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Mathematical Programming based Climate Crisis

Modelling of the Planet

Nidhika Yadav

Abstract- This paper aims to present a mathematical

programing-based model for climate change control. The

problem is modeled as a convex programming problem in form

of an optimal control problem, since the aim is to consider the

state of planet and at the same time minimize the key factors

involved in climate change globally as well as region wise. The

article aims to present a basic model to elaborate and illustrate

the use of mathematical modeling in climate crisis. This shall

help the experts with right solutions and right values of each

component, which is not limited to just global warming, but to

local warming and even excessive land mining.

Keywords: Mathematical Programming, Convex Programming,

Climate Change, Climate Modelling, Optimal Control Problem

1. Introduction

There are certain problems which can be well represented as optimization techniques.

Climate change and climate crisis fits well in to it. The main reason to use optimization

problems to solve climate crisis problems is that there are well developed algorithms to

solve optimization problems and climate crisis possess certain states, constraints and

objective. The objective is to reduce climate crisis. Further, recent state of art techniques

in evolutionary algorithms and greedy algorithms can also be used to solve an

optimization problem, once formulated. The optimization problem can be formulated as

a maximization problem or a minimization problem.

Optimization problem can be utilized to model the requirements needed to maintain a

particular threshold in a planned process to give the minimum waste, minimum pollution

and minimum local warming (Yadav, 2021) and even monitor global warmings. Climate

change can be modeled as an optimization problem (Vidale and Wolfe, 1957). An

optimization problem has an objective function which needs to be minimized under

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certain constraints. This fits well in the design of mathematical model for climate

objectives. The objectives are to reduce pollution, reduce green house gasses, reduce

carbon foot prints, reduce water impurities, to mention a few. Further, the climate

change modelling is not without constraints. However, this article is an introductory

article for illustration, hence we shall demonstrate the use of mathematical optimization

for sample constraints. The constrains can be increased after careful analysis with climate

experts. It even more than the ones presented in this article. The article aims to present

a basic model to elaborate the use of mathematical modeling in climate crisis and in

helping the experts with right solutions and right values of each component, which is not

limited to just global warming, but to local warming and even excessive land mining.

Every problem has certain variable components, the variable components satisfying the

constraints of the problem is said to lie in the feasible region of the solution space. The

objectives represented as function(s) chosen for the problem of modeling climate change

shall be referred to as objective function, henceforth in the article.

In the problem of modelling climate change, we propose the use of optimal control

problem. The following functions are pre-dominantly required for the climate crisis

modelling using optimal control problem (Vidale and Wolfe, 1957):

1. State Function: This function models the state of the planet, seen from the angle of

climate change, and is a function of time, t.

2. Control Function: The control function controls the input to the climate model, we

develop here, and makes it mandatory for the input variables (typically a vector) to

satisfy constraints. The control function makes sure to adhere to minimize the climate

change objective function(s).

The next section describes the proposed optimal control problem to find the solution to

climate crisis in terms of values needed region wise and property wise to sustain a balanced

climatic environment.

2. Proposed Optimal Control Problem

2.1. Planets State function modelling

Consider that at time t the state function p(t) which models the climatic conditions of

planet, where p(t) is a vector function. The components of p(t) are:

p(t) = [p1(t), p2(t),………,pn(t)]

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Here p1, p2,…pn measure the state of the planet in terms of climate sensitive topics.

Some of the candidates to the state vector are as follow:

1. Index of global warming

2. Average pollution on planet

3. Average water quality of planet

4. Soil Erosion

5. Planation and greenery on planet

6. Temperature on planet

7. Average atmospheric pressure, and so on…

The list can be increased or decreased without effecting the model and without effecting

the solution evaluation techniques. Outputs can be analyzed for various combinations of

importance from being part of state of planet from climate sensitivity consideration. The

aim is to maintain the safe level of each component of the vector p(t) and set a level that

is safe level, any quantity above safe level can be dangerous to the planet. Further, the

rate of change of the vector p(t) is an essential piece of information needed by the

optimal control system.

2.2. Planets Control function modelling

Control function can be earth resources region wise, which has constraints and the use of which

has to be put under the realms of an objective function. The earth resources can be computed

as follow:

u(t) = [u1(t), u2(t),……uk(t)],

where u1(t), u2(t),……uk(t) are the resources which are limited in quantity. One such

representation of u is:

u1(t)= amount of fossil fuel at time t

u2(t) = amount of silicon available in Earth’s crust

u3(t) = amount of rare metals in Earth crust

u4(t) = amount of fresh water available

u5(t)= amount of coal extracted, and so on…

In this way the function u can be defined.

The objective function is given as:

F(p, u) =

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Here, is the dirac delta function and represents the final state. The final state is the

ultimate goal one requires from the optimal control problem.

Further, we note that the control function needs to take care for region wise, climate index as

well. For this we first define what a climate index is. The climate index, c(t), at time t, is defined

as the aggerate of the following:

1. Average temperature at time t

2. Pollution at region at time t

3. Humidity in air at time t

4. Smog in air at time t

5. Difference in atmospheric pressure at time t. This is given as maximum atmospheric

pressure at time – minimum atmospheric pressure

6. Water quality at time t

7. Air quality at time t

A weighted average is computed of all these parameters and this is referred to as climate index,

CI, at region ri.

Let w(t) = [w1(t), w2(t),……..wb(t)] represent the vector of climatic index CI b regions, r1, r2,….rb

across the globe which are to be monitored. Then, the second objective function is given as:

G(p, w) =

The aim is to minimize the objective functions F(.,.) and G(.,.). This becomes a multi objective

optimization problem whose solution can be found using Pareto based optimization. Further, the

initial condition p(0) = p0, is set for the current time, when one start the global planet monitoring

control system. The constraints of the system can be given as follows:

The next section describes future work and conclusions of the proposed model. It is emphasized

here this is a base model for illustration purpose only. It needs to be enhanced with close

discussions with climate experts.

3. Conclusion and Future Work

The future work involving the evaluation of these models, using data, fitting the right methods

in the model. The undefined methods can also be estimated with help of a climate scientist.

Once the model is ready, these can be solved for the variables in the control problem and hence

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the gold standard solutions and limits to be maintained by the regions level and at global level

can be determined.

The future work, involve modelling it further, and using real time data, to keep the global level

systems up. The problem can be solved using any Lagrange’s multipliers using KKT. Further, once

the methods are rightly chosen, this makes it a convex programing problem and hence global

minima is guaranteed to be obtained.

References

1. Vidale, M. L. and Wolfe, H. B. (1957) An operation research study of sales response to

advertising, Operations Research, 5, 370-81

2. Yadav, Nidhika (Sept, 2021) Local Warming and Its Effect on Global Environment (DOI:

10.13140/RG.2.2.16972.54408 )