Constructing a Precise Method to Control Non-Linear Systems Employing Special Functions and Machine Learning

Abstract

This paper is about finding the best way to control certain types of equations that describe heat and diffusion using methods that are not exact but close to the best solution. A new way to control things better is suggested using patterns we've seen before and computer learning. By employing the data gathered from the EEFs are computed using the Karhunen-Loève decompose in the PDE computer. The EEFs are used to convert the PDE system into a high-order ODE system. A smaller model (ROM) that may depict the primary behavior of the PDE subsystem is created using the SP approach. This facilitates understanding of the system. In order to reduce its size, a more basic model (ROM) that demonstrates the primary motions of the PDE system is created using the SP approach. Next, the optimal controller for the ROM is designed using the HJB technique. This ensures that the high-order ODE system is stable using the SP theory. By splitting the best way to control something into two sections, we can solve a math problem to figure out the first part and come up with a new equation for the second part. Third, a method is suggested for updating and controlling based on small improvements to solve a specific equation, and it has been proven to work effectively. Moreover, we use a technique called the NN approach to estimate the cost function. Finally, the simulation results demonstrate the effectiveness of the proposed optimum control approach when applied to a diffusion-reaction process using a unique spatially component.