Control and Optimization in Applied Mathematics

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Exploration of Physics Informed Neural Network for Solving Optimal Tracking Control Problems

Articles in Press

Document Type : Research Article

Authors

1 Department of Mathematics, Universitas Sumatera Utara, Medan, 20155, Indonesia

2 Department of Data Science, Institut Teknologi Sumatera, Lampung, 35365, Indonesia

10.30473/coam.2025.75192.1322

Abstract

In this study, we examine solutions to Optimal Tracking Control (OTC) problems for both Linear Quadratic (LQ) and nonlinear systems. Classical approaches to OTC rely on formulating and solving the Hamilton-Jacobi-Bellman (HJB) equation, which typically requires numerical solutions of the state, co-state, and stationary equations using the forward-backward method. Such methods often involve intricate mathematical analysis and substantial computational effort. To address these challenges, we explored the use of Physics Informed Neural Networks (PINN) as an alternative framework for solving OTC problems. The PINN approach is implemented by constructing a problem-specific loss function that directly incorporates the governing dynamics and control objectives. This method is comparatively simpler and more flexible to implement. The performance of PINNs is evaluated through quantitative error analysis and benchmarked against the classical Runge-Kutta (RK) method. A detailed comparison is presented using tabulated error metrics and time-domain plots of absolute errors. Numerical results demonstrate that PINNs achieve lower approximation errors than Runge-Kutta method for both LQ and nonlinear tracking problems, indicating their effectiveness as a viable alternative solution strategy for OTC problems. 

Highlights

  • Physics-Informed Neural Networks (PINNs) are applied to optimal tracking control problems
  • Both linear quadratic and nonlinear tracking systems are addressed within a unified framework
  • PINNs eliminate the need for solving Hamilton–Jacobi–Bellman equations
  • Superior tracking accuracy is achieved compared to the Runge–Kutta method
  • SiLU activation function yields the lowest tracking errors among tested activations
  • PINNs offer a flexible data-driven alternative for complex nonlinear control problems

Keywords

Main Subjects

References
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Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 11 January 2026
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