Excellent student’s academic performance is the uppermost priority and goal of educators and facilitators. The dubious marginal rate between admission and graduation rates unveils the rates of dropout and withdrawal from school. To improve the academic performance of students, we optimize the performance indices to the dynamics describing the academic performance in the form of nonlinear system ODE. We established the uniform boundedness of the model and the existence and uniqueness result. The independence and interdependence equilibria were found to be locally and globally asymptotically stable. The optimal control analysis was carried out, and lastly, numerical simulation was run to visualize the impact of the performance index in optimizing academic performance.

The human capital theory perceives education and learning activities as an investment in people to increase the productivity of goods and services [

The academic performance of a student serves as the bedrock for knowledge acquisition and the development of skills that directly impact the socio-economic development of a country [

There are many factors that enhance and impede students’ academic performance attributed to students, parents, teachers and environments. The student’s factors include self- motivation, interest in a subject, punctuality in class, regular studying and access to learning materials. Class attendance and students’ attitudes toward their learning have an impact on academic performance. In [

Qualified teachers and facilitators render effective facilitation which enhances academic performance. However, performance target, completion of syllabus, paying attention to weak students, assignment and student evaluation have significant impact too [

Parental background and status have significant impact on student’s academic performance. Educated parent provide home school tutorial to their ward and are more encouraging as well. In [

Environmental factors that influence academic performance are enabling environment, infrastructure, adequate facilities and learning materials, well-equipped laboratories, etc. In [

Also, fairly disciplined schools perform better than less or no disciplined schools. Effective discipline is used to control students’ behavior, which has a direct impact on their academic performance [

Age has a significant impact on academic performance; older students are likely to drop out than younger ones. Reference [

Mathematical models play a significant role in solving real-life problems [

The paper is arranged as follows: Introduction is given in chapter one, followed by definitions of terms and important theorems in

A fairly general continuous time optimal control problem can be defined as follows:

This special type of optimal control problem is called the minimum time problem.

With a time varying Largrange’s multiplier function

Let

The above theorem states the conditions sufficient for the existence and uniqueness of a fixed point, which we will see in a point that is mapped to itself.

Let

Further,

If

If the functions

This theorem is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.

The model is constructed based on the assumption that the new intake is admitted into the average class at the rate

The performance dynamics is described by the nonlinear system of ODE

Let the population size be

Solving the linear ODE

The long-term behavior of

Meanwhile, as time increases without bound all the solutions converge to the equilibrium

Let the system

Analogously, we obtained

Composing the system

Since

Let

Since

Hence

Define the fixed point operator

So,

For contraction

Since

Since the state

The system

The independence equilibrium

From system

The eigenvalues of

For

Define

Since,

Then

The time derivative of

Applying the relation between arithmetic and geometric means:

Since,

Hence

Define

The time derivative of

Applying the relation between arithmetic and geometric means:

Since,

Then

Hence

The optimal control strategy is aimed at optimizing student’s academic performance which reflects in the increase of number of graduating students.

Let the control rates:

Then the control dynamics is described by the nonlinear system of ODE below:

We seek for optimal control

where

To derive the optimal academic performance of student, define Hamiltonian,

Applying (10),

Analogously,

Applying the optimality condition

Hence,

In this section numerical examples are given to support the analytic results. We use the following values of variables and parameters for the simulations.

To improve the academic performance of students, we optimize the performance indices to the dynamics describing the academic performance in the form of nonlinear system ODE. We established the uniform boundedness of the model and the existence and uniqueness result. The independence and interdependence equilibria were found to be locally and globally asymptotically stable. The optimal control analysis was carried out, and lastly, numerical simulation was run to visualize the impact of the performance index in optimizing academic performance.

From the numerical simulation result, it can be observed that the weak students’ population dominates other populations. This shows that when there is too much intermingling between Weak students and the other categories of students, it will be to the disadvantage of the other students.

The significance of the optimal control is also clearly shown. There is a drastic increase in the populations of Average, below–Average, Excellent and Graduating students’ population after the application of the control. On the other hand, there is a drastic decrease in the population of weak students after the application of the control.

In the future, the fractional analogue of the model should be considered and real data should be used to validate it.