Industrialization, financial activities, and intensive human activities have reduced continuous habitats to smaller patches, threatening the safety of the ecosystem. However, as technological innovation and digitization increase, this negative impact will be somewhat neutralized. To address this issue, the current study examined the role of economic, socioeconomic, and green indicators on the ecological footprint in the Gulf Cooperation Council (GCC). By using data from 1990–2019, we have applied multiple panel tests to determine the long-run and short-run relationships among the variables. The findings show that economic growth increases the long-term ecological footprint in the GCC. The human development index and financial inclusion coefficients are also positively and significantly linked with the ecological footprint. The socioeconomic index, however, reveals a negative relationship between ecological footprint and GCC. Similarly, digitalization and environmental technologies have a negative and major impact on the ecological footprint. It indicates that green growth factors contribute to long-term improvements in environmental quality. So, GCC nations should emphasize investing in green growth factors and enact strict environmental regulations to safeguard their country from environmental problems.
Citation: Majid Ibrahim Alsaggaf. The dynamic nexus between economic factors, socioeconomic factors, green growth factors, and ecological footprint: evidence from GCC economies[J]. AIMS Environmental Science, 2024, 11(5): 797-830. doi: 10.3934/environsci.2024040
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Industrialization, financial activities, and intensive human activities have reduced continuous habitats to smaller patches, threatening the safety of the ecosystem. However, as technological innovation and digitization increase, this negative impact will be somewhat neutralized. To address this issue, the current study examined the role of economic, socioeconomic, and green indicators on the ecological footprint in the Gulf Cooperation Council (GCC). By using data from 1990–2019, we have applied multiple panel tests to determine the long-run and short-run relationships among the variables. The findings show that economic growth increases the long-term ecological footprint in the GCC. The human development index and financial inclusion coefficients are also positively and significantly linked with the ecological footprint. The socioeconomic index, however, reveals a negative relationship between ecological footprint and GCC. Similarly, digitalization and environmental technologies have a negative and major impact on the ecological footprint. It indicates that green growth factors contribute to long-term improvements in environmental quality. So, GCC nations should emphasize investing in green growth factors and enact strict environmental regulations to safeguard their country from environmental problems.
It is worth to start by quoting Bradley [15] "For real progress, the mathematical modeller, as well as the epidemiologist must have mud on his boots"!
Indeed most of the pioneers in mathematical epidemiology have got "mud on their boots"; it is a duty and a pleasure to acknowledge here the ones who, apart from D. Bernoulli (1760) [14], established the roots of this field of research (in chronological order): W. Farr (1840) [41], W.H. Hamer (1906) [45], J. Brownlee (1911) [17], R. Ross (1911) [63], E. Martini (1921) [59], A. J. Lotka (1923) [57], W.O. Kermack and A. G. McKendrick (1927) [51], H. E. Soper (1929) [66], L. J. Reed and W. H. Frost (1930) [42], [1], M. Puma (1939) [62], E. B. Wilson and J. Worcester (1945) [69], M. S. Bartlett (1949) [12], G. MacDonald (1950) [58], N.T.J. Bailey (1950) [11], before many others; the pioneer work by En'ko (1989) [39] suffered from being written in Russian; historical accounts of epidemic theory can be found in [64], [35], [36]. After the late '70's there has been an explosion of interest in mathematical epidemiology, also thanks to the establishment of a number of new journals dedicated to mathematical biology. The above mentioned pioneers explored possible models to match real data, based on genuine epidemiological reasoning; further they did not choose a priori deterministic models as opposed to stochastic models. Unfortunately the most recent literature has suffered of a dramatic splitting in both approaches and methods, which has induced criticism among applied epidemiologists. About the relevance of mathematics in Life Sciences, Wilson and Worcester had since long [69] expressed a fundamental statement that we like to share: "Although mathematics is used to develop the logical inferences from known laws, it may also used to investigate the consequences of various assumptions when the laws are not known, .... one of the functions of mathematical and philosophical reasoning is to keep us alive to what may be only possibilities, when the actualities are not yet known".
The scheme of this presentation is the following: in Section 2 a general structure of mathematical models for epidemic systems is presented in the form of compartmental systems; in Paragraph 2.1 the possible derivation of deterministic models is presented as an approximation, for large populations, of stochastic models; in Paragraph 2.2 nonlinear models are discussed as opposed to the standard epidemic models based on the "law of mass" action assumption; in Paragraph 2.3 the concept of field of forces of infection is discussed for structured populations. In Section 3 the particular case of man-environment-man infection is discussed, and, with respect to these models, in Section 4 optimal control problems are presented in the case of boundary feedback. Finally in Section 5 the most important problem of global eradication via regional control is presented.
Model reduction for epidemic systems is obtained via the so-called compartmental models. In a compartmental model the total population (relevant to the epidemic process) is divided into a number (usually small) of discrete categories: susceptibles, infected but not yet infective (latent), infective, recovered and immune, without distinguishing different degrees of intensity of infection; possible structures in the relevant population can be superimposed when required (see e.g. Figure 1).
A key problem in modelling the evolution dynamics of infectious diseases is the mathematical representation of the mechanism of transmission of the contagion. The concepts of "force of infection" and "field of forces of infection" (when dealing with structured populations) will be the guideline of this presentation.
We may like to remark here (see also [19]) that this concept is not very far from the medieval idea that infectious diseases were induced into a human being by a flow of bad air ("mal aria" in Italian). On the other hand in quantum field theory any field of forces is due to an exchange of particles: in this case bacteria, viruses, etc., so that the corpuscular and the continuous concepts of field are conceptually unified.
It is of interest to identify the possible structures of the field of forces of infection which depend upon the specific mechanisms of transmission of the disease among different groups. This problem has been raised since the very first models when age and/or space dependence had to be taken into account.
Suppose at first that the population in each compartment does not exhibit any structure (space location, age, etc.). The infection process (
(f.i.)(t)=[g(I(.))](t) |
which acts upon each individual in the susceptible class. Thus a typical rate of the infection process is given by the
(incidence rate)(t)=(f.i.)(t)S(t). |
From this point of view, the so called "law of mass action" simply corresponds to choosing a linear dependence of
(f.i.)(t)=kI(t). |
The great advantage, from a mathematical point of view, is that the evolution of the epidemic is described (in the space and time homogeneous cases) by systems of ODE 's which contain at most bilinear terms.
Indeed, for several models of this kind it is possible to prove global stability of nontrivial equilibria. A general result in this direction has been proposed in [13] where it has been shown that many bilinear epidemic systems can be expressed in the general form
dzdt=diag(z)(e+Az)+b(z) |
where
b(z)=c+Bz |
with
bij≥0,i,j=1,…,n;bii=0,i=1,…,n. |
Once a strictly positive equilibrium
V(z):=n∑i=1wi(zi−z∗i−z∗ilnziz∗i),z∈Rn∗ |
where
Here we denote by
Rn∗+:={z∈Rn∣zi>0,i=1,…,n}, |
and clearly
V:=Rn∗+→R+. |
A discussion on
Actually for populations of a limited size, the stochastic version is more appropriate; but it is not difficult to show that for sufficiently large populations, the usual deterministic approximation can be gained via suitable laws of large numbers (see e.g. [40]).
The stochastic process modelling an
Considering the usual transitions
S→I→R, |
by assuming the law of mass action, the only nontrivial transition rates are usually taken as
q(S,I),(S−1,I+1)=κINS:infection; | (1) |
q(S,I),(S,I−1)=δI:removal, | (2) |
N=St+It+Rt=S0+I0=const. | (3) |
We may notice that the above transition rates can be rewritten as follows
q(S,I),(S−1,I+1)=NκINSN; | (4) |
q(S,I),(S,I−1)=NδIN. | (5) |
So that both transition rates are of the form
q(N)k,k+l=Nβl(kN) | (6) |
for
k=(S,I) | (7) |
and
k+l={(S,I−1),(S−1,I+1). | (8) |
Due to the constancy of the total population we may reduce the analysis to the Markov process
ˆX(N)(t)=ˆX(N)(0)+∑l∈Z2lYl(N∫t0βl(ˆX(N)(τ)N)dτ), | (9) |
for
Here the
By setting
F(x)=∑l∈Z2lβl(x),x∈R2 | (10) |
for the scaled process
X(N)=1NˆX(N), | (11) |
we have
X(N)(t)=X(N)(0)+∫t0F(X(N)(τ))dτ+∑l∈Z2lN˜Yl(N∫t0βl(X(N)(τ))dτ) | (12) |
where the
˜Yl(u)=Yl(u)−u | (13) |
are independent centered standard Poisson processes, so that the last term in the above equation is a zero-mean martingale.
Of interest is the asymptotic behavior of the system for a large value of the scale parameter
By the strong law of large numbers for Poisson processes (more generally for martingales), we know that
limN→∞supu≤v|1N˜Yl(Nu)|=0,a.s., | (14) |
for any
Theorem 2.1. Under suitable regularity assumptions on
limN→∞X(N)(0)=x0∈R2, | (15) |
then, for every
limN→∞supτ≤t|X(N)(τ)−x(τ)|=0,a.s., | (16) |
where
x(t)=x0+∫t0F(x(s))ds,t≥0, | (17) |
wherever it exists.
In our case the above deterministic system becomes the usual deterministic SIR model
{ds(t)dt=−κs(t)s(t)di(t)dt=κs(t)i(t)−δi(t) | (18) |
for
s(t):=limN→∞StN,i:=limN→∞ItN. |
A different scaling, may give rise to the diffusion approximation of the epidemic system (see [40], [22], and [67], for a variety of applications to Biology and Medicine).
An interesting "pathology" arises when the relevant populations are very small, so that the deterministic approximation of the epidemic system may fail. Indeed for many epidemic models, above threshold the infective fraction of the relevant deterministic equations, while tending eventually to large values of a possible endemic level, may get very close to zero, but still never becomes extinct. This situation had been analyzed in [47] by suitable perturbation methods on the Fokker-Planck equation associated with the diffusion approximation of a typical SIR epidemic model, which lead to a non trivial extinction probability of the infective population, whenever its deterministic counterpart may get close to zero.
It is worth mentioning that the discussion regarding the original stochastic model and its deterministic counterpart had involved J.L. Doob and others, who proposed (in 1945) [38] an algorithm for generating statistically correct trajectories of the stochastic system. It was presented by D. Gillespie in 1976 [43] as the Doob-Gillespie algorithm, well known in computational chemistry and physics.
Referring to the "Law of mass Action", Wilson and Worcester [69] stated the following:
"It would in fact be remarkable, in a situation so complex as that of the passage of an epidemic over a community, if any simple law adequately represented the phenomenon in detail... even to assume that the new case rate should be set equal to any function... might be questioned?".
Indeed Wilson and Worcester [69], and Severo [65] had been among the first epidemic modelers including nonlinear forces of infection of the form
(f.i.)(t)=κI(t)pS(t)q |
in their investigations. Here
Independently, during the analysis of data regarding the spread of a cholera epidemic in Southern Italy during 1973, in [28] the authors suggested the need to introduce a nonlinear force of infection in order to explain the specific behavior emerging from the available data.
A more extended analysis for a variety of proposed generalizations of the classical models known as Kermack-McKendrick models, appeared in [29], though nonlinear models became widely accepted in the literature only a decade later, after the paper [55].
Nowadays models with nonlinear forces of infection are analyzed within the study of various kinds of diseases; typical expressions include the so called Holling type functional responses (see e.g. [29], [48])
(f.i.)(t)=g(I(t)); |
with
g(I)=kIpα+βIq,p,q>0. | (19) |
Particular cases are
g(I)=kIp,p>0 | (20) |
For the case
A rather general analysis regarding existence and stability of nontrivial equilibria for model (19) has been carried out in a series of papers [61], [16], [56], [48] (see also [19], and [68]). The particular case
Additional shapes of
Further extensions include a nonlinear dependence upon both
When dealing with populations which exhibit some structure (identified here by a parameter
(incidencerate)(z;t)=(f.i.)(z;t)s(z;t). |
When dealing with populations with space structure the relevant quantities are spatial densities, such as
The corresponding total populations are given by
S(t)=∫Ωs(z;t)dz,I(t)=∫Ωi(z;t)dz |
In the law of mass action model, if only local interactions are allowed, the field at point
(f.i.)(z;t)=k(z)i(z;t). |
On the other hand if we wish to take distant interactions too into account, as proposed by D.G. Kendall in [50], the field at a point
(f.i.)(z;t)=∫Ωk(z,z′)i(z′;t)dz′. |
For this case the emergence of travelling waves has been shown in [50] and [9]. The analysis of the diffusion approximation of Kendall's model can be found in [49].
When dealing with populations with an age structure, we may interpret the parameter
A large literature on the subject can be found in [19].
A widely accepted model for the spatial spread of epidemics in an habitat
Typical real cases include typhoid fever, malaria, schistosomiasis, cholera, etc. (see e.g. [34], [6]).
{∂u1∂t(x,t)=d1Δu1(x,t)−a11u1(x,t)+∫Ωk(x,x′)u2(x′,t)dx′∂u2∂t(x,t)=−a22u2(x,t)+g(u1(x,t)) | (21) |
in
●
●
● The terms
● The total susceptible population is assumed to be sufficiently large with respect to the infective population, so that it can be taken as constant.
Environmental pollution is produced by the infective population, so that in the first equation of System (21), the integral term
∫Ωk(x,x′)u2(x′,t)dx′ |
expresses the fact that the pollution produced at any point
Model (21) includes spatial diffusion of the pollutant, due to uncontrolled additional causes of dispersion (with a constant diffusion coefficient to avoid purely technical complications); we assume that the infective population does not diffuse (the case with diffusion would be here a technical simplification). As such, System (21) can be adopted as a good model for the spatial propagation of an infection in agriculture and forests, too.
The above model is part of another important class of epidemics which exhibit a quasimonotone (cooperative) behavior (see [19]). For this class of problems stability of equilibria can be shown by monotone methods, such as the contracting rectangles technique (see [52], [53]).
The local "incidence rate" at point
(i.r.)(t)=g(u1(x,t)), |
depending upon the local concentration of the pollutant.
If we wish to model a large class of fecal-oral transmitted infectious diseases, such as typhoid fever, infectious hepatitis, cholera, etc., we may include the possible seasonal variability of the environmental conditions, and their impact on the habits of the susceptible population, so that the relevant parameters are assumed periodic in time, all with the same period
As a purely technical simplification, we may assume that only the incidence rate is periodic, and in particular that it can be expressed as
(i.r.)(x,t)=h(t,u1(x,t))=p(t)g(u1(x,t)), |
where
The explicit time dependence of the incidence rate is given via the function
p(t)=p(t+T). |
Remark 1. The results can be easily extended to the case in which also
In [21] the above model was studied, and sufficient conditions were given for either the asymptotic extinction of an epidemic outbreak, or the existence and stability of an endemic state; while in [27] the periodic case was additionally studied, and sufficient conditions were given for either the asymptotic extinction of an epidemic outbreak, or the existence and stability of a periodic endemic state with the same period of the parameters.
The choice of
{dz1dt(t)=−a11z1(t)+a12z2(t)du2dt(t)=−a22z2(t)+g(z1(t)) | (22) |
In [26] and [60] the bistable case (in which system (22) may admit two nontrivial steady states, one of which is a saddle point in the phase plane) was obtained by assuming that the force of infection, as a function of the concentration of the pollutant, is sigma shaped. In [60] this shape had been obtained as a consequence of the sexual reproductive behavior of the schistosomes. In [26] (see also [25]) the case of fecal-oral transmitted diseases was considered; an interpretation of the sigma shape of the force of infection was proposed to model the response of the immune system to environmental pollution: the probability of infection is negligible at low concentrations of the pollutant, but increases with larger concentrations; it then becomes concave and saturates to some finite level as the concentration of pollutant increases without limit.
Let us now refer to the following simplified form of System (21), where as kernel we have taken
{∂u1∂t(x,t)=d1Δu1(x,t)−a11u1(x,t)+a12u2(x,t)∂u2∂t(x,t)=−a22u2(x,t)+g(u1(x,t)) | (23) |
The concavity of
An interesting problem concerns the case of boundary feedback of the pollutant, which has been proposed in [24], and further analyzed in [30]; an optimal control problem has been later analyzed in [8].
In this case the reservoir of the pollutant generated by the human population is spatially separated from the habitat by a boundary through which the positive feedback occurs. A model of this kind has been proposed as an extension of the ODE model for fecal-oral transmitted infections in Mediterranean coastal regions presented in [28].
For this kind of epidemics the infectious agent is multiplied by the infective human population and then sent to the sea through the sewage system; because of the peculiar eating habits of the population of these regions, the agent may return via some diffusion-transport mechanism to any point of the habitat, where the infection process is restarted.
The mathematical model is based on the following system of evolution equations:
{∂u1∂t(x;t)=Δu1(x;t)−a11u1(x;t)∂u2∂t(x;t)=−a22u2(x;t)+g(u1(x;t)) |
in
∂u1∂ν(x;t)+αu1(x;t)=∫Ωk(x,x′)u2(x′;t)dx′ |
on
Here
H[u2(⋅,t)](x):=∫Ωk(x,x′)u2(x′;t)dx′ |
describes boundary feedback mechanisms, according to which the infectious agent produced by the human infective population at time
Clearly the boundary
The parameter
α(x),k(x,⋅)=0, forx∈Γ2. |
A relevant assumption, of great importance in the control problems that we have been facing later, is that the habitat
for anyx′∈Ω there exists somex∈Γ1 such thatk(x,x′)>0. |
This means that from any point of the habitat infective individuals contribute to polluting at least some point on the boundary (the sea shore).
In the above model delays had been neglected and the feedback process had been considered to be linear; various extensions have been considered in subsequent literature.
Let us now go back to System (21) in
The public health concern consists of providing methods for the eradication of the disease in the relevant population, as fast as possible. On the other hand, very often the entire domain
This has led the first author, in a discussion with Jacques Louis Lions in 1989, to suggest that it might be sufficient to implement such programmes only in a given subregion
In this section a review is presented of some results obtained by the authors, during 2002-2012, concerning stabilization (for both the time homogeneous case and the periodic case). Conditions have been provided for the exponential decay of the epidemic in the whole habitat
∂u1∂ν(x,t)+αu1(x,t)=0 on ∂Ω×(0,+∞), |
where
For the time homogeneous case the following assumptions have been taken:
(H1)
(H2)
∫Ωk(x,x′)dx>0 a.e. x′∈Ω; |
(H3)
Let
χω(x)h(x)=0,x∈RN∖¯ω, |
even if function
Our goal is to study the controlled system
{∂u1∂t(x,t)=d1Δu1(x,t)−a11u1(x,t)+∫Ωk(x,x′)u2(x′,t)dx′+χω(x)v(x,t),(x,t)∈Ω×(0,+∞)∂u1∂ν(x,t)+αu1(x,t)=0,(x,t)∈∂Ω×(0,+∞)∂u2∂t(x,t)=−a22u2(x,t)+g(u1(x,t)), (x,t)∈Ω×(0,+∞)u1(x,0)=u01(x),u2(x,0)=u02(x),x∈Ω, |
subject to a control
We have to mention that existence, uniqueness and nonnegativity of a solution to the above system can be proved as in [10]. The nonnegativity of
Definition 5.1. We say that our system is zero-stabilizable if for any
u1(x,t)≥0,u2(x,t)≥0, a.e. x∈Ω, for any t≥0 |
and
limt→∞‖u1(t)‖L∞(Ω)=limt→∞‖u2(t)‖L∞(Ω)=0. |
Definition 5.2. We say that our system is locally zero-stabilizable if there exists
Remark 2. It is obvious that if a system is zero-stabilizable, then it is also locally zero-stabilizable.
A stabilization result for our system, in the case of time independent
In [3], by Krein-Rutman Theorem, it has been shown that
{−d1Δφ+a11φ−a21a22∫Ωk(x,x′)φ(x′)dx′=λφ, x∈Ω∖¯ωφ(x)=0,x∈∂ω∂φ∂ν(x)+αφ(x)=0,x∈∂Ω, |
admits a principal (real) eigenvalue
K={φ∈L∞(Ω); φ(x)≥0 a.e. in Ω}. |
The following theorem holds [3]:
Theorem 5.3. If
Conversely, if
Moreover, the proof of the main result in [3] shows that for a given affordable sanitation effort
{−d1Δφ+a11φ−a21a22∫Ωk(x,x′)φ(x′)dx′+γχωφ=λφ, x∈Ω∂φ∂ν(x)+αφ(x)=0,x∈∂Ω. | (24) |
A natural question related to the practical implementation of the sanitation policy is the following: "For a given sanitation effort
So, the first problem to be treated is the estimation of
limt→+∞∫Ωyω(x,t)dx=ζ−λω1,γ, | (25) |
where
{∂y∂t−d1Δy+a11y+γχωy−a21a22∫Ωk(x,x′)y(x′,t)dx′ −ζy+(∫Ωy(x,t)dx)y=0,x∈Ω, t>0∂y∂ν(x,t)+αy(x,t)=0,x∈∂Ω, t>0y(x,0)=1,x∈Ω, | (26) |
and
Remark 3. Problem (26) is a logistic model for the population dynamics with diffusion and migration. Since the solutions to the logistic models rapidly stabilize, this means that (25) gives an efficient method to approximate
ζ−∫Ωyω(x,T)dx |
gives a very good approximation of
We may also remark that, if in (26)
y(x,0)=y0,x∈Ω, | (27) |
with
limt→+∞∫Ωyω1(x,t)dx=ζ−λω1,γ, |
where
Assume now that for a given sanitation effort
Let
Rω=∫ω[uω1(x,T)+uω2(x,T)]dx, |
at some given finite time
Here
{∂u1∂t(x,t)=d1Δu1(x,t)−a11u1(x,t)+∫Ωk(x,x′)u2(x′,t)dx′−γχω(x)u1(x,t),(x,t)∈Ω×(0,+∞)∂u1∂ν(x,t)+αu1(x,t)=0,(x,t)∈∂Ω×(0,+∞)∂u2∂t(x,t)=−a22u2(x,t)+g(u1(x,t)), (x,t)∈Ω×(0,+∞)u1(x,0)=u01(x),x∈Ωu2(x,0)=u02(x),x∈Ω. | (28) |
For this reason we are going to evaluate the derivative of
For any
dRω(V)=limε→0RεV+ω−Rωε. |
For basic results and methods in the optimal shape design theory we refer to [46].
Theorem 5.4. For any
dRω(V)=γ∫T0∫∂ωuω1(x,t)pω1(x,t)ν(x)⋅Vdσ dt, |
where
{∂p1∂t+d1Δp1−a11p1−γχωp1+g′(uω1)p2=0,x∈Ω, t>0∂p2∂t+∫Ωk(x′,x)p1(x′,t)dx′−a22p2=0,x∈Ω, t>0∂p1∂ν(x,t)+αp(x,t)=0,x∈∂Ω, t>0p1(x,T)=p2(x,T)=1,x∈Ω. | (29) |
Here
For the construction of the adjoint problems in optimal control theory we refer to [54].
Based on Theorem 5.4, in [5] the authors have proposed a conceptual iterative algorithm to improve the position (by translation) of
As a purely technical simplification, we have assumed that only the incidence rate is periodic, and in particular that it can be expressed as
(i.r.)(x,t)=h(t,u1(x,t))=p(t)g(u1(x,t)), |
were
In this case our goal is to study the controlled system
{∂u1∂t(x,t)=d1Δu1(x,t)−a11u1(x,t)+∫Ωk(x,x′)u2(x′,t)dx′+χω(x)v(x,t),(x,t)∈Ω×(0,+∞)∂u1∂ν(x,t)+αu1(x,t)=0,(x,t)∈∂Ω×(0,+∞)∂u2∂t(x,t)=−a22u2(x,t)+h(t,u1(x,t)), (x,t)∈Ω×(0,+∞)u1(x,0)=u01(x),u2(x,0)=u02(x),x∈Ω, | (30) |
with a control
The explicit time dependence of the incidence rate is given via the function
p(t)=p(t+T). |
Remark 4. The results can be easily extended to the case in which also
Consider the following (linear) eigenvalue problem
{∂φ∂t−d1Δφ+a11φ−∫Ωk(x,x′)ψ(x′,t)dx′=λφ,x∈Ω∖¯ω, t>0∂φ∂ν(x,t)+αφ(x,t)=0,x∈∂Ω, t>0φ(x,t)=0,x∈∂ω, t>0∂ψ∂t(x,t)+a22ψ(x,t)−a21p(t)φ(x,t)=0,x∈Ω∖¯ω, t>0φ(x,t)=φ(x,t+T),ψ(x,t)=ψ(x,t+T),x∈Ω∖¯ω, t≥0. | (31) |
By similar procedures, as in the time homogeneous case, Problem (31) admits a principal (real) eigenvalue
KT={φ∈L∞(Ω×(0,T)); φ(x,t)≥0 a.e. in Ω×(0,T)}. |
Theorem 5.5. If
Conversely, if
Theorem 5.6. Assume that
{∂φ∂t−d1Δφ+a11φ−∫Ωk(x,x′)ψ(x′,t)dx′=λφ,x∈Ω∖¯ω, t>0∂φ∂ν(x,t)+αφ(x,t)=0,x∈∂Ω, t>0φ(x,t)=0,x∈∂ω, t>0∂ψ∂t(x,t)+a22ψ(x,t)−g′(0)p(t)φ(x,t)=0,x∈Ω∖¯ω, t>0φ(x,t)=φ(x,t+T),ψ(x,t)=ψ(x,t+T),x∈Ω∖¯ω, t≥0 | (32) |
If
Conversely, if the system is locally zero stabilizable, then
Remark 5. Since
Remark 6. Future directions. Another interesting problem is that when
In a recently submitted paper [7], the problem of the best choice of the subregion
The work of V. Capasso was supported by the MIUR-PRIN grant
It is a pleasure to acknowledge the contribution by Klaus Dietz regarding the bibliography on the historical remarks reported in the Introduction.
Thanks are due to the Anonymous Referees for their precious advise and suggestions.
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