Abstract
CO2 emissions are one of the most critical and challenging problems impacting the atmosphere. In order to reduce CO2 emissions, significant efforts must be put forward in advanced research. In this research, we propose a fractional optimal control approach to effectively deal with the accessible resources and minimize the restriction of variables. The mathematical model is formulated in terms of the Caputo fractional derivative. The formulated problem is numerically solved using a forward-backward sweep approach with the generalized Euler method. We present a numerical problem to obtain the optimal solution and show that the suggested process is very efficient in science and engineering problems.
1. Introduction
CO2 emissions have been emerged as one of the most critical and complicated issues impacting the atmosphere. It has an indirect influence on the more obvious form of air pollution and smog. By increasing temperature and humidity, it produces smog, which has a detrimental effect on respiratory health and numerous skin illnesses. Fossil fuel combustion is the major source of new CO2 emissions because it discharges carbon dioxide into the atmosphere, which acts as a blanket over the Earth, absorbing solar heat and raising temperatures. CO2 emissions have had far-reaching implications, hurting both the ecosystem and the humans who live within it. Numerous authors [1, 2] have investigated fractional differential equations with integer and noninteger derivatives for use in mathematical modeling of CO2 emissions. Recently, there has been a lot of interest in the modeling of CO2 emissions utilizing time-dependent controls and optimal control theory [3, 4]. Scientists used mathematical models to describe the changes of CO2 emissions and deforestation and provided the mathematical modelling to explain the influence 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]. This is because fractional order derivatives include the memory effect, which is a significant attribute. Hertel and Rosch [10] created general optimum control issues that are motivated by the fractional derivative of the Riemann–Liouville equation. The 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 efficient models for doing so. They 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. The state model for an optimal control problem contained both first-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. Therefore, utilising fractional derivatives and optimal control theory to describe climatic changes has a lot of benefits [15, 16]. This research presents the formulation of a mathematical model with the help of fractional differential equations where the optimal solutions are determined by the application of Pontryagin’s principle. The paper is summarized as follows: we provide the formulation of fractional optimal control problem in Section 2. In Section 3, the formulation of controlled CO2 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 finally, Section 6 deals with the concluding remarks.
2. Fundamental Properties of Fractional Calculus
This section provides a quick overview of the fractional optimal control model’s mathematical formulation. Then, in Section 3, we will put the modelling approach to build our CO2 emissions model. There 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.
Definition 1. The left R-L fractional derivative is given as [17]where .
Definition 2. The right R-L fractional derivative is given asHere order fulfills , where denotes Euler’s Gamma function.
Definition 3. The left Caputo fractional derivative is termed as
Definition 4. The right Caputo fractional derivative is termed aswhere is the order of the Caputo derivative.
3. Fractional Optimal Control Formulation
Agrawal [18] presented the formulation of fractional optimal control for a class of distributed systems. The main goal is to find an optimal control which minimizes objective functional,subject to fractional dynamics constraintswith initial conditionwhere is the state variable, and are the two arbitrary functions, and is the state variable at time . 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,withwhere represents the Lagrange multiplier or co-state variable.
4. Controlled CO2 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 find an optimal control that reduces the emission of CO2 and increase forested area which means to find an optimal control that reduces the objective functional,subject to fractional dynamics constraintswith initial conditionwhereso, we getwhere represents a state vector and represents a control vector. Initial conditions are and . Manabe and Stouffer [19] formulated the classical optimal control problem at and therefore the dynamic constraint equations (14)–(16) with above initial condition become as = = 0.
It is important to point out that there are thorough justifications in the literature for the formulation of necessary conditions for optimality of various fractional dynamical systems [15, 16]. Therefore,where = is the co-state vector.
We can obtain CO2 emissions system in an enlarged version by applying the compact form of the prerequisites listed previously.
Therefore,
Moreover, the optimal controls are given bywhere denotes the optimal values of and . x(0) = , z(0) = and y(0) = , = 0, = 0, = 0.
5. Numerical Results and Discussion
In this segment, we provide an explanation of a few numerical simulations and their results. We develop a forward-backward 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. The generalised Euler method has recently been used in a TB model along with the forward-backward 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.
The numerical simulations are performed using the initial conditions, and model parameter values are listed as follows: million tons of carbon dioxide, billion international dollars, million·m3, , , , = 0.0006, , and . The graphical representations demonstrate that we provide a significant reduction in the rate of CO2 and reforestation increases with decrease in GDP when = 1 and noninteger orders with time dependent control. The model’s formulation for fractional optimal control demonstrates that by maximizing investments in clean technology research and reforestation activities, significant reductions in CO2 emissions can be attained. Figure 1 represents the graphical result of million tons of CO2, Figure 2 shows the graphical of million·m3/year, Figure 3 shows the graphical representation of billion US dollars, Figure 4 represents emission of CO2 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 CO2 without control with different values of , Figure 8 represents comparison of the forestation rate without control with different values of , Figure 9 represents comparison of GDP without control with different values of , Figure 10 represents comparison of the amount of CO2 with control with different values of , Figure 11 represents comparison of the forestation rate with control with different values of , and Figure 12 represents comparison of GDP with control with different values of .
6. Conclusion
In this study, a mathematical model that displays CO2 emissions, deforestation rate, and GDP has been taken into consideration. The model is characterized by fractional derivatives. The forward-backward sweep approach combined with the extended Euler method was used to numerically solve the optimality problem. The results of the models demonstrate that when control is implemented, the rate of CO2 emissions is reduced and the reforestation rate increases. The amount of CO2 is reduced more than with any other method when all time dependent controls are employed in simulations. The numerical solutions show that the optimum control problem for fractional orders is significantly more precise than involving integer orders.
Data Availability
All the data are available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.