Design of an integral sliding mode controller for reducing CO₂ emissions in the transport sector to control global warming

Abstract

Carbon dioxide (CO₂) is the significant contributor to greenhouse gases and plays a crucial role in the greenhouse effect and climate change. The primary source of CO₂ emissions is fossil fuel combustion, basically due to human activities and transportation activities. The objective of this research is to develop a dynamic model aimed at mitigating global warming by reducing atmospheric CO₂ emissions resulting from the transportation sector. The model includes equations for atmospheric CO₂ emissions, human population, vehicle population, and global warming. Initially, the stability of the model at each equilibrium point is determined by analyzing the eigenvalues of the Jacobian matrix. Subsequently, sensitivity analysis is performed to predict the impact of any parameter of a vehicle population and CO₂ emissions causing global warming. The vehicle parameters are then optimized by applying an integral sliding mode controller (ISMC) to decrease CO₂ emissions and minimize global warming. The ISMC method effectively reduces CO₂ emissions and offers stability for human and vehicle populations, ultimately leading to a reduction in global warming. It is has been found that reducing the vehicle population by 20% can lead to about 4% reduction in CO₂ emissions. This study integrates optimization control techniques to develop a comprehensive model to address CO₂ emissions and global warming, providing a robust framework for sustainable environmental management.

Introduction

Greenhouse gases, including carbon dioxide (CO₂), methane (CH₄), nitrous oxide (N₂O), and fluorinated gases, are essential components of the Earth’s atmosphere. These gases have the special ability to absorb and emit infrared radiation in a way that helps trap heat inside the Earth’s atmosphere, keeping it warm. This process is known as the greenhouse effect. Without it, the Earth would be too cold for life to exist. However, the issue arises when the concentration of these gases, particularly CO₂, increases significantly. An increase in CO₂ causes a natural greenhouse effect, resulting in global warming. This enhanced warming indeed leads to rising temperatures globally, melting of polar ice caps, increased sea levels, and many other unpredictable and harsh weather phenomena. The consequences include not only rising temperatures but also damaged ecosystems, loss of biodiversity, and threats to human health, increasing significant challenges for future generations.

The exponential growth of the human population has greatly contributed to CO₂ emissions. As the global population increases, so does the demand for energy, food, and land. The more energy, primarily from burning fossil fuels, is used, the greater the volume of CO₂ emissions^1,2,3. Fossil fuel combustion for energy production is the highest source. It includes the burning of coal, oil, and natural gas for electricity generation, transportation, industrial processes, and heating contributing approximately 75% of total CO₂ emissions⁴. The remaining 25% of CO₂ emissions come from other sources, such as deforestation, land-use changes, and industrial processes. York et al.⁵ found a strong correlation between population size and CO₂ emissions, suggesting that population growth leads to increased energy consumption and, consequently, increased emissions. According to the United Nations⁶, unless significant changes are made to the energy consumption patterns, global population growth will continue to be a major source of CO₂ emissions. In addition, deforestation of agricultural land reduces the number of trees that could have absorbed CO₂, reducing the Earth’s capacity to absorb CO₂⁷. Industrialization is an additional source of CO₂ emissions that continues to emit a substantial amount of this gas.

The transport sector is also a major contributor to CO₂ emissions due to the burning of fossil fuels^8,9,10,11. Urban transportation alone represented 23% of the energy-related emissions, while the transportation sector contributes roughly a quarter of the total CO₂ emissions worldwide¹². It is necessary to reduce CO₂ emissions from the transport sector. One effective strategy is promoting the adoption of electric vehicles (EVs). Shifting from usual gasoline and diesel vehicles to EVs can significantly reduce CO₂ emissions. The government can encourage EV adoption through subsidies, tax fees, and infrastructure development, such as expanding charging networks. Remodeling public transport infrastructure and encouraging the use of public transport, cycling, and walking also reduce CO₂ emissions from the transport sector. Additionally, improving fuel efficiency standards and regulations for vehicles can help to reduce emissions from the existing fleet of gasoline and diesel vehicles. Technological advancements such as hybridization and engine optimization can improve fuel efficiency and decrease CO₂ emissions per kilometer of vehicle traveled.

Optimal control theory (OCT) offers various methods to handle different types of dynamic systems and optimization problems. The main approaches include deterministic control, stochastic control, robust control, hybrid control, distributed control, etc. Very few studies are available on the application of OCT to the dynamics of CO₂ capture and emission. Alex et al.¹³ developed a mathematical model to investigate the dynamics of CO₂ emissions from forest plantations. The model included the effects of live biomass, intrinsic growth, and burned area. They applied Pontryagin’s maximum technique of deterministic control to maximize carbon capture from forest plantations. The optimal results in both the optimistic and pessimistic scenarios show that the maximum reforestation effort should be applied after felling. Nassef et al.¹⁴ developed a model to investigate the dynamics of CO₂ capture. They applied the fuzzy optimizer to the model to maximize CO₂ capture capacity. Results show that the proposed method increases CO₂ capture capacity by 10.08% and 9.39%, respectively, compared to surface methodology methods. Verma and Misra¹⁵ investigated the nonlinear model of CO₂ emissions from fossil fuels and industry. They used deterministic optimal control on the model to reduce CO₂ levels and minimized the significant reduction in CO₂ levels through optimizing the parameters. Kurniawan et al.¹⁶ theoretically examined the impact of CO₂ emissions on global warming and melting polar ice caps. They utilized Pontryagin’s Maximum Principle on the model to reduce CO₂ emissions, and simulations showed that controls significantly reduced CO₂ levels and the rate of polar ice cap melting. In conclusion, optimal control theory is a powerful framework for making decisions over time to achieve the best possible outcome for complex dynamic systems.

This study integrates advanced mathematical optimization and control techniques, specifically the Integral Sliding Mode Controller (ISMC), to address the serious issue of CO₂ emissions from the transport sector and their impact on global warming. Unlike traditional models that are essentially descriptive models or isolated control methods. The use of ISMC not only optimizes vehicle production parameters to reduce CO₂ emissions but also ensures stability in human and vehicle populations, providing a robust framework for environmentally sustainable management.

The study is investigated in the following manner: First, the model incorporates equations for atmospheric CO₂ emissions, population, vehicle population, and global warming. Its stability at equilibrium points is analyzed through the behavior of the eigenvalues of the Jacobian matrix. Next, a sensitivity analysis is conducted to investigate the impact of vehicle population and CO₂ emission parameters on global warming. Finally, optimization techniques are applied to vehicle production parameters to reduce CO₂ emissions, leading to the control of global warming. In addition, the results are matched with real-time data. The findings of this research have significant implications for policymakers and environmental planners, offering a scientifically grounded method to achieve long-term climate stability and sustainability. The flow chart for investigation is displayed in Fig. 1.

Fig. 1

Flow chart of the problem.

Mathematical model

The increase in $CO_{2}$ proportionally increased with the increase in the human population and number of vehicles and decreased due to natural phenomena, such as plants absorbing a large amount of $CO_{2}$ during photosynthesis. Under these assumptions, the equation for the $CO_{2}$ level $C\left( t \right)$ is written as

$$\frac{dC}{{dt}} = Q + \delta_{1} V + \eta N - \sigma C.$$

(1)

Here, $Q$ is the constant emission of $CO_{2}$ from certain natural sources. $\delta_{1}$ is the linear growth rate coefficient for the emissions of $CO_{2}$ from vehicles. $\eta$ is the ejection rate coefficient of $CO_{2}$ due to different human activities. $\sigma$ is the decreasing rate of $CO_{2}$ due to natural phenomena.

It is considered that the human population $N\left( t \right)$ grows logistically. An increase in the $CO_{2}$ concentration causes global warming, which adversely affects the human population. Thus, the dynamics of $N\left( t \right)$ are developed as

$$\frac{dN}{{dt}} = rN\left( {1 - \frac{N}{L}} \right) - \theta CN.$$

(2)

Here, $r$ is the intrinsic growth rate, $L$ is the carrying capacity of the population, and $\theta$ is the mortality rate coefficient of the population due to adverse impacts posed by enhanced $CO_{2}$ levels.

The production of vehicles is closely tied to the growth of the human population. As the number of people increases, so does the demand for transportation. This heightened demand has led to a rise in the number of vehicles on the road, as more individuals and families require personal and public transport. Thus, the equation of vehicle population $V\left( t \right)$ is taken as human population growth as

$$\frac{dV}{{dt}} = \alpha V\left( {1 - \frac{V}{{L_{1} }}} \right) - \omega V,$$

(3)

where $\alpha$ is the intrinsic growth rate, $L_{1}$ is the carrying capacity of the vehicle population and $\omega$ is the depletion rate coefficient of vehicles due to accidents, scrappage programs, etc.

Global warming refers to the increase in Earth’s average surface temperature due to increased levels of greenhouse gases, primarily carbon dioxide. This increase is largely driven by human activities such as fossil fuel combustion, deforestation, and industrial processes. Thus, the equation of global warming $G\left( t \right)$ under these conditions is written as

$$\frac{dG}{{dt}} = A + \beta C + \lambda N - dG.$$

(4)

In the above equation, $A$ shows the global warming through different natural sources. The $\beta$ and $\lambda$ are the coefficients of growth rate of global warming caused by $CO_{2}$ and human population. The $d$ is the natural reduction rate of global warming.

Considering Eqs. (1)–(4), the proposed problem is expressed in Fig. 2.

Fig. 2

A representation of the problem.

The mathematical model of the problem includes Eqs. (1)–(4) related to $C\left( t \right)$, $N\left( t \right)$, $V\left( t \right)$ and $G\left( t \right)$. It is expressed as^17,18,19

$$\begin{aligned} & \frac{dC}{{dt}} = Q + \delta_{1} V + \eta N - \sigma C, \\ & \frac{dN}{{dt}} = rN\left( {1 - \frac{N}{L}} \right) - \theta CN, \\ & \frac{dV}{{dt}} = \alpha V\left( {1 - \frac{V}{{L_{1} }}} \right) - \omega V, \\ & \frac{dG}{{dt}} = A + \beta C + \lambda N - dG, \\ \end{aligned}$$

(5)

where $C\left( 0 \right) \ge Q,\, \, N\left( 0 \right) \ge 0,\, \, V\left( 0 \right) \ge 0,\, \, G\left( 0 \right) \ge 0.$ Region of attraction for the dynamical system Eq. (5) is given by

$$\Omega = \left\{ {\left( {C,N,E,G} \right) \in R^{4} :0 \le C \le C_{m} ,0 \le N \le N_{m} ,0 \le V \le V_{m} ,0 \le G \le G_{m} } \right\},$$

where $C_{m} = \frac{{Q + \delta_{1} V_{m} + \eta N_{m} }}{\sigma },\,$ $N_{m} = \frac{r}{\theta C},\,$ $\,V_{m} = \frac{\alpha }{\omega }$ and $G_{m} = \frac{{A + \beta C_{m} + \lambda N_{m} }}{d}$.

Stability analysis

Stability analysis examines how the solutions of a system behave as time progresses, particularly in response to changes in initial conditions or parameters. Following are the key steps involved in stability analysis of system Eq. (5).

Local stability

The possible equilibria of model Eq. (5) are given below as

• 1. $S_{1} \left( {\frac{Q}{\sigma },0,0,\frac{Q\beta + A\sigma }{{d\sigma }}} \right)$,

• 2. $S_{2} = \left( {\frac{{r\left( {Q + L\eta } \right)}}{L\eta \theta + r\sigma },\,\frac{{L\left( {r\sigma - Q\theta } \right)}}{L\eta \theta + r\sigma },\,0,\,\frac{Qr\beta + Lr\beta \eta + AL\eta \theta - LQ\theta \lambda + Ar\sigma + Lr\lambda \sigma }{{d\left( {L\eta \theta + r\sigma } \right)}}} \right)$,

• 3. $S_{3} = \left( {C_{3} ,\,0,\,V_{3} ,\,G_{3} } \right)$,

• 4. $S_{4} = \left( {C_{4} ,\,N_{4} ,\,V_{4} ,\,G_{4} } \right)$,

where

$C_{3} = \frac{{Q\alpha + L_{1} \delta_{1} \left( {\alpha - \omega } \right)}}{\alpha \sigma }$, $V_{3} = L_{1} - \frac{{L_{1} \omega }}{\alpha }$, $G_{3} = \frac{{Q\alpha \beta + A\alpha \sigma + L_{1} \beta \delta_{1} \left( {\alpha - \omega } \right)}}{d\alpha \sigma }$, $C_{4} = \frac{{r\left( {\left. {Q\alpha + L\alpha \eta + L_{1} \delta_{1} \left( \alpha \right. - \omega } \right)} \right)}}{{\alpha \left( {L\eta \theta + r\sigma } \right)}}$, $C_{4} = \frac{{L\left( {\left. { - Q\alpha \theta + r\alpha \sigma + L_{1} \delta_{1} \theta \left( \alpha \right. - \omega } \right)} \right)}}{{\alpha \left( {L\eta \theta + r\sigma } \right)}}$, and $C_{4} = \frac{{Q\alpha \left( {\left. {r\beta - L\theta \lambda } \right) + \sigma \left( {Lr\beta \eta + AL\eta \theta + Ar\sigma + Lr\lambda \sigma } \right) + L_{1} \delta_{1} \left( {r\beta - L\theta \lambda } \right)\left( {\alpha - \omega } \right.} \right)}}{{\alpha d\left( {L\eta \theta + r\sigma } \right)}}$.

The local stability at equilibrium points $S_{1}$, $S_{2}$, $S_{3}$ and $S_{4}$ is determined through the sign of eigenvalues of the following Jacobian matrix $J$ that is defined as

$$J = \left[ {\begin{array}{*{20}l} { - \sigma } \hfill & \eta \hfill & {\delta_{1} } \hfill & 0 \hfill \\ { - N\theta } \hfill & {\frac{ - Nr}{L} + \left( {1 - \frac{N}{L}} \right) - C\theta } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\frac{ - V\alpha }{{L_{1} }} + \left( {1 - \frac{V}{{L_{1} }}} \right)\alpha - \omega } \hfill & 0 \hfill \\ \beta \hfill & \lambda \hfill & 0 \hfill & { - d} \hfill \\ \end{array} } \right].$$

(6)

Take $J_{i}$ $\left( {i = 1,2,3,4} \right)$ Jacobian matrix that is determined at equilibrium point $S_{i} \left( {i = 1,2,3,4} \right)$.

For $S_{1}$, the eigenvalues of $J_{1}$ are $\left( { - d,\,\, - \sigma ,\,\,\frac{r\sigma - Q\theta }{\sigma },\,\,\alpha - \omega } \right)$, and the system is unstable at this equilibrium point if $\alpha > \omega$. Moreover, the eigenvalues of the $J_{2}$ are

$$\left( { - d,\,\frac{{ - b + \sqrt {b^{2} - 4ac} }}{2a},\frac{{ - b - \sqrt {b^{2} - 4ac} }}{2a},\alpha - \omega } \right),$$

(7)

where $a = L\eta \theta + r\sigma$, $b = Qr\theta - r^{2} \sigma - L\eta \theta \sigma - r\sigma^{2}$, and $c = - LQ\eta \theta^{2} - Qr\theta \sigma + Lr\eta \theta \sigma + r^{2} \sigma^{2}$. Thus, the equilibrium point $S_{2}$ is unstable if $\alpha > \omega$.

The characteristic equation of $J_{i} (i = 3,4)$ for $S_{i} (i = 3,4)$ is given by

$$x^{4} + A_{1} x^{3} + A_{2} x^{2} + A_{3} x + A_{4} = 0,$$

(8)

where

$$A_{1} = \frac{1}{{LL_{1} }}\left( {dLL_{1} - LL_{1} r + 2L_{1} N_{i} r + LL_{1} \sigma + C_{i} LL_{1} \sigma + LV\left( {V_{i} - L_{1} } \right)\alpha \omega } \right),$$

$$A_{2} = \frac{1}{{LL_{1} }}\left( \begin{gathered} - dLL_{1} r + 2dL_{1} N_{i} r + dLL_{1} \sigma + C_{i} dLL_{1} \sigma - LLr\sigma + 2L_{1} N_{i} r\sigma + LL_{1} N_{i} \eta \sigma \hfill \\ + C_{i} LL_{1} \sigma^{2} + dLV_{i} \left( {V_{i} - L_{1} } \right)\alpha \omega - LrV_{i} \left( {V_{i} - L_{1} } \right)\alpha \omega + 2N_{i} rV_{i} \left( {V_{i} - L_{1} } \right)\alpha \omega \hfill \\ + LV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega + C_{i} LV\left( {V_{i} - L_{1} } \right)\alpha \sigma \omega \hfill \\ \end{gathered} \right),$$

$$A_{3} = \frac{1}{{LL_{1} }}\left( \begin{gathered} - dLL_{1} r\sigma + 2dL_{1} N_{i} r\sigma + dLL_{1} N_{i} \eta \sigma + C_{i} dLL_{1} \sigma^{2} - dLrV_{i} \left( {V_{i} - L_{1} } \right)\alpha \omega \hfill \\ + 2dN_{i} rV_{i} \left( {V_{i} - L_{1} } \right)\alpha \omega + dLV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega + C_{i} dLV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega \hfill \\ - LrV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega + 2N_{i} rV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega + LN_{i} V_{i} \left( {V_{i} - L_{1} } \right)\alpha \eta \sigma \omega \hfill \\ + C_{i} LV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma^{2} \omega \hfill \\ \end{gathered} \right),$$

$$A_{4} = \frac{1}{{LL_{1} }}\left( \begin{gathered} - dLrV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega + 2dN_{i} rV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma \omega \hfill \\ + dLN_{i} V_{i} \left( {V_{i} - L_{1} } \right)\alpha \eta \sigma \omega + C_{i} dLV_{i} \left( {V_{i} - L_{1} } \right)\alpha \sigma^{2} \omega \hfill \\ \end{gathered} \right),$$

where the coefficients $A_{i} \left( {i = 1,2,3,4} \right)$ are positive. Equation (8) has either negative or positive eigenvalues iff the following Routh–Hurwitz condition is satisfied.

$$A_{3} \left( {A_{1} A_{2} - A_{3} } \right) - A_{1}^{2} A_{4} > 0$$

(9)

Therefore, the system is stable at $S_{i} (i = 3,4)$ if Eq. (9) holds.

Theorem 1

The system in Eq. (5) at equilibriums $S_{1}$ and $S_{2}$ is always unstable under the condition $\alpha > \omega$. The system at equilibrium points $S_{i}$ is locally asymptotically stable iff Eq. (9) is hold.

Global stability

Now, stability of the Liapunov function is applied to determine the global stability. The following theorem illustrates the conditions of global stability.

Theorem 2

The Eq. (5) is globally stable in region $\Omega$ if following conditions are hold

$$m_{2} < \frac{{2\delta_{1}^{2} L_{1} }}{\alpha \sigma },$$

(10)

$$m_{3} < \min \left\{ {\frac{\eta rd}{{2\theta L\lambda^{2} }},\frac{d\sigma }{{4\beta^{2} }}} \right\}.$$

(11)

Proof

Consider the following positive function:

$$V = \frac{1}{2}\left( {C - C^{ * } } \right)^{2} + m_{1} \left( {N - N^{ * } - N^{ * } \ln \frac{N}{{N^{ * } }}} \right) + m_{2} \left( {V - V^{ * } - V^{ * } \ln \frac{V}{{V^{ * } }}} \right) + \frac{{m_{3} }}{2}\left( {G - G^{ * } } \right)^{2} ,$$

(12)

where $m_{1}$, $m_{2}$ and $m_{3}$ are positive constants. Equation (5) is globally stable if $\frac{dV}{{dt}} < 0$ at all equilibrium points. Therefore, the derivative of Eq. (11) is calculated as

$$\frac{dV}{{dt}} = \left( {C - C^{ * } } \right)\frac{dC}{{dt}} + m_{1} \frac{{\left( {N - N^{ * } } \right)}}{N}\frac{dN}{{dt}} + m_{2} \frac{{\left( {V - V^{ * } } \right)}}{V}\frac{dV}{{dt}} + m_{3} \left( {G - G^{ * } } \right)\frac{dG}{{dt}}$$

(13)

Using Eqs. (1–4), we obtain

$$\begin{aligned} & \frac{dV}{{dt}} = \left( {C - C^{ * } } \right)\left[ {Q + \delta_{1} V + \eta N - \sigma C} \right] + m_{1} \left( {N - N^{ * } } \right)\left[ {r\left( {1 - \frac{N}{L}} \right) - \theta C} \right] \\ & \quad + \,m_{2} \left( {V - V^{ * } } \right)\left[ {\alpha \left( {1 - \frac{V}{{L_{1} }}} \right) - \omega } \right] + m_{3} \left( {G - G^{ * } } \right)\left[ {A + \beta C + \lambda N - dG} \right]. \\ \end{aligned}$$

(14)

Equation (14) is rewritten when the condition for finding the equilibrium point $S^{*} \left( {C^{*} ,N^{*} ,V^{*} ,G^{*} } \right)$ for Eq. (5) is

$$\begin{aligned} \frac{dV}{{dt}} & = \left( {C - C^{ * } } \right)\left[ {Q + \delta_{1} V + \eta N - \sigma C - \left\{ {Q + \delta_{1} V^{ * } + \eta N^{ * } - \sigma C^{ * } } \right\}} \right] \\ & \quad + \,m_{1} \left( {N - N^{ * } } \right)\left[ {r\left( {1 - \frac{N}{L}} \right) - \theta C - \left\{ {r\left( {1 - \frac{{N^{ * } }}{L}} \right) - \theta C^{ * } } \right\}} \right]. \\ & \quad + \,m_{2} \left( {V - V^{ * } } \right)\left[ {\alpha \left( {1 - \frac{V}{{L_{1} }}} \right) - \omega - \left\{ {\alpha \left( {1 - \frac{{V^{ * } }}{{L_{1} }}} \right) - \omega } \right\}} \right] \\ & \quad + \,m_{3} \left( {G - G^{ * } } \right)\left[ {A + \beta C - dG + \lambda N - \left\{ {A + \beta C^{ * } - dG^{ * } + \lambda N^{ * } } \right\}} \right]. \\ \end{aligned}$$

(15)

Rewriting Eq. (15) as

$$\begin{aligned} \frac{dV}{{dt}} & = \left( {C - C^{ * } } \right)\left[ {\delta_{1} \left( {V - V^{ * } } \right) + \eta \left( {N - N^{ * } } \right) - \sigma \left( {C - C^{ * } } \right)} \right] \\ & \quad + \,m_{1} \left( {N - N^{ * } } \right)\left[ {\frac{ - r}{L}\left( {N - N^{ * } } \right) - \theta \left( {C - C^{ * } } \right)} \right] + m_{2} \left( {V - V^{ * } } \right)\left[ {\frac{ - \alpha }{{L_{1} }}\left( {V - V^{ * } } \right)} \right] \\ & \quad + \,m_{3} \left( {G - G^{ * } } \right)\left[ {\beta \left( {C - C^{ * } } \right) - d\left( {G - G^{ * } } \right) + \lambda \left( {N - N^{ * } } \right)} \right], \\ \end{aligned}$$

(16)

$$\begin{aligned} \frac{dV}{{dt}} & = \delta_{1} \left( {V - V^{ * } } \right)\left( {C - C^{ * } } \right) + \left( {\eta - m_{1} \theta } \right)\left( {N - N^{ * } } \right)\left( {C - C^{ * } } \right) - \sigma \left( {C - C^{ * } } \right)^{2} \\ & \quad - \frac{{m_{1} r}}{L}\left( {N - N^{ * } } \right)^{2} - \frac{{m_{2} \alpha }}{{L_{1} }}\left( {V - V^{ * } } \right)^{2} + m_{3} \beta \left( {G - G^{ * } } \right)\left( {C - C^{ * } } \right) \\ & \quad + \,m_{3} \lambda \left( {G - G^{ * } } \right)\left( {N - N^{ * } } \right) - m_{3} d\left( {G - G^{ * } } \right)^{2} . \\ \end{aligned}$$

(17)

Choosing $m_{1} = \frac{\eta }{\theta }$, Eq. (17) becomes

$$\begin{aligned} \frac{dV}{{dt}} & = - \frac{\sigma }{2}\left( {C - C^{ * } } \right)^{2} + m_{3} \beta \left( {G - G^{ * } } \right)\left( {C - C^{ * } } \right) - \frac{{m_{3} d}}{2}\left( {G - G^{ * } } \right)^{2} \\ & \quad - \frac{\sigma }{2}\left( {C - C^{ * } } \right)^{2} + \delta_{1} \left( {V - V^{ * } } \right)\left( {C - C^{ * } } \right) - \frac{{m_{2} \alpha }}{{L_{1} }}\left( {V - V^{ * } } \right)^{2} \\ & \quad - \frac{\eta r}{{\theta L}}\left( {N - N^{ * } } \right)^{2} + m_{3} \lambda \left( {G - G^{ * } } \right)\left( {N - N^{ * } } \right) - \frac{{m_{3} d}}{2}\left( {G - G^{ * } } \right)^{2} . \\ \end{aligned}$$

(18)

Note that $a_{1} x^{2} + a_{2} xy + a_{3} y^{2}$ is negatively defined if $a_{1} < 0$ and $a_{2}^{2} < 4a_{1} a_{3}$. Using this condition, $\frac{dV}{{dt}}$ is negatively defined within the region of attraction $\Omega$ when the provided conditions in Eqs. (10) and (11) for $m_{2}$ and $m_{3}$ hold.

Parameter estimation

To find the values of model’s parameters, fitting the Eq. (5) with the annual data on the atmospheric $CO_{2}$ concentration²⁰, global population²¹, vehicle population²², and global warming²³ from 2000 to 2023. The values of variables $C,$ $N,$ $V$ and $G$ at year 2000 are taken as initial conditions and follow as

$$\left. \begin{gathered} C\left( {2000} \right) = 370.10 \hfill \\ N\left( {2000} \right) = 6.151 \hfill \\ V\left( {2000} \right) = 58.948 \hfill \\ G\left( {2000} \right) = 0.43 \hfill \\ \end{gathered} \right\}.$$

(19)

For the curve fitting, the FindFit command in Mathematica 13.2 was used. The values of the model’s parameters are obtained as follows

$$\begin{aligned} & Q = 4.6,\,\,\eta = 0.16,\,\,\sigma = 0.01621,\,\,r = 0.045,\,\,L = 12,\delta_{1} = 0.029,\,\,\theta = 2.5 \times 10^{ - 6} ,\, \\ & \alpha = 0.033,\,\,L_{1} = 200,\,\omega = 0.0001,\,\,A = 0.0005,\,\,\beta = 0.00007,\,\,\lambda = 0.00001,\, \\ & d = 0.001. \\ \end{aligned}$$

(20)

The curve fitting results are displayed in Figs. 3, 4, 5 and 6, and good agreement is found between the Eq. (5) and actual data.

Fig. 3

Fitting Eq. (5) with data of carbon dioxide data.

Fig. 4

Fitting Eq. (5) with data of population.

Fig. 5

Fitting Eq. (5) with data of vehicle population.

Fig. 6

Fitting the Eq. (5) with data of global warming.

Sensitivity analysis

Sensitivity analysis is a technique used to evaluating a dynamic system in response of parameter or initial condition variations. To evaluate the sensitivity of the model in respect of parameters, $w\frac{dX}{{dw}}$ is needed to calculate, where $X$ is the dependent function and parameter $w$ is an independent function. The sensitivity expressions of Eq. (5) for parameters $\delta_{1}$ and $\alpha$ are expressed as

$$\left. \begin{aligned} & \dot{C}_{{\delta_{1} }} = V + \eta N_{{\delta_{1} }} - \sigma C_{{\delta_{1} }} , \\ & \dot{N}_{{\delta_{1} }} = r\left( {1 - \frac{2N}{L}} \right)N_{{\delta_{1} }} - \theta C_{{\delta_{1} }} N - \theta CN_{{\delta_{1} }} , \\ & \dot{V}_{{\delta_{1} }} = \alpha \left( {1 - \frac{2V}{{L_{1} }}} \right)V_{{\delta_{1} }} - \omega V_{{\delta_{1} }} , \\ & \dot{G}_{{\delta_{1} }} = \beta C_{{\delta_{1} }} + \lambda N_{{\delta_{1} }} - dG_{{\delta_{1} }} . \\ \end{aligned} \right\},$$

(21)

$$\left. \begin{aligned} & \dot{C}_{\alpha } = V + \eta N_{\alpha } - \sigma C_{\alpha } , \\ & \dot{N}_{\alpha } = r\left( {1 - \frac{2N}{L}} \right)N_{\alpha } - \theta C_{\alpha } N - \theta CN_{\alpha } , \\ & \dot{V}_{\alpha } = \alpha \left( {1 - \frac{2V}{{L_{1} }}} \right)V_{\alpha } - \omega V_{\alpha } , \\ & \dot{G}_{\alpha } = \beta C_{\alpha } + \lambda N_{\alpha } - dG_{\alpha } . \\ \end{aligned} \right\}.$$

(22)

The sensitivity analysis results over time $\left[ {t_{o} , \, t_{1} } \right]$ are calculated as $\int\limits_{{t_{o} }}^{{t_{1} }} {w\frac{dX}{{dw}}} dt$ and shown in Figs. 6, 7, 8 and 9, provide valuable insights into the influence of the parameters $\delta_{1}$ and $\alpha$ on $C\left( t \right)$, $N\left( t \right)$, $V\left( t \right)$ and $G\left( t \right)$. These figures provide critical understandings into how changes in vehicle growth rates and CO₂ emission influence the dynamics of the system.

Fig. 7

Results of sensitivity analysis of C in variation of parameters α and $\delta_{1}$.

Fig. 8

Results of sensitivity analysis of N in variation of parameters α and $\delta_{1}$.

Fig. 9

Results of sensitivity analysis of V in variation of parameters α and $\delta_{1}$.

Figure 7 shows the growth rate $\alpha$ of vehicles and the linear growth rate coefficient of CO₂ $\delta_{1}$ by vehicle at the $C\left( t \right)$ level. $\alpha$ has a significantly greater impact than $\delta_{1}$ on increasing $C\left( t \right).$ These results show that as the population of vehicles increases $\alpha$, this leads to increased use and consequently higher emissions. In contrast, $\delta_{1}$ affects emissions growth more linearly, resulting in a lesser impact on $C\left( t \right)$.

Figure 8 shows the impact of the parameters $\delta_{1}$ and $\alpha$ on the human population $N\left( t \right)$. $\alpha$ has a more negative effect than $\delta_{1}$ on $N\left( t \right)$. This negative sensitivity illustrates that an increase in $\alpha$ leads to a significant decrease in the population due to the adverse impacts of increased CO₂ levels on health. $\delta_{1}$ decreases the population, but its effect is less prominent because it affects CO₂ emissions indirectly through vehicle emissions rather than directly through population growth.

Figure 9 illustrates the influence of parameters $\alpha$ and $\delta_{1}$ on vehicle population $V\left( t \right)$. $\alpha$ does not affect $V\left( t \right)$, whereas $\delta_{1}$ has a positive and considerable impact. This shows that the growth rate of vehicles $\alpha$ is more strongly associated with the number of vehicles already on the road, while $\delta_{1}$ affects the growth rate of CO₂ emissions from vehicles, which indirectly influences the vehicle population through policies, regulations, and environmental awareness.

Figure 10 shows the effects of $\alpha$ and $\delta_{1}$ on global warming $G\left( t \right)$. Both have positive impacts on $G\left( t \right)$, but $\alpha$ shows a dominant effect. This is because increased vehicle usage contributes significantly to global warming through elevated CO₂ emissions. Although $\delta_{1}$ also affects global warming, its impact is mediated through changes in emission rates rather than direct changes in vehicle numbers.

Fig. 10

Results of sensitivity analysis of G in variation of parameters α and $\delta_{1}$.

In short, the analysis of sensitivity identifies vehicle numbers increase as the most important factor that affects the rate of CO₂ emissions, population of humans, and global warming. Although the coefficient of the linear growth rate of the rate of CO₂ emissions is important too, its impact is less pronounced compared to the vehicle population increase. This trend of the results is similar to trend found in study¹⁹.

Control scheme

In this section, the integral sliding mode controller (ISMC) is applied to obtain a balance between the production of vehicles and carbon dioxide to control global warming.

Design of an integral sliding mode controller

Sliding surface

First, the surface is defined in the form of tracking error as

$$S = \tilde{e} + \int\limits_{0}^{t} {\tilde{e} \, dt} ,$$

(23)

where $\tilde{e}$ is the tracking error and is defined as

$$\tilde{e} = V - V_{d} .$$

(24)

Here, $V_{d}$ is the desired trajectory. Equation (23) is written as

$$\dot{S} = \dot{\tilde{e}} + \tilde{e}.$$

(25)

Equation (24) is expressed as

$$\dot{S} = \dot{V} - \dot{V}_{d} + V - V_{d} .$$

(26)

Desired trajectory

Since $V$ linearly increases over time, we define the desired trajectory $V_{d}$ in the form of a linear function as

$$V_{d} = V_{0} + \frac{{\left( {V_{f} - V_{0} } \right)}}{T}t,$$

(27)

where $V_{0}$ is the initial value and $V_{f}$ is the final value.

Equation (26), by using Eqs. (27) and (3), is expressed as

$$\dot{S} = \alpha V\left( {1 - \frac{V}{{L_{1} }}} \right) + \left( {1 - \omega } \right)V - V_{0} - \left( {1 + t} \right)\frac{{\left( {V_{f} - V_{0} } \right)}}{T}.$$

(28)

Equivalent control

Now, calculate the equivalent control $\alpha_{eq}$ by setting $\dot{s} = 0$ as

$$\alpha_{eq} = \frac{{\left( {\omega - 1} \right)V + V_{0} + \left( {1 + t} \right)\frac{{\left( {V_{f} - V_{0} } \right)}}{T}}}{{V\left( {1 - \frac{V}{{L_{1} }}} \right)}}.$$

(29)

Control law

The control law $\alpha$ consists of the equivalent control $\alpha_{eq}$​ and a corrective term to handle disturbances and uncertainties as

$$\alpha = \alpha_{eq} - \frac{k \, }{{V\left( {1 - \frac{V}{{L_{1} }}} \right)}}S\tanh \left( S \right).$$

(30)

where $k$ is a positive constant to ensure the robustness of the control law. The flow chart of control scheme is shown in Fig. 11.

Fig. 11

Flow chart of control scheme.

Stability of Lyapunov

Now, consider a quadratic Lyapunov function V in terms of the sliding surface as follows:

$$f = \frac{1}{2}S^{2} .$$

(31)

This function is positive $f > 0$ for all $S \ne 0$ and $f = 0$ if $S = 0$. To show the stability, it is necessary to show that the time derivative of the Lyapunov function is $f^{\prime} < 0$.

The take the time derivative of $f$ as

$$\dot{f} = S\dot{S},$$

(32)

where $\dot{S}$ is expressed as

$$\dot{S} = \left( {\alpha_{eq} - \frac{k}{{V\left( {1 - \frac{V}{{L_{1} }}} \right)}}S\tanh \left( S \right)} \right)V\left( {1 - \frac{V}{{L_{1} }}} \right) + \left( {1 - \omega } \right)V - V_{0} - \left( {1 + t} \right)\frac{{\left( {V_{f} - V_{0} } \right)}}{T}.$$

(33)

After simplification, we have

$$\dot{S} = - k \, S\tanh \left( S \right).$$

(34)

By using Eqs. (34), (31) is expressed as

$$\dot{V} = - ks^{2} \tanh \left( S \right), \, \Rightarrow \, \dot{V} \le 0.$$

(35)

This indicates that the Lyapunov function $V\left( S \right)$ is not increasing over time, implying that the sliding surface $S$ is stable.

The desired function plays a central role in the application of the ISMC. By controlling the vehicle population to follow the desired function Eq. (27), the ISMC indirectly influences CO₂ emissions, human population dynamics, and global warming. It’s results are presented in Figs. 12, 13, 14, 15 and 16 and compared with the results of system Eq. (5).

Fig. 12

Comparison of C with and without ISMC.

Fig. 13

Comparison of N with and without ISMC.

Fig. 14

Comparison of V with and without ISMC.

Fig. 15

Comparison of G with and without ISMC.

Fig. 16

The result of the integral sliding surface S.

Figure 12 shows the levels of CO₂ in the atmosphere from 2000 to 2045. When compared with the system without ISMC, the application of ISMC significantly reduced CO2 levels. The controller effectively maintains CO₂ emissions within desired limits by optimizing vehicle population growth and emissions.

Figure 13 compares human population dynamics with and without ISMC. The ISMC helps stabilize the human population by maintaining lower CO₂ levels, reducing the adverse health effects associated with high CO₂ concentrations. As a result, the mortality rate due to environmental degradation is minimized, leading to more stable and sustainable population growth.

Figure 14 shows vehicle population dynamics over the given period. With ISMC, the vehicle population follows the desired trajectory $V_{f} = 130$ more closely. The controller ensures the vehicle population does not exceed the optimal level, balancing the need for transportation with environmental sustainability. This optimal vehicle population contributes to reduced CO₂ emissions and aligns with desired environmental goals.

Figure 15 shows the effects of ISMC on global warming. The reduction in CO₂ emissions and stabilization of human and vehicle population have resulted in a significant decrease in global warming rates. ISMC ensures global warming levels are controlled, mitigating long-term impact on the environment and promoting a healthier climate.

Figure 16 illustrates the performance of the sliding surface under ISMC. The sliding surface shows the system remains stable and follows the desired trajectory efficiently. The controller’s ability to maintain the system on the sliding surface indicates robust performance in the presence of disturbances and uncertainties. This stability is crucial for achieving long-term sustainability in vehicle population management and CO₂ emission control.

Conclusion

This study presents a detailed dynamic model in a robust framework to see the impact of atmospheric CO₂ emissions from the transport sector on global warming. The results highlight the adoption of advanced mathematical control technique to develop effective strategies to control global warming. The ISMC is applied qualitatively effectively for optimization with regard to vehicle population growth and CO₂ emissions and showed high potential for reducing atmospheric CO₂ levels and, hence, global warming. ISMC effectively reduces CO₂ emissions, with a 20% reduction in vehicle population leading to a 4% drop in emissions. It stabilizes human and vehicle populations by minimizing health risks and balancing transportation needs with environmental sustainability. Additionally, the stability and sensitivity of the model are analyzed and conditions for stability are found at the equilibrium points, as well as critical factors that influence CO₂ levels, population, and global warming.