Document Type : Research Article
Author
Department of Mathematics, MaS.C., Islamic Azad University, Masjed Soleiman, Iran.
Abstract
The orthogonal polynomials approximation method is widely regarded as a highly effective and versatile technique for solving optimal control problems in nonlinear systems. This powerful approach has found extensive applications in both theoretical research and practical engineering, demonstrating its capability to address complex dynamical behaviors. In this paper, we thoroughly investigate the optimal control problem of the Van der Pol oscillator, a classic nonlinear system with broad scientific and engineering relevance. The proposed solution follows two distinct and systematic steps. First, the state and control functions are approximated by linear combinations of shifted Chelyshkov polynomials, whose coefficients are treated as unknown parameters to be determined. Second, the resulting transformed problem is formulated as a nonlinear optimization problem and efficiently solved using advanced numerical optimization tools implemented in \textsc{Matlab}. To demonstrate the accuracy and robustness of the proposed approach, we present and analyze numerical results across several representative scenarios.
Highlights
- Introduces a direct numerical scheme for nonlinear optimal control of the Van der Pol oscillator based on finite Chelyshkov-polynomial series, converting a two-point boundary-value problem into a finite-dimensional nonlinear program solvable with standard interior-point solvers.
- State and control trajectories are represented as linear combinations of shifted Chelyshkov polynomials, enabling projection onto an orthogonal basis and delivering accurate approximations with relatively few basis functions.
- Numerical experiments show extreme precision gains with modest increases in polynomial order (e.g., from m=7 to m=8), yielding negligible objective-value changes on the order of 8×10ˆ(-12), confirming robustness across representative scenarios.
- The Chelyshkov framework achieves comparable or superior accuracy with fewer basis functions and reduced computational cost, highlighting practical efficiency for nonlinear OCPs.
- The approach naturally enforces both standard and multipoint boundary conditions, offering simplicity and transparency often lacking in indirect methods such as Pontryagin’s Maximum Principle or dynamic programming.
- The framework is well-suited to high-dimensional systems. It can be extended to handle parameter uncertainty, measurement noise, and partially unknown models, with potential integration of adaptive-dynamic-programming ideas.
- Implemented in MATLAB, the solution framework provides a scalable platform for applying orthogonal polynomial collocation to a broad class of nonlinear optimal control problems, extending beyond the Van der Pol oscillator.
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