فهرست نشریات علمی دسترسی آزاد ایران

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.