. S. Ulbrich, Optimal Control of Nonlinear Hyperbolic Conservation Laws with Source Terms, Fakultät für Mathematik,Technische Universität München, 2002.
  9. D. Armbruster, D. Marthaler, and C. Ringhofer, “Kinetic and fluid model hierarchies for supply chains,” SIAM Journal on Multiscale Modeling and Simulation, vol. 2, pp. 43–61, 2004. View at Google Scholar
  10. S. Göttlich, M. Herty, and A. Klar, “Modelling and optimization of supply chains on complex networks,” Communications in Mathematical Sciences, vol. 4, no. 2, pp. 315–330, 2006. View at Google Scholar
  11. C. Kirchner, M. Herty, S. Göttlich, and A. Klar, “Optimal control for continuous supply network models,” Networks and Heterogeneous Media, vol. 1, no. 4, pp. 675–688, 2006. View at Google Scholar
  12. J. L. Lions, Some Methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, Paris, France, 1969.
  13. V. A. Solonnikov, “A priori estimates for equations of second order parabolic type,” Trudy Matematicheskogo Instituta imeni V.A. Steklova, vol. 70, pp. 133–212, 1960. View at Google Scholar