Stokastik optimal kontrol problemleri için Runge-Kutta yöntemi.

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Türkiye

Tezin Onay Tarihi: 2017

Tezin Dili: İngilizce

Öğrenci: Hacer Öz Bakan

Eş Danışman: Gerhard Weber, GERHARD WİLHELM WEBER

Özet:

In this thesis, we analyze Runge-Kutta scheme for the numerical solutions of stochastic optimal control problems by using discretize-then-optimize approach. Firstly, we dis- cretize the cost functional and the state equation with the help of Runge-Kutta schemes. Then, we state the discrete Lagrangian and take the partial derivative of it with respect to its variables to get the discrete optimality system. By comparing the continuous and discrete optimality conditions, we find a relationship between the Runge-Kutta coef- ficients of the state and adjoint equation, so that we present Runge-Kutta scheme for the adjoint pair (p(t),q(t)). Similar to the deterministic setting, the issue of conver- gence is important when dealing with a numerical scheme. In stochastic case, this can be achieved either by using the strong-order convergence or weak-order convergence criteria. We match the stochastic Taylor expansion on the exact solution of continuous optimality system with the stochastic Taylor expansion of approximate solution of our discrete optimality system, term by term, in order to get both strong and weak-order conditions. The thesis ends with a conclusion and a future outlook to forthcoming research and application.