Compressive control-vector parameterization with discrete cosine transform for the solution of optimal-control problems - Hallaçeli, Ahmet. (Thesis (M.A.) - Bogazici University. Institute for Graduate Studies in the Social Sciences, 2017., 2017)
In this thesis, a novel numerical method for the solution of open-loop optimal-control problems is proposed. The method combines the flexibility of the standard Control-Vector Parameterization (CVP) technique with the compressive power of the Discrete Cosine Transform (DCT). Thus, the proposed method is termed as the Compressive Control-Vector Parameterization with Discrete Cosine Transform (CVP-DCT). In the CVP-DCT method, the control input is parameterized in terms of only few DCT Coefficients (DCTCs). The CVP-DCT method transcribes the optimal-control problem to a Nonlinear Programming (NLP) problem where the coefficients selected from the early elements of the DCTC vector are the optimization decision variables. Terminal and path constraints, as well as control bounds, are handled by the penalty-function method. Several problems are solved using the CVP-DCT, standard CVP, and Control-Vector Optimization (CVO) methods for comparison and demonstration of the pros and cons of the proposed CVP-DCT method. The method does not require a priori knowledge of the shape and complexity of the control trajectory and it can be used in any optimal control problem without prespecification. With only a few parameters, the CVP-DCT method can provide a good initial-guess trajectory to other sophisticated optimal control software packages. Especially if the control trajectory is smooth, the CVP-DCT method can provide solutions which are very close to the global solution using just a few decision variables. The performance, required number of DCTCs, and number of optimization decision variables of the method is independent of the dimension of the states and the number of time grids. Even very few DCTCs are enough for the reconstruction of the control vector to hundreds or even thousands of time grids without affecting the CPU time noticeably. Therefore, the proposed method is a viable technique for the fast solution of generic open-loop optimal-control problems with efficient lowdimensional parameterization.
Keywords: Optimal stopping (Mathematical statistics); Discrete cosine transforms.