Hidayatullah, Alvian Alif (2026) Penyelesaian Numerik Persamaan Stochastic Hamilton-Jacobi-Bellman Partial Differential Equations (SHJB PDEs) Menggunakan Polinomial Chebyshev Dan Polinomial Shifted Legendre. Doctoral thesis, Institut Teknologi Sepuluh Nopember.
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Abstract
Stochastic Hamilton--Jacobi--Bellman Partial Differential Equations (SHJB PDEs) merupakan persamaan diferensial parsial Hamilton--Jacobi--Bellman orde dua deterministik yang diperoleh dari formulasi masalah kontrol optimal stokastik. Persamaan ini berperan penting dalam mengarakterisasi fungsi nilai dan kontrol optimal pada sistem dinamis yang dipengaruhi oleh ketidakpastian. Komponen stokastik pada dinamika state direpresentasikan melalui suku difusi yang melibatkan turunan orde dua terhadap variabel state. Disertasi ini membahas penyelesaian numerik SHJB PDEs menggunakan dua skema aproksimasi hibrida, yaitu shifted Legendre--fractional-order Chebyshev (SLFC) dan shifted Chebyshev--fractional-order Legendre (SCFL). Metode SLFC menggunakan polinomial shifted Legendre pada variabel waktu dan polinomial fractional-order Chebyshev pada variabel state, sedangkan metode SCFL menggunakan polinomial shifted Chebyshev pada variabel waktu dan polinomial fractional order Legendre pada variabel state. Kedua skema tersebut digunakan untuk membangun aproksimasi fungsi nilai berbasis hasil kali tensor. Analisis konvergensi dilakukan dalam kerangka ruang fungsi berbobot untuk menunjukkan bahwa aproksimasi yang dibangun dapat mendekati solusi SHJB PDEs ketika derajat aproksimasi ditingkatkan. Selanjutnya, disusun prosedur numerik berbasis residual kolokasi untuk mereduksi SHJB PDEs menjadi sistem persamaan aljabar nonlinier. Kinerja metode diuji pada beberapa studi kasus kontrol optimal stokastik dengan memvariasikan derajat aproksimasi waktu, derajat aproksimasi state, dan orde fraksional. Hasil numerik menunjukkan bahwa metode SLFC dan SCFL menghasilkan akurasi yang baik dan memiliki kinerja kompetitif dibandingkan metode shifted Legendre--fractional-order Legendre (SLFL).
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Stochastic Hamilton--Jacobi--Bellman Partial Differential Equations (SHJB PDEs) are second-order deterministic Hamilton--Jacobi--Bellman partial differential equations derived from the formulation of stochastic optimal control problems. These equations play an important role in characterizing the value function and the optimal control of dynamic systems affected by uncertainty. The stochastic component in the state dynamics is represented through a diffusion term involving the second-order derivative with respect to the state variable. This dissertation discusses the numerical solution of SHJB PDEs using two hybrid approximation schemes, namely shifted Legendre--fractional-order Chebyshev (SLFC) and shifted Chebyshev--fractional-order Legendre (SCFL). The SLFC method employs shifted Legendre polynomials for the time variable and fractional-order Chebyshev polynomials for the state variable, whereas the SCFL method employs shifted Chebyshev polynomials for the time variable and fractional-order Legendre polynomials for the state variable. Both schemes are used to construct a tensor-product-based approximation of the value function. The convergence analysis is carried out within a weighted function space framework to show that the constructed approximation can approach the solution of SHJB PDEs as the approximation degree increases. Furthermore, a numerical procedure based on collocation residuals is developed to reduce the SHJB PDEs into a system of nonlinear algebraic equations. The performance of the proposed methods is tested on several stochastic optimal control case studies by varying the approximation degree in time, the approximation degree in the state variable, and the fractional order. The numerical results show that the SLFC and SCFL methods provide good accuracy and have competitive performance compared with the shifted Legendre--fractional-order Legendre (SLFL) method.
| Item Type: | Thesis (Doctoral) |
|---|---|
| Uncontrolled Keywords: | Stochastic Hamilton--Jacobi--Bellman partial differential equations, kontrol optimal stokastik, aproksimasi hibrida, polinomial orde fraksional, metode kolokasi, Stochastic Hamilton-Jacobi-Bellman Partial Differential Equations, stochastic optimal control, hybrid approximation, fractional-order polynomials, collocation method |
| Subjects: | Q Science > QA Mathematics > QA274.2 Stochastic analysis Q Science > QA Mathematics > QA371 Differential equations--Numerical solutions Q Science > QA Mathematics > QA401 Mathematical models. Q Science > QA Mathematics > QA614.8 Differentiable dynamical systems Q Science > QA Mathematics > QA76.9 Computer algorithms. Virtual Reality. Computer simulation. |
| Divisions: | Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44002-(S3) PhD Thesis |
| Depositing User: | Alvian Alif Hidayatullah |
| Date Deposited: | 22 Jul 2026 02:59 |
| Last Modified: | 22 Jul 2026 02:59 |
| URI: | http://repository.its.ac.id/id/eprint/136168 |
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