
Globally, scholars and practitioners are becoming increasingly interested in determining the interaction between finance and environmental sustainability. However, a few studies have investigated and organized existing information in the context of the green finance of banks in developing countries. The purpose of our study is to find major dimensions of green finance and research gaps from a thorough evaluation of the literature. As a result, existing research on green finance in the banking industry has been evaluated in this paper with a focus on green finance and sustainable development. This study employs the content analysis method and it analyzes and summarizes a total of 53 relevant previous studies in the field of green finance. The findings of this research reveal 21 crucial dimensions of green finance in Bangladesh. The primary green finance products of Bangladeshi banks include green securities, green investments, climate finance, green insurance, green credit, green bonds and green infrastructure. The other factors include environmental performance and green economic growth, energy efficiency, green finance policy and environmental protection and the risk impact of bank policy formulation. The findings of this study will help policymakers to understand the green finance concept and its associated variables, which need to be considered when adopting and implementing green finance.
Citation: Shahinur Rahman, Iqbal Hossain Moral, Mehedi Hassan, Gazi Shakhawat Hossain, Rumana Perveen. 2022: A systematic review of green finance in the banking industry: perspectives from a developing country, Green Finance, 4(3): 347-363. doi: 10.3934/GF.2022017
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Globally, scholars and practitioners are becoming increasingly interested in determining the interaction between finance and environmental sustainability. However, a few studies have investigated and organized existing information in the context of the green finance of banks in developing countries. The purpose of our study is to find major dimensions of green finance and research gaps from a thorough evaluation of the literature. As a result, existing research on green finance in the banking industry has been evaluated in this paper with a focus on green finance and sustainable development. This study employs the content analysis method and it analyzes and summarizes a total of 53 relevant previous studies in the field of green finance. The findings of this research reveal 21 crucial dimensions of green finance in Bangladesh. The primary green finance products of Bangladeshi banks include green securities, green investments, climate finance, green insurance, green credit, green bonds and green infrastructure. The other factors include environmental performance and green economic growth, energy efficiency, green finance policy and environmental protection and the risk impact of bank policy formulation. The findings of this study will help policymakers to understand the green finance concept and its associated variables, which need to be considered when adopting and implementing green finance.
Freeway traffic modeling and management have been intensively investigated in recent decades. It is of great importance to reduce stop-and-go oscillations in traffic control. The traffic flow can be represented by the typical Aw-Rascle-Zhang(ARZ) model [1,2], consisting of a set of nonlinear hyperbolic PDEs that describe the evolution of traffic density and velocity evolution.
In 1984, in the framework of the C1 solution, the stability of homogeneous nonlinear 2×2 systems was obtained using the characteristic method in [3]. In 1999, another method was introduced: the quadratic Lyapunov function. This function was first used to analyze the exponential stability of linear hyperbolic equations in [4]. Then, it was extended to the stability analysis of a nonlinear hyperbolic system under the C1 norms in [5], and sufficient conditions for system stability had been obtained. Prieur et al. studied a class of hyperbolic balance law systems and proved its well-posedness under boundary feedback control; then a sufficient exponential stability condition was derived using operator theory in [6].
Also, for the hyperbolic balance law system, Wang et al. constructed a suitable Lyapunov function in [7], which led to the exponential stability of the equilibrium of the H2 solution. This method was applied to the ARZ traffic flow equation in [8]. In [9], asymptotic spectral analysis was conducted on a 1-D 2×2 constant-coefficients linear hyperbolic equation with proportional feedback control, the spectrum determined the growth condition held, and then the exponential stability of the system was established. For the case of variable coefficients, the exponential stability of the system under the H2 norm was obtained by constructing the weighted Lyapunov function in [10]. Hayat et al. proposed proportional-integral (PI) control and used the Lyapunov method to study the exponential stability and output regulation of closed-loop systems in [11]. In [12], the input delay compensation design of the ARZ traffic flow model based on first-order 2×2 nonlinear hyperbolic systems was studied, and the exponential stability of the closed-loop system under the L2 norm was proved.
The most basic control objective of freeway traffic is to maintain the stability of traffic volume, speed, and density at a steady value and to suppress the oscillation as much as possible. A multi-value cellular automata model under Lagrange coordinates was proposed, and the traffic flow was simulated on the basis of the evolution equation of the model. And it is concluded that the lower the density, the more lanes there are, and the greater the traffic flow in [13].
The common control measures include on-ramp metering, which means that the vehicles entering the expressway are regulated by the traffic lights on the on-ramp, and the variable speed limits (VSL), that is, through variable message sign (VMS), which controls the speed limit of passing vehicles [14]. In [17], the influence of ramp metering control strategy on single-segment road traffic flow was analyzed. Based on the backstepping method, the results showed that this method reduces the stop-and-go waves in congested traffic and shortens the adjustment time. Then, in [18] and [19], the ramp metering strategy was used to design the output feedback controller to ensure that the traffic flow on the two connected roads was stable simultaneously to suppress traffic oscillations. In [20], the variable speed limit control strategy was used to design the controller combined with the backstepping transformation, and the exponential stability of the traffic flow of a single section is achieved using the Lyapunov function. For the problem of traffic congestion on a one-way, two-lane freeway, two different VSLs were applied at the exit boundary, and finally converged to the equilibrium point in finite time [21]. To the authors' knowledge, the latest research progress on the ARZ model indicates that the current control strategies for ARZ traffic flow remain ramp metering [22], variable speed limits [23], or a combination of both [24]. Zhang et al. studied a system of linear hyperbolic equations with constant coefficients, which was derived from the ARZ traffic flow model, with ramp metering and variable speed control as boundary conditions. The proportional control and PI control are designed in [24] and [25], respectively. Based on the Lyapunov function, sufficient conditions for exponential stability under the L2 norm and sufficient conditions for parameters are obtained.
However, to the authors' knowledge, there has been limited research on the exponential stability of variable-coefficient hyperbolic systems under varying steady states concerning the spatial domain x when linearizing the ARZ model, with ramp metering and variable speed limit control serving as boundary conditions. In this paper, the exponential stability of the ARZ traffic flow model based on a 2×2 variable-coefficient hyperbolic system under proportional control is studied.
The contribution is as follows: first, considering that the steady-state values of state quantity density and speed in ARZ traffic flow are variables related to position x, combining ramp metering and variable speed limit as boundary control, a variable-coefficient one-dimensional 2×2 hyperbolic system is obtained. Second, a new Lyapunov function is chosen, where the coefficients of the Lyapunov function are constructed from the solution of a partial differential equation. It is derived that when the feedback parameters satisfy the constraint conditions, the system achieves exponential stability. Third, through numerical simulations, we concluded that the velocity value v(L,t) on the right-hand side of the system converges to the steady-state value v∗(L).
The outline is as follows: In Section 2, the linearized ARZ model and boundary conditions are introduced. First, the Riemann coordinate transformation is defined, and after considering that the system steady-state is variable concerning position x, we obtain a hyperbolic system of equations with variable coefficients in the free/congestion region. For congestion, a proportional feedback controller combining on-ramp metering and variable speed limits is proposed, which is rewritten as an abstract evolution equation. In Section 3, the well-posedness of the closed-loop system is proved using the operator semigroup theory and Sobolev embedding theorem. In Section 4, a strict Lyapunov function is constructed to prove the exponential stability of the system in the L2 norm, and the stability region for the feedback gain value is given. Finally, the numerical simulation is given to illustrate the effectiveness of the developed boundary feedback control.
The macroscopic traffic flow dynamics of the freeway is generally described by the ARZ model:
{∂tρ+∂x(ρv)=0,∂tv+(v−ρp′(ρ))∂xv=V(ρ)−vτ0, | (2.1) |
where the state variable ρ(x,t) is the density of the traffic, and v(x,t) is the speed of the traffic; (x,t)∈[0,L]×[0,∞), x and t represent the position and time, respectively, and τ0 is the relaxation time related to driving behavior. The variable p(ρ) is defined as the traffic pressure, an increasing function of density,
p(ρ)=vf(ρρm)γ, | (2.2) |
and V(ρ) represents the equilibrium speed curve and satisfies
V(ρ)=vf−p(ρ)=vf(1−(ρρm)γ), | (2.3) |
where vf is the free flow velocity, ρm is the maximum density, and γ>0 is generally a constant of about 1 (see [2] and its references for a detailed description of the model).
We define new variables (w,z) in Riemann coordinates, let
{w=v+vf(ρρm)γ,z=v. | (2.4) |
Then the ARZ model (2.1) can be described under the Riemann coordinate as
{∂tw+z∂xw=vf−wτ0,∂tz+[(1+γ)z−γw]∂xz=vf−wτ0. | (2.5) |
To obtain the linearized ARZ model, assume (ρ∗(x),v∗(x)) and (w∗(x),z∗(x)) are respectively steady states of system (2.1) and system (2.4), and satisfy z∗(x)=v∗(x),w∗(x)=v∗(x)+vf(ρ∗(x)ρm)γ. Furthermore, (w∗(x),z∗(x)) satisfy the system (2.5) such that we have
{z∗(x)∂xw∗(x)=vf−w∗(x)τ0,[(1+γ)z∗(x)−γw∗(x)]∂xz∗(x)=vf−w∗(x)τ0. | (2.6) |
Define the deviations of the state (w,z) with respect to the steady state (w∗(x),z∗(x)) as:
{˜w=w−w∗(x),˜z=z−z∗(x), | (2.7) |
therefore, the linearized ARZ model can be obtained from a 2×2 hyperbolic system with variable coefficients:
{∂t˜w+λ1(x)∂x˜w+δ˜w=0,∂t˜z−λ2(x)∂x˜z+δ˜w=0, | (2.8) |
where
λ1(x)=z∗(x)=v∗(x)>0,λ2(x)=−(1+γ)z∗(x)+γw∗(x),δ=1τ0>0. | (2.9) |
Moreover, by (2.4) and (2.9), it has
λ2(x)=−(1+γ)z∗(x)+γw∗(x)=−v∗(x)+γvf(ρ∗(x)ρm)γ. | (2.10) |
It is obvious that λ2(x) can be positive or negative. Therefore, the speed-density relationship diagram can be divided into two parts:
Free-flow regime: λ2(x)<0, that is, v∗(x)>γvf(ρ∗(x)ρm)γ. The speed information of the linearized ARZ model of (2.8) is transmitted from the left boundary x=0 to the right boundary x=L.
Congested regime: λ2(x)>0, that is, v∗(x)<γvf(ρ∗(x)ρm)γ. The speed information of the linearized ARZ model of (2.8) is transmitted from the right boundary x=L to the left boundary x=0. So, the hetero-directional propagations of traffic flow might lead to the shock waves of stop-and-go traffic.
In this paper, we focus on the controller design for the congested regime.
To regulate freeway traffic, we designed the on-ramp metering controller r(t) and the variable speed limit controller v(L,t), based on the regimes in which traffic lies, as shown in Figure 1.
On-ramp metering: we should regulate the upstream on-ramp flow rate r(t) based on the measurements collected from the downstream boundary x=L:
r(t)=r∗+kρ(ρ(L,t)−ρ∗(L)). | (2.11) |
Variable speed limit: As the traffic lies in the congestion regime, the characteristic velocity of speed propagating is from downstream to upstream, we should regulate the downstream speed v(L,t) based on the measurement v(0,t) at the upstream boundary, i.e.,
v(L,t)=v∗(L)+kv(v(0,t)−v∗(0)), | (2.12) |
where kρ,kv are the feedback gains, and r∗ is the normal regulation rate of the on-ramp.
Let ˜ρ=ρ−ρ∗(x),˜v=v−v∗(x); then (2.11) and (2.12) become
{r(t)=r∗+kρ˜ρ(L,t),˜v(L,t)=kv˜v(0,t). | (2.13) |
Assume that the conservation conditions satisfied by the traffic flow at the upstream entrance boundary and the conservation conditions at the steady state of the traffic flow are, respectively,
pin+r(t)=ρ(0,t)v(0,t), | (2.14) |
pin+r∗=ρ∗(0)v∗(0), | (2.15) |
where pin is the traffic demand of the mainline.
Combining (2.13)–(2.15), we have
kρ˜ρ(L,t)=ρ(0,t)v(0,t)−ρ∗(0)v∗(0), |
and linearizing by the first-order Taylor formula of a binary function, we have the following linearized boundary condition:
kρ˜ρ(L,t)=v∗(0)˜ρ(0,t)+ρ∗(0)˜v(0,t). | (2.16) |
Furthermore, assume that γ=1 and let α=vfρm; we could rewrite the boundary conditions for system (2.8) in the Riemann coordinates as:
˜w(0,t)=˜v(0,t)+α˜ρ(0,t)=(1−αρ∗(0)v∗(0))˜v(0,t)+αkρv∗(0)˜ρ(L,t)=kρv∗(0)˜w(L,t)+(1−αρ∗(0)v∗(0)−kρkvv∗(0))˜z(0,t)=k1˜w(L,t)+k2˜z(0,t), | (2.17) |
and
˜z(L,t)=k3˜z(0,t), | (2.18) |
where
k1=kρv∗(0),k2=1−αρ∗(0)v∗(0)−kρkvv∗(0),k3=kv. | (2.19) |
Therefore, we have a PDE system under a proportional controller
{∂t˜w+λ1(x)∂x˜w+δ˜w=0,∂t˜z−λ2(x)∂x˜z+δ˜w=0,˜w(0,t)=k1˜w(L,t)+k2˜z(0,t),˜z(L,t)=k3˜z(0,t). | (2.20) |
Without loss of generality, let L=1 for convenience. Assume that the Hilbert state space
H=L2(0,1)×L2(0,1), | (2.21) |
equipped with the following inner product
⟨X1,X2⟩=∫10[f1(x)¯f2(x)+g1(x)¯g2(x)]dx, | (2.22) |
where Xi=(fi,gi)∈H(i=1,2), and ˉf is the conjugate of f. Moreover, the norm of Xi is induced by the inner product
∥Xi∥2=∫10[|fi(x)|2+|gi(x)|2]dx,i=1,2. | (2.23) |
Define linear operator A:D(A)⊆H→H by
AX=(−λ1(x)∂∂x−δ0−δλ2(x)∂∂x)(fg), | (2.24) |
D(A)={(f,g)∈(H1(0,1))2∣f(0)=k1f(1)+k2g(0),g(1)=k3g(0)}. | (2.25) |
Then system (2.20) can be written as an abstract evolution equation in H
{˙X(t)=AX(t),t>0,X(0)=X0, | (2.26) |
where X(t)=(w(⋅,t),z(⋅,t)).
Theorem 3.1. Let A be given by (2.24) and (2.25). Then A−1 exists and is compact, if the feedback parameters k1, k2, and k3 satisfy
(k3−1)(1−k1e−∫10δλ1(s)ds)−k2∫10δλ2(s)e−∫s0δλ1(σ)dσds≠0. |
Hence, σ(A), the spectrum of A, consists of isolated eigenvalues of finite algebraic multiplicity only.
Proof. For X1=(f1,g1)∈H, solve
AX=X1,X=(f,g)∈D(A), | (3.1) |
we can obtain
A(fg)=(−λ1(x)f′−δfλ2(x)g′−δf)=(f1g1), | (3.2) |
i.e.,
{λ1(x)f′+δf+f1=0,λ2(x)g′−δf−g1=0,f(0)=k1f(1)+k2g(0),g(1)=k3g(0). | (3.3) |
Solving the first differential equation of (3.3), we have
f(x)=e−∫x0δλ1(s)ds[−∫x0f1(s)λ1(s)e∫s0δλ1(σ)dσds+f(0)]. | (3.4) |
Let F(x)=−e−∫x0δλ1(s)ds∫x0f1(s)λ1(s)e∫s0δλ1(σ)dσds, we can get
f(x)=F(x)+e−∫x0δλ1(s)dsf(0). | (3.5) |
Combining with the second equation of (3.3), we have
g′(x)=δλ2(x)f(x)+g1(x)λ2(x). | (3.6) |
Integrating both sides of (3.6) yields the following
g(x)=g(0)+∫x0δλ2(s)f(s)ds+∫x0g1(s)λ2(s)ds=g(0)+∫x0δλ2(s)[F(s)+e−∫s0δλ1(σ)dσf(0)]ds+∫x0g1(s)λ2(s)ds=g(0)+∫x0δλ2(s)F(s)ds+f(0)∫x0δλ2(s)e−∫s0δλ1(σ)dσds+∫x0g1(s)λ2(s)ds. | (3.7) |
According to the third equation of (3.3), (3.5), and (3.7), we have
(1−k1e−∫10δλ1(s)ds)f(0)=k2g(0)+k1F(1). | (3.8) |
Similarly, according to the fourth equation of (3.3), (3.5), and (3.7), we have
∫10δλ2(s)e−∫s0δλ1(σ)dσdsf(0)=(k3−1)g(0)−∫10δλ2(s)F(s)ds−∫10g1(s)λ2(s)ds. | (3.9) |
Combining (3.8) and (3.9), we obtain
((k3−1)(1−k1e−∫10δλ1(s)ds)−k2∫10δλ2(s)e−∫s0δλ1(σ)dσds)f(0)=k1(k3−1)F(1)+k2[∫10δλ2(s)F(s)ds+∫10g1(s)λ2(s)ds]. | (3.10) |
Let
{M1=k1(k3−1)F(1)+k2[∫10δλ2(s)F(s)ds+∫10g1(s)λ2(s)ds],M2=(k3−1)(1−k1e−∫10δλ1(s)ds)−k2∫10δλ2(s)e−∫s0δλ1(σ)dσds, | (3.11) |
where M2≠0. We can obtain f(0)=M1M2,g(0)=1k2M1M2−k1k2f(1). So, we have the expressions of f(x) and g(x):
{f(x)=−e−∫x0δλ1(s)ds∫x0f1(s)λ1(s)e∫s0δλ1(σ)dσds+M1M2e−∫x0δλ1(s)ds,g(x)=1k2M1M2−k1k2f(1)+∫x0δλ2(s)f(s)ds+∫x0g1(s)λ2(s)ds. | (3.12) |
Hence, A−1 exists and is compact by the Sobolev embedding theorem. Therefore, σ(A) consists only of isolated eigenvalues of finite algebraic multiplicity.
In this section, we focus on constructing an appropriate Lyapunov function to analyze the exponential stability of system (2.20).
Definition 4.1. The closed-loop system (2.20) is exponentially stable (in L2 norm) if there exist ϑ>0 and C>0 such that, for every initial condition (˜w0(x),˜z0(x))∈L2((0,L);R2), the system solution to the Cauchy problem (2.20) satisfies
‖˜w(⋅,t),˜z(⋅,t)‖L2((0,L);R2)≤Ce−ϑt‖˜w0,˜z0‖L2((0,L);R2). |
Lemma 4.2. The function η(x) defined by
η(x)=1−∫x0δλ2(s)ds+1+∫10δλ2(s)ds | (4.1) |
is a solution of the differential equation
η′(x)=δλ2(x)η2(x), | (4.2) |
where λ2(x),δ are given by (2.9), respectively.
Remark 1. It is obvious that the function η(x) satisfies η(x)>0,η′(x)>0,∀x∈[0,1], and η(x) is bounded on [0,1].
Theorem 4.3. The nonlinear ARZ systems (2.1), (2.11), and (2.12) are exponentially stable for the L2-norm provided that the boundary conditions satisfy
k21≤12b,2k22b+k23≤1b, | (4.3) |
where k1,k2, and k3 are feedback parameters,
b=1+∫10δλ2(s)ds. | (4.4) |
Proof. We construct the following candidate Lyapunov function:
V(t)=∫10[p1(x)˜w2(x,t)+p2(x)˜z2(x,t)]dx, | (4.5) |
where functions p1(x)∈C1([0,L];(0,+∞)) and p2(x)∈C1([0,L];(0,+∞)) are to be determined. Along the solution of (2.20), combining the integral formula of the distribution, the derivative of time of V(t) can be obtained:
˙V(t)=2∫10[p1(x)˜w∂t˜w+p2(x)˜z∂t˜z]dx=2∫10[p1(x)(−λ1(x)∂x˜w−δ˜w)˜w+p2(x)(λ2(x)∂x˜z−δ˜w)˜z]dx=2∫10[−p1(x)λ1(x)˜w∂x˜w+p2(x)λ2(x)˜z∂x˜z]dx−2∫10[δp1(x)˜w2+δp2(x)˜w˜z]dx=−∫10p1(x)λ1(x)d˜w2+∫10p2(x)λ2(x)d˜z2−2∫10[δp1(x)˜w2+δp2(x)˜w˜z]dx=−p1(1)λ1(1)˜w2(1,t)+p2(1)λ2(1)˜z2(1,t)+p1(0)λ1(0)˜w2(0,t)−p2(0)λ2(0)˜z2(0,t)+∫10(p1(x)λ1(x))x˜w2dx−∫10(p2(x)λ2(x))x˜z2dx−2∫10[δp1(x)˜w2+δp2(x)˜w˜z]dx≐−V1−V2, | (4.6) |
where
V1=∫10(˜w,˜z)Λ(˜w˜z)dx,Λ=(−(p1(x)λ1(x))x+2δp1(x)δp2(x)δp2(x)p2(x)λ2(x)), | (4.7) |
V2=p1(1)λ1(1)˜w2(1,t)−p2(1)λ2(1)˜z2(1,t)−p1(0)λ1(0)˜w2(0,t)+p2(0)λ2(0)˜z2(0,t). | (4.8) |
It is obvious that the exponential stability is guaranteed if V1 and V2 are positive definite quadratic forms.
For ∀x∈[0,1], we assume
p1(x)=1λ1(x)η(x),p2(x)=η(x)λ2(x). | (4.9) |
It is easy to verify that
−(p1(x)λ1(x))x+2δp1(x)=η′(x)η2(x)+2δλ1(x)η(x)>0,(p2(x)λ2(x))x=η′(x)>0,[−(p1(x)λ1(x))x+2δp1(x)](p2(x)λ2(x))x−δ2p22(x)=2δη′(x)λ1(x)η(x)>0. | (4.10) |
Then in this case, by continuity, V1 is positive.
Next, we consider the boundary term V2. Substituting the boundary conditions ˜w(0,t)=k1˜w(1,t)+k2˜z(0,t),˜z(1,t)=k3˜z(0,t) into V2, we have
V2=(p1(1)λ1(1)−2k21p1(0)λ1(0))˜w2(1,t)+(p2(0)λ2(0)−k23p2(1)λ2(1)−2k22p1(0)λ1(0))˜z2(0,t)=(1η(1)−2k211η(0))˜w2(1,t)+(η(0)−k23η(1)−2k221η(0))˜z2(0,t)=(1−2(1+∫10δλ2(s)ds)k21)˜w2(1,t)+(11+∫10δλ2(s)ds−k23−2(1+∫10δλ2(s)ds)k22)˜z2(0,t)=(1−2bk21)˜w2(1,t)+(1b−k23−2bk22)˜z2(0,t), | (4.11) |
where b is defined in (4.4). If the system parameters satisfy condition (4.3), we have V2>0.
Therefore, there must exist a positive constant c such that ˙V(t)<−cV(t).
Remark 2. Combining Eq (2.19), it can be concluded that when the feedback gains kρ and kv in controllers (2.11) and (2.12) satisfy the following conditions, the nonlinear ARZ systems (2.1), (2.11), and (2.12) are exponentially stable for the L2-norm
k2ρ≤v∗2(0)2b,2(β−kρkvv∗(0))2b+k2v≤1b, | (4.12) |
where b is defined in (4.4), β=1−αρ∗(0)v∗(0).
In this section, we illustrate Theorem 4.3 using MATLAB numerical simulations. Without loss of generality and for computational convenience. The road parameters are shown in Table 1.
Road length | Relaxation time | Maximum density | Free flow velocity |
L=1km | τ0=120h | ρm=150Vehicle/km | vf=150km/h |
The steady-state is chosen with initial conditions prescribed as v∗(0)=60km/h, ρ∗(0)=90Vehicle/km, and the non-uniform steady-state is v∗(x)=60+5x, ρ∗(x)=90−10x. Based on Table 1, the initial steady-state and the constraint condition (4.3) of Theorem 4.3, we can obtain the values of the control parameters as follows: k1=−0.5, k2=0.3, k3=0.4. Combined with equation (4.11), we take the feedback gain kρ=−30, kv=0.4.
In Figure 2, Figures 2(a) and 2(b) respectively represent the changes of disturbance state variables ˜w and ˜z over time and space, where the initial conditions are ˜w(x,0)=cos(2πx)+2sin(2πx) and ˜z(x,0)=5cos(2πx)+sin(2πx)−3x. It is evident from Figure 2(a) that the state variable ˜w exhibits significant fluctuations above and below zero at the initial moment but eventually converges to zero. Similarly, Figure 2(b) shows that the state variable ˜z also initially exhibits significant fluctuations above and below zero but similarly converges to zero. Figure 3 shows that the velocity value v(L,t) at the right boundary (represented by the red line) experiences large fluctuations at the initial moment but ultimately converges to its steady-state value v∗(L) (represented by the blue line).
This paper considers the stability of the linearized variable-coefficient ARZ equation with boundary control. The proportional boundary feedback controller that combined ramp metering and variable speed limits was designed to regulate stop-and-go traffic flow oscillations caused by congestion. Constructing a suitable Lyapunov function, it has been proven that the system is exponentially stable when the feedback parameters satisfy certain constraints. Numerical simulations demonstrate the effectiveness of the proposed boundary control and the feasibility of the selected parameters. In future work, we attempt to study the eigenvalue problem AX=μX and obtain the distribution of eigenvalues. The spectrum-determined growth condition and Riesz basis property are also of interest.
All authors contributed equally to this work. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported by the National Natural Science Foundation of China under Grant No.12001343.
The authors declare that they have no conflict of interest.
[1] |
Akomea-Frimpong I, Adeabah D, Ofosu D, et al. (2021) A review of studies on green finance of banks, research gaps and future directions. J Sust Financ Investment, 1–24. https://doi.org/10.1080/20430795.2020.1870202 doi: 10.1080/20430795.2020.1870202
![]() |
[2] |
Al-Qudah AA, Hamdan A, Al-Okaily M, et al. (2022) The impact of green lending on credit risk: evidence from UAE's banks. Environ Sci Pollut Res, 1–24. https://doi.org/10.1007/s11356-021-18224-5 doi: 10.1007/s11356-021-18224-5
![]() |
[3] |
Amidjaya PG, Widagdo AK (2020) Sustainability reporting in Indonesian listed banks: Do corporate governance, ownership structure and digital banking matter? J Appl Account Res 21: 231–247. https://doi.org/10.1108/JAAR-09-2018-0149 doi: 10.1108/JAAR-09-2018-0149
![]() |
[4] |
Amighini A, Giudici P, Ruet J (2022) Green finance: An empirical analysis of the Green Climate Fund portfolio structure. J Clean Prod 350: 131383. https://doi.org/10.1016/J.JCLEPRO.2022.131383 doi: 10.1016/j.jclepro.2022.131383
![]() |
[5] | Barua S, Aziz S (2022) Making green finance work for the sustainable energy transition in emerging economies. Energy-Growth Nexus in an Era of Globalization, Elsevier, 353–382. https://doi.org/10.1016/b978-0-12-824440-1.00014-x |
[6] | Berensmann K, Volz U, Bak C, et al. (2020) Fostering Sustainable Global Growth Through Green Finance – What Role for The G20? G20-Insights. Org 1–8. Available from: https://collaboration.worldbank.org/content/usergenerated/asi/cloud/attachments/sites/collaboration-for-development/en/groups/green-finance-community-of-practice/documents/jcr:content/content/primary/blog/green_finance_educat-LVNC/Climate_Green-Finance_V2 |
[7] | Bhatia A, Ey YLLP, Technology C, et al. (2017) Governance for Green Growth in Bangladesh: Policies, Institutions, and Political Economy. Econ Dialogue Green Growth (EDGG) 1: 1–65. https://www.greengrowthknowledge.org/sites/default/files/downloads/resource/Governance for Green Growth in Bangladesh_Policies, Institutions, and Political Economy_0.pdf |
[8] |
Chen J, Siddik AB, Zheng GW, et al. (2022) The Effect of Green Banking Practices on Banks' Environmental Performance and Green Financing: An Empirical Study. Energies 15: 1292. https://doi.org/10.3390/en15041292 doi: 10.3390/en15041292
![]() |
[9] |
Chen Y, Cheng L, Lee CC, et al. (2021) The impact of regional banks on environmental pollution: Evidence from China's city commercial banks. Energy Econ 102: 105492. https://doi.org/10.1016/j.eneco.2021.105492 doi: 10.1016/j.eneco.2021.105492
![]() |
[10] |
Debrah C, Chan APC, Darko A (2022) Green finance gap in green buildings: A scoping review and future research needs. Build Environ 207: 108443. https://doi.org/10.1016/j.buildenv.2021.108443 doi: 10.1016/j.buildenv.2021.108443
![]() |
[11] |
Díaz-García C, González-Moreno Á, Sáez-Martínez FJ (2015) Eco-innovation: Insights from a literature review. Innovation Manage Policy Pract 17: 6–23. https://doi.org/10.1080/14479338.2015.1011060 doi: 10.1080/14479338.2015.1011060
![]() |
[12] | Dikau S, Volz U (2018) Central Banking, Climate Change and Green Finance. |
[13] |
Dikau S, Volz U (2021) Central bank mandates, sustainability objectives and the promotion of green finance. Ecol Econ 184: 107022. https://doi.org/10.1016/j.ecolecon.2021.107022 doi: 10.1016/j.ecolecon.2021.107022
![]() |
[14] |
Dörry S, Schulz C (2018) Green financing, interrupted. Potential directions for sustainable finance in Luxembourg. Local Environ 23: 717–733. https://doi.org/10.1080/13549839.2018.1428792 doi: 10.1080/13549839.2018.1428792
![]() |
[15] |
Durrani A, Rosmin M, Volz U (2020) The role of central banks in scaling up sustainable finance–what do monetary authorities in the Asia-Pacific region think? J Sust Financ Investment 10: 92–112. https://doi.org/10.1080/20430795.2020.1715095 doi: 10.1080/20430795.2020.1715095
![]() |
[16] |
Falagas ME, Pitsouni EI, Malietzis GA, et al. (2008) Comparison of PubMed, Scopus, Web of Science, and Google Scholar: strengths and weaknesses. FASEB J 22: 338–342. https://doi.org/10.1096/fj.07-9492lsf doi: 10.1096/fj.07-9492LSF
![]() |
[17] |
Fedorova EP (2020) Role of the State in the Resolution of Green Finance Development Issues. Financ J 12: 37–51. https://doi.org/10.31107/2075-1990-2020-4-37-51 doi: 10.31107/2075-1990-2020-4-37-51
![]() |
[18] |
Giudici P (2018) Fintech Risk Management: A Research Challenge for Artificial Intelligence in Finance. Frontiers in Artif Intell 1: 1. https://doi.org/10.3389/frai.2018.00001 doi: 10.3389/frai.2018.00001
![]() |
[19] |
Giudici P, Hadji-Misheva B, Spelta A (2020) Network based credit risk models. Qual Eng 32: 199–211. https://doi.org/10.1080/08982112.2019.1655159 doi: 10.1080/08982112.2019.1655159
![]() |
[20] |
Gunningham N (2020) A Quiet Revolution: Central Banks, Financial Regulators, and Climate Finance. Sustainability 12: 9596. https://doi.org/10.3390/SU12229596 doi: 10.3390/su12229596
![]() |
[21] |
Gusenbauer M, Haddaway NR (2020) Which academic search systems are suitable for systematic reviews or meta-analyses? Evaluating retrieval qualities of Google Scholar, PubMed, and 26 other resources. Res Synth Methods 11: 181–217. https://doi.org/10.1002/JRSM.1378 doi: 10.1002/jrsm.1378
![]() |
[22] | Haque MS, Murtaz M (2018) Green Financing in Bangladesh. International Conference on Finance for Sustainable Growth and Development, 82–89. Available from: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=+Haque%2C+M.S.%3B+Murtaz%2C+M.+Green+Financing+in+Bangladesh.+In+Proceedings+of+the+International+Conference+on+Finance+for+Sustainable+Growth+and+Development%2C+Chittagong%2C+Bangladesh&btnG= |
[23] | Hasan HR, Bakkar SA, Akter A (2019) Corporate Bond Market: The Case of Bangladesh. World Rev Bus Res 9: 20–38. https://zantworldpress.com/wp-content/uploads/2019/04/9.-Afia.pdf |
[24] |
He L, Liu R, Zhong Z, et al. (2019) Can green financial development promote renewable energy investment efficiency? A consideration of bank credit. Renewable Energy 143: 974–984. https://doi.org/10.1016/j.renene.2019.05.059 doi: 10.1016/j.renene.2019.05.059
![]() |
[25] | Hossain, M (2018) Green Finance in Bangladesh: Policies, Institutions, and Challenges. ADBI Working Paper Series 892: 1–24. https://www.econstor.eu/handle/10419/190313 |
[26] |
Islam MS, Das PC (2013). Green Banking practices in Bangladesh. IOSR J Bus Manage 8: 39–44. https://doi.org/10.9790/487x-0833944 doi: 10.9790/487X-0833944
![]() |
[27] |
Jin Y, Gao X, Wang M (2021) The financing efficiency of listed energy conservation and environmental protection firms: Evidence and implications for green finance in China. Energy Policy, 153: 112254. https://doi.org/10.1016/j.enpol.2021.112254 doi: 10.1016/j.enpol.2021.112254
![]() |
[28] |
Julia T, Kassim S (2020) Exploring green banking performance of Islamic banks vs conventional banks in Bangladesh based on Maqasid Shariah framework. J Islamic Mark 11: 729–744. https://doi.org/10.1108/JIMA-10-2017-0105 doi: 10.1108/JIMA-10-2017-0105
![]() |
[29] |
Khairunnessa F, Vazquez-Brust DA, Yakovleva N (2021) A review of the recent developments of green banking in bangladesh. Sustainability (Switzerland) 13: 1–21. https://doi.org/10.3390/su13041904 doi: 10.3390/su13041904
![]() |
[30] |
Khatun MN, Sarker MNI, Mitra S (2021) Green Banking and Sustainable Development in Bangladesh. Sust Clim Change 14: 262–271. Mary Ann Liebert Inc. https://doi.org/10.1089/scc.2020.0065 doi: 10.1089/scc.2020.0065
![]() |
[31] |
Lee CC, Lee CC (2022) How does green finance affect green total factor productivity? Evidence from China. Energy Econ 107: 105863. https://doi.org/10.1016/j.eneco.2022.105863 doi: 10.1016/j.eneco.2022.105863
![]() |
[32] |
Li Z, Kuo TH, Siao-Yun W, et al. (2022) Role of green finance, volatility and risk in promoting the investments in Renewable Energy Resources in the post-covid-19. Resour Policy 76: 102563. https://doi.org/10.1016/j.resourpol.2022.102563 doi: 10.1016/j.resourpol.2022.102563
![]() |
[33] |
Li Z, Liao G, Wang Z, et al. (2018) Green loan and subsidy for promoting clean production innovation. J Clean Prod 187: 421–431. https://doi.org/10.1016/j.jclepro.2018.03.066 doi: 10.1016/j.jclepro.2018.03.066
![]() |
[34] | Lindenberg N (2014) Definition of Green Finance. German Development Institute, 3. Available from: https://scholar.archive.org/work/tgtjvkykqrfkpek5wcyvtlauue/access/wayback/https://www.die-gdi.de/uploads/media/Lindenberg_Definition_green_finance.pdf |
[35] |
Liu H, Yao P, Latif S, et al. (2022) Impact of Green financing, FinTech, and financial inclusion on energy efficiency. Environ Sci Pollut Res 29: 18955–18966. https://doi.org/10.1007/s11356-021-16949-x doi: 10.1007/s11356-021-16949-x
![]() |
[36] |
Liu N, Liu C, Xia Y, et al. (2020) Examining the coordination between green finance and green economy aiming for sustainable development: A case study of China. Sustainability (Switzerland), 12. https://doi.org/10.3390/su12093717 doi: 10.3390/su12093717
![]() |
[37] |
Liu R, Wang D, Zhang L, et al.(2019) Can green financial development promote regional ecological efficiency? A case study of China. Nat Hazards 95: 325–341. https://doi.org/10.1007/s11069-018-3502-x doi: 10.1007/s11069-018-3502-x
![]() |
[38] |
Macpherson M, Gasperini A, Bosco M (2021) Artificial Intelligence and FinTech Technologies for ESG Data and Analysis. SSRN Electronic J. https://doi.org/10.2139/ssrn.3790774 doi: 10.2139/ssrn.3790774
![]() |
[39] |
Mengze H, Wei L (2015) A comparative study on environment credit risk management of commercial banks in the Asia-Pacific Region. Bus Strategy Environ 24: 159–174. https://doi.org/10.1002/bse.1810 doi: 10.1002/bse.1810
![]() |
[40] |
Mohd S, Kaushal VK (2018) Green Finance: A Step towards Sustainable Development. MUDRA: J Financ Account 5: 59–74. https://doi.org/10.17492/mudra.v5i01.13036 doi: 10.17492/mudra.v5i01.13036
![]() |
[41] |
Nawaz MA, Seshadri U, Kumar P, et al. (2021) Nexus between green finance and climate change mitigation in N-11 and BRICS countries: empirical estimation through difference in differences (DID) approach. Environ Sci Pollut Res 28: 6504–6519. https://doi.org/10.1007/s11356-020-10920-y doi: 10.1007/s11356-020-10920-y
![]() |
[42] |
Ngwenya N, Simatele MD (2020) The emergence of green bonds as an integral component of climate finance in South Africa. S Afr J Sci 116: 1–3. https://doi.org/10.17159/sajs.2020/6522 doi: 10.17159/sajs.2020/6522
![]() |
[43] |
Sadiq M, Amayri MA, Paramaiah C, et al. (2022) How green finance and financial development promote green economic growth: deployment of clean energy sources in South Asia. Environ Sci Pollut Res 29: 65521–65534. https://doi.org/10.1007/S11356-022-19947-9 doi: 10.1007/s11356-022-19947-9
![]() |
[44] |
Sadiq M, Nonthapot S, Mohamad S, et al. (2022) Does green finance matter for sustainable entrepreneurship and environmental corporate social responsibility during COVID-19? China Financ Rev Int 12: 317–333. https://doi.org/10.1108/CFRI-02-2021-0038 doi: 10.1108/CFRI-02-2021-0038
![]() |
[45] |
Saeed MM, Karim MZA (2022) The role of green finance in reducing CO2 emissions: An empirical analysis. Borsa Istanbul Rev 22: 169–178. https://doi.org/10.1016/j.bir.2021.03.002 doi: 10.1016/j.bir.2021.03.002
![]() |
[46] |
Sanchez-Roger M, Oliver-Alfonso MD, Sanchís-Pedregosa C (2018) Bail-In: A sustainable mechanism for rescuing banks. Sustainability (Switzerland) 10: 3789. https://doi.org/10.3390/su10103789 doi: 10.3390/su10103789
![]() |
[47] |
Sarma P, Roy A (2021) A Scientometric analysis of literature on Green Banking (1995-March 2019). J Sust Financ Investment 11: 143–162. https://doi.org/10.1080/20430795.2020.1711500 doi: 10.1080/20430795.2020.1711500
![]() |
[48] |
Schäfer H (2018) Germany: The 'greenhorn' in the green finance revolution. Environment 60: 19–27. https://doi.org/10.1080/00139157.2018.1397472 doi: 10.1080/00139157.2018.1397472
![]() |
[49] |
Sharma GD, Sarker T, Rao A, et al. (2022) Revisting conventional and green finance spillover in post-COVID world: Evidence from robust econometric models. Global Financ J 51: 100691. https://doi.org/10.1016/j.gfj.2021.100691 doi: 10.1016/j.gfj.2021.100691
![]() |
[50] |
Suh W, Kim K (2019) Artificial Intelligence and Financial Law. Gachon Law Rev 12: 179–214. https://doi.org/10.15335/glr.2019.12.4.006 doi: 10.15335/glr.2019.12.4.006
![]() |
[51] |
Umar M, Ji X, Mirza N, et al. (2021) Carbon neutrality, bank lending, and credit risk: Evidence from the Eurozone. J Environ Manage 296: 113156. https://doi.org/10.1016/j.jenvman.2021.113156 doi: 10.1016/j.jenvman.2021.113156
![]() |
[52] |
Urban MA, Wójcik D (2019) Dirty banking: Probing the gap in sustainable finance. Sustainability (Switzerland) 11: 1745. https://doi.org/10.3390/su11061745 doi: 10.3390/su11061745
![]() |
[53] |
Wang F, Yang S, Reisner A, et al. (2019) Does green credit policy work in China? The correlation between green credit and corporate environmental information disclosure quality. Sustainability (Switzerland) 11: 737. https://doi.org/10.3390/su11030733 doi: 10.3390/su11030733
![]() |
[54] |
Wang R, Zhao X, Zhang L (2022) Research on the impact of green finance and abundance of natural resources on China's regional eco-efficiency. Resour Policy 76: 102579. https://doi.org/10.1016/j.resourpol.2022.102579 doi: 10.1016/j.resourpol.2022.102579
![]() |
[55] |
Wang Y, Zhi Q (2016) The Role of Green Finance in Environmental Protection: Two Aspects of Market Mechanism and Policies. Energy Procedia 104: 311–316. https://doi.org/10.1016/j.egypro.2016.12.053 doi: 10.1016/j.egypro.2016.12.053
![]() |
[56] |
Wang Z, Shahid MS, Binh AN, et al. (2022) Does green finance facilitate firms in achieving corporate social responsibility goals? Econ Res-Ekon Istrazivanja, 1–20. https://doi.org/10.1080/1331677X.2022.2027259 doi: 10.1080/1331677X.2022.2027259
![]() |
[57] | Weber O, ElAlfy A (2019) The Development of Green Finance by Sector, 53–78. https://doi.org/10.1007/978-3-030-22510-0_3 |
[58] |
Yin X, Xu Z (2022) An empirical analysis of the coupling and coordinative development of China's green finance and economic growth. Resour Policy 75: 102476. https://doi.org/10.1016/j.resourpol.2021.102476 doi: 10.1016/j.resourpol.2021.102476
![]() |
[59] |
Yip AWH, Bocken NMP (2018) Sustainable business model archetypes for the banking industry. J Clean Prod 174: 150–169. https://doi.org/10.1016/j.jclepro.2017.10.190 doi: 10.1016/j.jclepro.2017.10.190
![]() |
[60] |
Yuan F, Gallagher KP (2018) Greening Development Lending in the Americas: Trends and Determinants. Ecol Econ 154: 189–200. https://doi.org/10.1016/j.ecolecon.2018.07.009 doi: 10.1016/j.ecolecon.2018.07.009
![]() |
[61] |
Zhang H, Geng C, Wei J (2022) Coordinated development between green finance and environmental performance in China: The spatial-temporal difference and driving factors. J Clean Prod 346: 131150. https://doi.org/10.1016/j.jclepro.2022.131150 doi: 10.1016/j.jclepro.2022.131150
![]() |
[62] |
Zhang X, Wang Z, Zhong X, et al. (2022) Do Green Banking Activities Improve the Banks' Environmental Performance? The Mediating Effect of Green Financing. Sustainability (Switzerland) 14: 989. https://doi.org/10.3390/su14020989 doi: 10.3390/su14020989
![]() |
[63] |
Zheng GW, Siddik AB, Masukujjaman M, et al. (2021a) Factors affecting the sustainability performance of financial institutions in Bangladesh: The role of green finance. Sustainability (Switzerland) 13: 10165. https://doi.org/10.3390/su131810165 doi: 10.3390/su131810165
![]() |
[64] |
Zheng GW, Siddik AB, Masukujjaman M, et al. (2021b) Green finance development in Bangladesh: The role of private commercial banks (PCBs). Sustainability (Switzerland) 13: 1–17. https://doi.org/10.3390/su13020795 doi: 10.3390/su13020795
![]() |
[65] |
Zhixia C, Hossen MM, Muzafary SS, et al. (2018) Green banking for environmental sustainability-present status and future agenda: Experience from Bangladesh. Asian Econ Financ Rev 8: 571–585. https://doi.org/10.18488/journal.aefr.2018.85.571.585 doi: 10.18488/journal.aefr.2018.85.571.585
![]() |
[66] |
Zhou X, Tang X, Zhang R (2020) Impact of green finance on economic development and environmental quality: a study based on provincial panel data from China. Environ Sci Pollut Res 27: 19915–19932. https://doi.org/10.1007/s11356-020-08383-2 doi: 10.1007/s11356-020-08383-2
![]() |
[67] |
Ziolo M, Filipiak BZ, Bak I, et al. (2019) How to design more sustainable financial systems: The roles of environmental, social, and governance factors in the decision-making process. Sustainability (Switzerland) 11: 5604. https://doi.org/10.3390/su11205604 doi: 10.3390/su11205604
![]() |
![]() |
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Road length | Relaxation time | Maximum density | Free flow velocity |
L=1km | τ0=120h | ρm=150Vehicle/km | vf=150km/h |