Analysis of Climatic Model Using Fractional Optimal Control

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Introduction
CO 2 emissions have been emerged as one of the most critical and complicated issues impacting the atmosphere.It has an indirect infuence on the more obvious form of air pollution and smog.By increasing temperature and humidity, it produces smog, which has a detrimental efect on respiratory health and numerous skin illnesses.Fossil fuel combustion is the major source of new CO 2 emissions because it discharges carbon dioxide into the atmosphere, which acts as a blanket over the Earth, absorbing solar heat and raising temperatures.CO 2 emissions have had far-reaching implications, hurting both the ecosystem and the humans who live within it.Numerous authors [1,2] have investigated fractional diferential equations with integer and noninteger derivatives for use in mathematical modeling of CO 2 emissions.Recently, there has been a lot of interest in the modeling of CO 2 emissions utilizing time-dependent controls and optimal control theory [3,4].Scientists used mathematical models to describe the changes of CO 2 emissions and deforestation and provided the mathematical modelling to explain the infuence of greenhouse gas emissions on the ecosystem [5,6].Using the same mathematical model, Nordhaus [7] analyzed the optimal taxing systems to stabilize climate and carbon dioxide emissions.Fractional calculus has been extensively utilized to represent dynamical processes in many various disciplines, including science, engineering, and many more [8,9].Tis is because fractional order derivatives include the memory effect, which is a signifcant attribute.Hertel and Rosch [10] created general optimum control issues that are motivated by the fractional derivative of the Riemann-Liouville equation.Te same authors developed a reliable numerical framework for the mathematical model and presented related optimal control problems using Caputo derivatives [11].
Numerous scholars have been working on identifying climate changes in a certain geographical area and have created multiple efcient models for doing so.Tey used a numerical scheme to derive the conditions for the optimality system for a general control problem with Caputo derivatives.Verma et al. [12] suggested various nonlinear dynamical models to identify the optimum solutions for carbon dioxide emission reduction.Te state model for an optimal control problem contained both frst-order and noninteger derivatives, and necessary optimality conditions have been derived for this problem [13,14].
According to research in the literature, environmental systems and other processes with nonlinear behaviour are good candidates for optimal control problems.Terefore, utilising fractional derivatives and optimal control theory to describe climatic changes has a lot of benefts [15,16].Tis research presents the formulation of a mathematical model with the help of fractional diferential equations where the optimal solutions are determined by the application of Pontryagin's principle.Te paper is summarized as follows: we provide the formulation of fractional optimal control problem in Section 2. In Section 3, the formulation of controlled CO 2 emissions model is considered and necessary condition for the optimality of model problem is derived.Sections 4 and 5 describe the numerical scheme for the solution of the problem and the experimentation of the results in the form of simulation, and fnally, Section 6 deals with the concluding remarks.

Fundamental Properties of Fractional Calculus
Tis section provides a quick overview of the fractional optimal control model's mathematical formulation.Ten, in Section 3, we will put the modelling approach to build our CO 2 emissions model.Tere are a lot of other kinds of fractional derivatives, but Riemann-Liouville (R-L) derivative and Caputo derivative are the two that are most frequently used in engineering and mathematical modelling.
We construct the optimization model for this task using Caputo fractional derivatives.

Defnition 3. Te left Caputo fractional derivative is termed as
Defnition 4. Te right Caputo fractional derivative is termed as where α is the order of the Caputo derivative.

Fractional Optimal Control Formulation
Agrawal [18] presented the formulation of fractional optimal control for a class of distributed systems.Te main goal is to fnd an optimal control u * which minimizes objective functional, subject to fractional dynamics constraints with initial condition where x(t) is the state variable, F(x, u, t) and W(x, u, t) are the two arbitrary functions, and x 0 is the state variable at time t � 0. We can obtain the necessary condition only if we manipulate equations ( 5) and ( 6) using Lagrange multiplier approach, the variations of calculus, and integration by parts which shows that this equation is now independent from variation of a derivative such that, with where λ represents the Lagrange multiplier or co-state variable.

Controlled CO 2 Emissions Model with Fractional Derivatives
In order to obtain the essential conditions or equations for the fractional optimal control model's optimality, we will use the data from the formulation of the fractional optimal control problem.Our aim is to fnd an optimal control that reduces the emission of CO 2 and increase forested area which means to fnd an optimal control u * that reduces the objective functional, where so, we get 0 where X(t) represents a state vector and U(t) represents a control vector.Initial conditions are x(0) � x 0 , z(0) � z 0 and y(0) � y 0 .Manabe and Stoufer [19] formulated the classical optimal control problem at α � 1 and therefore the dynamic constraint equations ( 14)-( 16) with above initial condition become as It is important to point out that there are thorough justifcations in the literature for the formulation of necessary conditions for optimality of various fractional dynamical systems [15,16].Terefore, 0 where λ(t) � (λ 1 , λ 2 , λ 3 ) T is the co-state vector.
We can obtain CO 2 emissions system in an enlarged version by applying the compact form of the prerequisites listed previously. Terefore, Moreover, the optimal controls are given by where ( * ) denotes the optimal values of u 1 and u 2 .

Numerical Results and Discussion
In this segment, we provide an explanation of a few numerical simulations and their results.We develop a forwardbackward sweep algorithm using the RK approach to analyze the biological model in optimality system.In this work, a forward-backward approach using the generalized Euler scheme is employed to compute the numerical solution of the optimality systems.Te generalised Euler method has recently been used in a TB model along with the forwardbackward algorithm.We use Mathematica software 11 to illustrate the graphical results of various fractional order.Table 1 shows the description of variables and control functions, whereas Table 2 demonstrate the parameters of the model.Te numerical simulations are performed using the initial conditions, and model parameter values are listed as follows: x(0) � 398 million tons of carbon dioxide, y(0) � 2787 billion international dollars, z(0 and α 2 � 0.00005.Te graphical representations demonstrate that we provide a signifcant reduction in the rate of CO 2 and reforestation increases with decrease in GDP when α � 1 and noninteger orders (α � 0.98, 0.96, 0.94, 0.92, 0.9, 0.88, 0.86) with time dependent control.Te model's formulation for fractional optimal control demonstrates that by maximizing investments in clean technology research and reforestation activities, signifcant reductions in CO 2 emissions can be attained.Figure 1 represents the graphical result of million tons of CO 2 , Figure 2 shows the graphical of million•m 3 / year, Figure 3 shows the graphical representation of billion US dollars, Figure 4 represents emission of CO 2 with and without control, Figure 5 represents optimal solutions for forest with and without control, Figure 6 shows the comparison of GDP with and without control, Figure 7 shows the comparison of the amount of CO 2 without control with diferent values of α, Figure 8 represents Represents GDP (gross domestics product) u 1 (t) Represents optimal reforestation efort u 2 (t) Represents optimal technological efort Journal of Mathematics

. Conclusion
In this study, a mathematical model that displays CO 2 emissions, deforestation rate, and GDP has been taken into consideration.Te model is characterized by fractional derivatives.Te forward-backward sweep approach combined with the extended Euler method was used to numerically solve the optimality problem.Te results of the models demonstrate that when control is implemented, the rate of CO 2 emissions is reduced and the reforestation rate increases.Te amount of CO 2 is reduced more than with any other method when all time dependent controls are employed in simulations.Te numerical solutions show that the optimum control problem for fractional orders is signifcantly more precise than involving integer orders.

Figure 6 :
Figure 6: Comparison of GDP with and without control.

Figure 7 :
Figure 7: Comparison of the amount of CO 2 without control with diferent values of alpha.

Figure 8 :
Figure 8: Comparison of the forestation rate without control with diferent values of alpha.

Figure 9 :
Figure 9: Comparison of GDP without control with diferent values of alpha.

Figure 10 :
Figure 10: Comparison of the amount of CO 2 with control with diferent values of alpha.

Table 1 :
Description of variables and control functions.

Table 2 :
Description of parameters.