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

Authors: Ciro D'Apice; Peter I. Kogut; Rosanna Manzo
Journal: Journal of Control Science and Engineering
 (2010)

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## 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.


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## Body

## 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 laws(1)ρt+(f(μ,ρ))x=0,(t,x)∈ΩT=(0,T)×R,μt-μx=u(t,x),(t,x)∈ΩT,ρ(0,x)=ρ0(x),μ(0,x)=μ0(x),x∈R.
Throughout this paper we suppose that the structure of the source term u(t,x) is prescribed, namely,(2)u=u(t,x)=∑i=1Nui(t)δτi(x),with-∞<a<τ1<⋯<τN<b<+∞,
where the functions {ui∈L2(0,T)}i=1N 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 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(3)Iε(vε,ρε)=‖vε‖L2(0,T;M(O))2+ε-1‖(f2(με))x-vε‖L2(0,T;H-1(O))⟶inf⁡
subject to the constraints(4)ρtε(t,x)-ερxxε(t,x)+(f1(ρε(t,x)))x=v(t,x),(t,x)∈ΩT,μtε(t,x)-εμxxε(t,x)-μxε(t,x)=u(t,x),(t,x)∈ΩT,ρε(0,x)=ρ0(x),με(0,x)=μ0(x),x∈R,u(t,x)=u1(t)δτ1(x)+u2(t)δτ2(x)+⋯+uN(t)δτN(x),ui∈L2(0,T),∀i={1,…,N},v∈L2(0,T;M(R)),ρε(t,x)=0,με(t,x)=0,on(0,T)×∂O,
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 {(v0ε,ρ0ε)∈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

Leta and b be two fixed constants such that -∞<a<b<+∞. For a given T>0 we set ΩT=(0,T)×ℝ and Ω=(0,T)×(a,b). Let Llocp(ΩT), with 1≤p≤∞, be the locally convex space of all measurable functions g:ΩT→ℝ such that g|(0,T)×K∈Lp((0,T)×K) for all compact sets K⊂ℝ.Letℳ(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 {μk}k∈ℕ converges weakly-* to a measure μ∈ℳ(ℝ) (in symbols μk⇀*μ) if(5)lim⁡k→∞∫Rφdμk=∫Rφdμ,∀φ∈C0(R).A subset𝔐 of ℳ(ℝ) is called to be bounded if for every compact set K⊂ℝ we have(6)sup⁡μ∈M|μ|(K)<+∞,
where |μ| denotes the total variation of μ. The following compactness result for measures is well known.Proposition 1.
Let{μk}k∈ℕ be a bounded sequence of Radon measures on ℝ. Then there exist a subsequence {μkj}j∈ℕ and a Radon measure μ∈ℳ(ℝ) such that μkj⇀*μ.According to the Riesz theory, every Radon measureμ 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(7)|〈μ,f〉|≤C‖f‖C(K),∀f∈C0(R)withsupp⁡f⊆K.
As an example of a Radon measure on ℝ, we consider the following one. Let {ak}k∈ℕ and {bk}k∈ℕ be two sequences in ℝ such that ∑k=1∞|ak|≤C<+∞. Let δc be the Dirac measure located at the point c∈ℝ, that is, this measure is defined as follows(8)〈δc,φ〉=∫Rδc(x)φ(x)dx∶=φ(c),∀φ∈C0(R).
Since(9)|∑k=1∞akφ(bk)|≤(∑k=1∞|ak|)‖φ‖C(R)≤C‖φ‖C(R)
for every continuous function with compact support φ∈C0(ℝ), it follows that the linear form(10)μ*=∑k=1∞akδbk
is continuous on C0(ℝ). Hence μ* is an element of the space of Radon measures ℳ(ℝ).Let𝒪 be a bounded open subset of ℝ. Let f:𝒪→ℝ be an element of L1(𝒪). Define (11)∫O|Df|=sup⁡{∫Ofφ′dx:φ∈C01(O),|φ(x)|≤1forx∈O}.
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 Dsf, singular with respect to the one-dimensional Lebesgue measure ℒ⌊𝒪 restricted to 𝒪, such that (12)Df=f′L⌊O+Dsf.Definition 2.
A functionf∈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(13)‖f‖BV(O)=‖f‖L1(O)+∫O|Df|,BV(𝒪) is a Banach space. The following compactness result for BV-functions is well known.Proposition 3.
The uniformly bounded sets inBV-norm are relatively compact in L1(𝒪), that is, if {fk}k=1∞⊂BV(𝒪) and supk∈ℕ∥fk∥BV(𝒪)<+∞, then there exists a subsequence of {fk}k=1∞ strongly converging in L1(𝒪) to some f∈BV(𝒪).Definition 4.
A sequence{fk}k=1∞⊂BV(𝒪) weakly converges to some f∈BV(𝒪), and we write fk⇀f if and only if the two following conditions hold: fk→f strongly in L1(𝒪), and Dfk⇀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{fk}k=1∞ be a sequence in BV(𝒪) strongly converging to some f in L1(𝒪) and satisfying supk∈ℕ∫𝒪|Dfk|<+∞. Then  (i)
f∈BV(𝒪) and ∫𝒪|Df|≤liminfk→∞∫𝒪|Dfk|;(ii)
fk⇀f in BV(𝒪).


## 3. Statement of Problem and Main Motivation

Let{τk}k=1N⊂ℝ 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:(14)ρt+(f(μ,ρ))x=0,(t,x)∈ΩT,(15)μt-μx=u(t,x),(t,x)∈ΩT,(16)ρ(0,x)=ρ0(x),μ(0,x)=μ0(x),x∈R,
where the source term is subjected to the following constraints:(17)u(t,x)=u1(t)δτ1(x)+u2(t)δτ2(x)+⋯+uN(t)δτN(x),(18)ui∈L2(0,T),∀i={1,…,N}.
Here ui∈L2(0,T) are some external distributed sources located at the corresponding points τi∈(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(19)∫0T‖u(t,⋅)‖M(R)2dt=∫0T(sup⁡∥φ∥C(R)=1φ∈C0(R)〈u(t,⋅),φ(⋅)〉M(R),C0(R))2dt=∫0T(sup⁡∥φ∥C(R)=1φ∈C0(R)∑k=1Nuk(t)∫Rδτk(x)φ(x)dx)2dt≤∫0T(∑k=1Nuk(t))2dt≤N∑k=1N‖uk‖L2(0,T)2.Hence, it is natural to define the following class:(20)Uad={u∈L2(0,T;M(R))∣u=u(t,x)satisfy(25)-(26)forsomea<τ1<…<τN<b}.Definition 6.
Letu∈𝒰ad be a fixed source term. We say that a vector value function Y=[ρμ]∈[L2(0,T;Lloc2(ℝ))]2 is a weak solution to (14)–(16) if the identities
(21)∫0T∫R(Y⊙∂φ∂t+F(Y)⊙∂φ∂x)dxdt+∫0T∑k=1NUk(t)⊙φ(τk)dt=0,lim⁡t→0+1t∫0t|∫RY(t,x)⊙ψ(x)dx-∫RY0⊙ψ(x)dx|R2dt=0
hold true for all C0∞-functions φ:[0,T]×ℝ→ℝ2 and ψ:ℝ→ℝ2 with compact supports in (0,T)×ℝ and ℝ, respectively. Here
(22)Y0=[ρ0μ0],Uk(t)=[0uk(t)],F(Y)=[f(μ,ρ)-μ],
and the symbol ⊙ denotes the tensor product  [a1a2]⊙[b1b2]=[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.
AC1-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
(23)Dη(v)⋅DF(v)=Dq(v),∀v∈R2.
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 theC1-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 {ην: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 {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 u∈𝒰ad can be defined as follows.Definition 9.
Letu∈𝒰ad be a given source term with prescribed location a<τ1<⋯<τN<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
(24)∫0T∫R(νl(ρ)ψt+gl(ρ)ψx)dxdt-∫0T∫Rsign⁡(ρ-l)(f2(μ))xψdxdt≥0,∫0T∫R(νk(μ)φt-qk(μ)φx)dxdt+∑i=1T∫0Tsign⁡(μ(t,τi)-k)ui(t)φ(t,τi)dt≥0
hold true for all positive functions φ,ψ∈C0∞(ΩT), provided that
(25)νk(μ)∶=|μ-k|,qk(μ)∶=(μ-k)sign⁡(μ-k),gl(ρ)∶=(f1(ρ)-f1(l))sign⁡(ρ-l).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
(26)[C([0,T];L1(a,b))∩L∞(Ω)∩L∞(0,T;BV(a,b))]2
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.
Letu∈𝒰ad be a given source term. We say that a vector value function Y=[ρμ]∈[L2(0,T;Lloc2(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 {Yε=[ρεμε]}ε>0⊂[L2(0,T;L2(𝒪))]2 such that  (B1)
ρε⇀ρ and με⇀μ in L2(0,T;L2(𝒪)) as ε→0;(B2)
for any constantsk,l∈ℝ and for all positive concave functions φ∈C0∞((0,T)×𝒪) the entropy inequalities
(27)∫0T∫O(νl(ρε)φt+gl(ρε)φx)dxdt+∫0T〈(f2(με(t,⋅))xsign⁡(ρε(t,⋅)-l)×(f2(με(t,⋅))x,φ(t,⋅)〉M(O),C0(O)dt≥0,(28)∫0T∫O(νk(με)φt-qk(με)φx)dxdt+∑i=1N∫0Tsign⁡(με(t,τi)-k)ui(t)φ(t,τi)dt≥0
it hold true for every ε>0 with νl(ρ)∶=|ρ-l|, gl(ρ)∶=(f1(ρ)-f1(l))sign(ρ-l), νk(μ)∶=|μ-k|, and qk(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 ui∈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:(29)f(μ,ρ)=f1(ρ)+f2(μ),(A1)  the functionf1:ℝ→ℝ is locally Lipschitz, that is,
(30)|f1(ρ1)-f1(ρ2)|≤Lρ|ρ1-ρ2|,∀ρ1,ρ2∈[-Mρ,Mρ],f1(0)=0, and f2′: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:
(31)f(μ,ρ)=f1(ρ)+f2(μ)={ρ,ifρ<μ0,ifρ≥μ}+{0,ifρ<μμ,ifρ≥μ}.
Hence, the fulfilment of Hypothesis (A1) is obvious in this case.Letε 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:(32)ρtε(t,x)-ερxxε(t,x)+(f1(ρε(t,x)))x=v(t,x),(t,x)∈ΩT,(33)μtε(t,x)-εμxxε(t,x)-μxε(t,x)=u(t,x),(t,x)∈ΩT,(34)ρε(0,x)=ρ0(x),με(0,x)=μ0(x),x∈R
subjected to the constraints(35)u(t,x)=u1(t)δτ1(x)+u2(t)δτ2(x)+⋯+uN(t)δτN(x),(36)ui∈L2(0,T),∀i={1,…,N},(37)v∈L2(0,T;M(R)),
where v=v(t,x) is a fictitious control. By 𝒱ad we denote the set of all fictitious controls satisfying conditions (37).Sinceρ0,μ0∈L1(ℝN)∩L∞(ℝN), 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 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):(38)ρε(t,x)=0,με(t,x)=0,on(0,T)×∂O.Since by the initial assumptions the influx-rateu=∑k=1Nuk(t)δk 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:(39)W={y:y∈L2(0,T;H01(O)),∂y∂t∈L2(0,T;H-1(O))},(40)W1={y:y∈L2(0,T;L2(O)),∂y∂t∈L2(0,T;H-1(O))},
equipped with the norm of the graph. Then, the following properties hold true:  (1)
the embeddings𝒲↪L2(0,T;L2(𝒪)), 𝒲1↪L2(0,T;H-1(𝒪)) are compact;(2)
one has the embedding(41)W↪C([0,T];L2(O)),W1↪([0,T];H-1(O)),
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 anyu,v∈𝒲(42)ddt∫Ou(t,x)v(t,x)dx=〈u′(t,⋅),v(t,⋅)〉H-1(O),H01(O)+〈v′(t,⋅),u(t,⋅)〉H-1(O),H01(O);(4)
lety∈L2(0,T;H01(𝒪))∩C([0,T];L2(𝒪)). Then the following density result holds: for any δ>0 there exists Φ∈C∞([0,T];C0∞(𝒪)), such that(43)‖y-Φ‖C([0,T];L2(O))≤δ,‖∇y-∇Φ‖L2((0,T)×O)≤δ.Further we note that by the Friedrichs inequality, we have(44)|∫Oyφ′dx|≤c∫Oy2dx∫O|φ′|2dx≤c1∫O|y′|2dx∫O|φ′|2dx,∀y,φ∈H01(O).
Hence the bilinear form ∫𝕆yφ′dx is bounded on H01(𝒪). Moreover, this form is skew-symmetric by the identity(45)∫Oyφ′dx=∫O(yφ)′dx-∫Oy′φdx=-∫Oy′φdx,∀y,φ∈C0∞(O),
which remains valid for all y,φ∈H01(𝒪) by continuity. Then, we come to the following classical result (see [12, 13]).Theorem 14.
Assume thatμ0∈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 (ρε,με)∈𝒲×𝒲 satisfying the integral identities:
(46)∫0T∫O[∂ρε∂tψ+ε∂ρε∂x∂ψ∂x-f1(ρε)∂ψ∂x]dxdt=∫0T〈v(t,⋅),ψ(t,⋅)〉H-1(O),H01(O)dt,∀ψ∈L2(0,T;H01(O)),(47)∫0T∫O[∂με∂tφ+ε∂με∂x∂φ∂x+με∂φ∂x]dxdt=∫0T〈u(t,⋅),φ(t,⋅)〉H-1(O),H01(O)dt,∀φ∈L2(0,T;H01(O)),
with a priori estimates
(48)∫0T∫O[|ρε(t,x)|2+ε|ρxε(t,x)|2+|ρtε(t,x)|2]dxdt≤C∫0T∫O[|ĝt(t,x)|2+|ĝx(t,x)|2]dxdt+C‖v‖L2(0,T;H-1(O))2,(49)∫0T∫O[|με(t,x)|2+ε|μxε(t,x)|2+|μtε(t,x)|2]dxdt≤C∫0T∫O[|gt(t,x)|2+|gx(t,x)|2]dxdt+C‖u‖L2(0,T;H-1(O))2,(50)∫0T‖ρtε(t,⋅)‖H-1(O)2dt≤C‖ĝ‖W-1,2((0,T)×O)2+C‖v‖L2(0,T;H-1(O))2,(51)∫0T‖μtε(t,⋅)‖H-1(O)2dt≤C‖g‖W-1,2((0,T)×O)2+C‖u‖L2(0,T;H-1(O))2,
where C>0 is a constant independent of ε and g,ĝ∈W1,2((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(52)ρε∈C([0,T];L2(O)),με∈C([0,T];L2(O))
by the embedding (41), and the terms in the right-hand sides of (46)-(47) are well defined, because H01(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 (53)u(t,x)=∑k=1Nuk(t)δk(x)
can be rewritten as follows:(54)∫0T∫O[∂με∂tφ+ε∂με∂x∂φ∂x+με∂φ∂x]dxdt=∑k=1N∫0Tuk(t)φ(t,τk)dt,∀φ∈L2(0,T;H01(O)).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.
Letu*=∑k=1Nuk*(t)δτk∈𝒰ad 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 {vε∈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∈C0∞((0,T)×𝒪), each of the pairs ((ρε,με)) satisfies the following integral inequalities:
(55)∫0T∫O(νl(ρε)φt+gl(ρε)φx)dxdt+∫0T〈sign⁡(ρε(t,⋅)-l)vε(t,⋅),φ(t,⋅)〉M(O),C0(O)dt≥,(56)∫0T∫O(νk(με)φt-qk(με)φx)dxdt+∑i=1N∫0Tsign⁡(με(t,τi)-k)ui*(t)φ(t,τi)dt≥0
with νl(ρ):=|ρ-l|, gl(ρ)∶=(f1(ρ)-f1(l))sign(ρ-l), νk(μ):=|μ-k|, and qk(μ):=(μ-k)sign(μ-k).Proof.
LetE=E(ρ)∈C2(ℝ) be any convex function. We multiply (32) by E′(ρ). Then the equalities
(57)E′(ρ)ρt=∂E(ρ(t,x))∂t,f1′(ρ)E′(ρ)ρx=∂∂x(∫kρ(t,x)f1′(ξ)E′(ξ)dξ),E′(ρ)ρxx=(E(ρ))xx-E′′(ρ)ρx2,
imply the following relation
(58)(E(ρε))t+(∫kρεf1′(ξ)E′(ξ)dξ)x=ε(E(ρε))xx-εE′′(ρε)(ρε)x2+E′(ρε)vε,inD′((0,T)×O).
By the initial assumptions, for every ε>0 the functions ρε∈𝒲 can be zero-extended to the domain ΩT=(0,T)×ℝ. Now let us multiply equality (58) by a test function φ∈C0∞(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:
(59)-∫0T∫R[E(ρε)φt+∫kρεf1′(ξ)E′(ξ)dξφx]dxdt=ε∫0T∫R(E(ρε))φxxdxdt-ε∫0T∫RE′′(ρε)(ρε)x2φdxdt+∫0T∫RE′(ρε)vεφdxdt≤ε∫0T∫RE(ρε)φxxdxdt+∫0T∫RE′(ρε)vεφdxdt.
Since ρε∈L2(0,T;H01(𝒪)) for all ε>0 and H01(𝒪)↪C(𝒪¯) by the classical Sobolev Embedding Theorem, it follows that the following term is well defined:
(60)∫0T∫RE′(ρε)vεφdxdt=∫0T〈E′(ρε)vε,φ〉M(R),C0(R)dt.
Further we use the well-known trick. Let {Em}m∈ℕ be a sequence of C2-functions approximating the function ξ↦|ξ-k| uniformly on ℝ. 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(ξ-k) for all ξ∈ℝ, ξ≠k. Since φxx≤0 in ΩT and
(61)∫kρεf1′(ξ)Em′(ξ)dξ⟶∫kρεf1′(ξ)sign⁡(ξ-k)dξ=sign⁡(ρε-k)∫kρεf1′(ξ)dξ=sign⁡(ρε-k)(f1(ρε)-f1(k)),
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 everyε>0 and a given influx-rate uε∈𝒰ad, we analyze the following penalized optimization problem associated with the singular perturbed initial-boundary value problem (32)–(38):(62)Iε(vε,ρε)=‖vε‖L2(0,T;M(O))2+ε-1‖(f2(με))x-vε‖L2(0,T;H-1(O))⟶inf⁡(63)subjecttotheconstraints(33)–(39).Definition 16.
We say that a pair(vε,ρε)  is admissible to the optimization problem (62)-(63) if vε∈L2(0,T;ℳ(ℝ)) and ρε=ρε(vε)∈𝒲 is the corresponding weak solution to the initial boundary value problem (32), (34)1, and (38)1.LetΞε 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:(64)Y=L2(0,T;M(O))×L2((0,T)×O).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′:ℝ→ℝ  is a piecewise linear mapping and με∈𝒲. Hence (f2′(με))μxε is in L2((0,T)×𝒪), and vε∈L2(0,T;ℳ(𝒪)) by the definition of the class 𝒱ad. Since L2((0,T)×𝒪)⊂L2(0,T;ℳ(𝒪)), we come to the required conclusion.We define theτ-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 Ξε of admissible solutions to the perturbed optimization problem (62)-(63).Lemma 18.
Assume that supposition (A1) holds true. Then the setΞε is nonempty and sequentially τ-closed for every ε>0.Proof.
For a fixedε>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
(65)με,ρε∈W∶={y:y∈L2(0,T;H01(O)),∂y∂t∈L2(0,T;H-1(O))}
and 𝒲↪L2((0,T)×𝒪), we conclude that (vε,ρε)∈Ξε and hence Ξε≠∅.
To establish theτ-closedness of Ξε, we fix an arbitrary τ-converging sequence of admissible solutions to the perturbed problem (32)–(38) and (62) {(vkε,ρkε)∈Ξε}k=1∞ and show that (v*ε,ρ*ε)∈Ξε, where (v*ε,ρ*ε) is its τ-limit.
We have thatvkε⇀*v*ε in L2(0,T;ℳ(𝒪)) and ρkε→ρ*ε in L2((0,T)×𝒪). Hence v*ε∈𝒱ad 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:
(66)f1(ρkε)⟶f1(ρ*ε),inL2(0,T;L2(O))ask⟶∞,(ρkε)x⇀(ρ*ε)x,inL2(0,T;L2(O))ask⟶∞,(ρkε)t⇀(ρ*ε)t,inL2(0,T;H-1(O))ask⟶∞,ρkε,ρ*ε∈C([0,T];L2(O)),ρkε(0,x)=ρ0(x)inO,∀k∈N.
This enables us to pass to the limit in the integral identity (46) as k→∞ with ρε=ρkε and v=vkε, 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 everyε>0 and uε∈𝒰ad there exists at least one pair (v0ε,ρ0ε)∈Ξε such that
(67)Iε(v0ε,ρ0ε)=inf⁡(vε,ρε)∈ΞεIε(vε,ρε),
that is, the problem (32)–(38), (62) is solvable.Proof.
SinceΞε≠∅ and the cost functional Iε is bounded below on Ξε, it follows that there exists a sequence {(vkε,ρkε)}k∈ℕ⊂Ξε such that
(68)Iε(vkε,ρkε)⟶k→∞⁡Iεmin⁡≡inf⁡(vε,ρε)∈ΞεIε(vε,ρε)≥0,
that is, {(vkε,ρkε)}k∈ℕ⊂Ξε is a minimizing sequence for the problem (32)–(38), (62).
To begin with, we show that for anyλ>0 the set
(69)Ξελ={(vε,ρε)∈Ξε:Iε(vε,ρε)≤λ}
is bounded in L2(0,T;ℳ(𝒪))×𝒲. Indeed, as follows from inequality (68), the sequence of fictitious controls {vkε}k∈ℕ is bounded in L2(0,T;ℳ(𝒪)). Hence, we may assume that there exists an element v0ε∈𝒱ad such that vkε⇀*v0ε in L2(0,T;ℳ(𝒪)) as k→∞, that is,
(70)lim⁡k→∞∫0T〈vkε(t,⋅),φ〉M(O),C0(O)ψ(t)dt=∫0T〈v0ε(t,⋅),φ〉M(O),C0(O)ψ(t)dt,∀φ∈C0(O),∀ψ∈C0∞(0,T).
Then having used the a priori estimate (48), we see that {ρkε=ρε(vkε)}k∈ℕ form a uniformly bounded sequence in 𝒲. Hence, we may again assume that, up to a subsequence, there exists an element ρ0ε∈𝒲 such that ρkε⇀ρ0ε weakly in 𝒲 and strongly in L2((0,T)×𝒪). As a result, (v0ε,ρ0ε)∈Ξε by Lemma 18.
Let us show that theτ-limit pair (v0ε,ρ0ε) is an optimal solution to the penalized problem (32)–(38), (62). Indeed, taking into account supposition (A1) and Theorem 14, we have
(71)[(f2(με))x-vkε]⇀*(f2(με))x-v0ε,inL2(0,T;M(O)).
Using the property of lower semi-continuity for Iε with respect to the τ-topology, we get
(72)0≤Iε(v0ε,ρ0ε)≤lim⁡k→∞I(vε,ρε)=Iεmin⁡.
Thus the pair (u0ε,ρ0ε) 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), where(73)Y1=L2(0,T;M(O))×L2(0,T;H-1(O)),
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.
Letu*=∑k=1Nuk*(t)δτk∈𝒰ad 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 {(v0ε,ρ0ε)∈Ξε}ε>0 be a sequence of optimal solutions to the penalized problem (32)–(38), (62). Assume that the fictitious controls {v0ε}ε>0 are bounded in L2(0,T;ℳ(𝒪)) and supposition (A1) holds true. Then subsequences of {με}ε>0 and of {(v0ε,ρ0ε)}ε>0, still denoted by the suffix ε, can be extracted such that  (a)
v0ε⇀*v* in L2(0,T;ℳ(𝒪));(b)
ρ0ε→ρ* and με→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:(74)ρt*+(f1(ρ*))x=v*,ρ*(0,⋅)=ρ0,(75)μt*-μx*=u*,μ*(0,⋅)=μ0.Proof.
As follows from the a priori estimates (50)-(51), the sequences {ρ0ε}ε>0 and {με}ε>0 are bounded in
(76)W1={y:y∈L2(0,T;L2(O)),∂y∂t∈L2(0,T;H-1(O))}.
Hence the compactness properties (a)-(b) of the sequences {v0ε}ε>0, {ρ0ε}ε>0, and {με}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 {(ρ0ε,με)}ε>0 is bounded in L2(0,T;L2(𝒪)). So, we can suppose that
(77)ρ0ε⟶ρ*,με⟶μ*,asε⟶0
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
(78)ε(ρ0ε)x⇀η,ε(με)x⇀η̂inL2((0,T)×O),asε⟶0.
In order to verify the item (c), we note that the integral identity (46) leads us to the following relation:
(79)∫0T∫O[-ρ0ε∂φ∂t+ε(ε∂ρ0ε∂x)∂φ∂x-f1(ρ0ε)∂φ∂x]dxdt=∫0T〈v0ε(t,⋅),φ(t,⋅)〉M(O),C0(O)dt,
which holds true for every ε>0 and any test function φ∈C0∞((0,T)×𝒪). Since v0ε⇀*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
(80)∫0T∫O[-ρ*∂φ∂t-f1(ρ*)∂φ∂x]dxdt=∫0T〈v*(t,⋅),φ(t,⋅)〉M(O),C0(O)dt,
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
(81)lim⁡t→0+1t∫0t|∫O(ρ0ε(s,⋅)-ρ0)ψdx|ds=0,∀ψ∈C0∞(O)
is valid for every ε>0. So, we can pass to the limit in (81) as ε→0 using the weak convergence of ρ0ε→ρ* in L2((0,T)×𝒪). As a result, the initial condition for the limit function ρ* is satisfied in the following sense:
(82)lim⁡t→0+1t∫0t|∫O(ρ*(s,⋅)-ρ0)ψdx|ds=0,∀ψ∈C0∞(O).
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.
Letu=∑k=1Nuk(t)δτk∈𝒰ad be a given source term with prescribed location a<τ1<⋯<τN<b. Assume that there exists a sequence of pairs {(v̂ε,ρ̂ε)∈Ξε}ε>0 satisfying the following relation:
(83)limsup⁡ε→0Iε(v̂ε,ρ̂ε)<+∞.
Let {(v0ε,ρ0ε)∈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 {(v0ε,ρ0ε)∈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{(v̂ε,ρ̂ε)∈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 {(v̂ε,ρ̂ε)∈Ξε}ε>0 can be constructed as follows: v̂ε=(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 Lemma20 indicates, the sequence {(με)∈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
(84)με⟶μ*,inL2(0,T;H-1(O)),με⇀μ*,inL2(0,T;L2(O)),
where μ*∈L2((0,T)×𝒪) is a weak solution to the Cauchy problem (75). For our further analysis we have to show that
(85)(f2(με))x⇀(f2(μ*))x,inL2(0,T;H-1(O)).
Indeed, let φ∈C0∞((0,T)×𝒪) be a fixed test function. Then the following estimate holds:
(86)|∫0T∫Oμxεφdxdt|=|∫0T∫O(μtεφ+μxεφx-uφ)dxdt|≤∫0T(|〈μtε,φ〉H-1(O),H01(O)|+ε∫O|εμxεφx|dx+|〈u,φ〉H-1(O),H01(O)|∫O|εμxεφx|)dt≤by(59),(61)(C+‖u‖L2(0,T;H-1(O)))‖φ‖L2(0,T;H01(O)).
Hence the sequence {μxε}ε>0 is uniformly bounded in L2(0,T;H-1(𝒪). Therefore, in view of (84), we can suppose that μx*∈L2(0,T;H-1(𝒪)) and
(87)μxε⇀*μx*,inL2(0,T;H-1(O)).
As a result, applying the arguments of Remark 17, we come to the required conclusion (85).
Let{(v̂ε,ρ̂ε)∈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:
(88)‖v0ε‖L2(0,T;M(O))2+ε-1‖(f2(με))x-v0ε‖L2(0,T;H-1(O))≤Iε(v̂ε,ρ̂ε)≤c,∀ε∈(0,ε0).
Hence the sequence of optimal fictitious controls {v0ε}ε>0 is bounded in L2(0,T;ℳ(𝒪)). Therefore, by Lemma 20 the sequence of optimal pairs {(v0ε,ρ0ε)∈Ξε}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
(89)0≤‖(f2(με))x-v0ε‖L2(0,T;H-1(O))≤εc,∀ε∈(0,ε0).
Since (f2(με))x-v0ε⇀(f2(μ*))x-v* in L2(0,T;H-1(𝒪)) (see (85)), v0ε⇀*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
(90)0≤‖(f2(μ*))x-v*‖L2(0,T;H-1(O))≤lim⁡inf⁡ε→0‖(f2(με))x-v0ε‖L2(0,T;H-1(O))≤0.
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 {(v0ε,ρ0ε)∈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).



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*Source: 982369-2011-01-26.xml*