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Dehghan, Reza. (1404). Optimal Control of the Van der Pol Oscillator Problem by Using Orthogonal Polynomial-Based Optimization. فصلنامه علمی زیست فناوری گیاهان زراعی, 10(2), 295-305. doi: 10.30473/coam.2025.73586.1286
Reza Dehghan. "Optimal Control of the Van der Pol Oscillator Problem by Using Orthogonal Polynomial-Based Optimization". فصلنامه علمی زیست فناوری گیاهان زراعی, 10, 2, 1404, 295-305. doi: 10.30473/coam.2025.73586.1286
Dehghan, Reza. (1404). 'Optimal Control of the Van der Pol Oscillator Problem by Using Orthogonal Polynomial-Based Optimization', فصلنامه علمی زیست فناوری گیاهان زراعی, 10(2), pp. 295-305. doi: 10.30473/coam.2025.73586.1286
Dehghan, Reza. Optimal Control of the Van der Pol Oscillator Problem by Using Orthogonal Polynomial-Based Optimization. فصلنامه علمی زیست فناوری گیاهان زراعی, 1404; 10(2): 295-305. doi: 10.30473/coam.2025.73586.1286

Optimal Control of the Van der Pol Oscillator Problem by Using Orthogonal Polynomial-Based Optimization

Control and Optimization in Applied Mathematics
مقاله 12، دوره 10، شماره 2 - شماره پیاپی 20، مهر 2025، صفحه 295-305    اصل مقاله (629.29 K)
نوع مقاله: Research Article
شناسه دیجیتال (DOI): 10.30473/coam.2025.73586.1286
نویسنده
Reza Dehghan*
Department of Mathematics, MaS.C., Islamic Azad University, Masjed Soleiman, Iran‎.
چکیده
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.

تازه های تحقیق
  • 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.
کلیدواژه‌ها
Nonlinear optimization؛ Numerical approximation؛ Nonlinear control system؛ Chelyshkov polynomials
مراجع

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