Galerkin optimal control
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A Galerkin-based family of numerical formulations is presented for solving nonlinear optimal control problems. This dissertation introduces a family of direct methods that calculate optimal trajectories by discretizing the system dy-namics using Galerkin numerical techniques and approximate the cost function with Gaussian quadrature. In this numerical approach, the analysis is based on L2-norms. An important result in the theoretical foundation is that the feasibility and consistency theorems are proved for problems with continuous and/or piecewise continuous controls. Galerkin methods may be formulated in a number of ways that allow for efficiency and/or improved accuracy while solving a wide range of optimal control problems with a variety of state and control constraints. Numerical formula-tions using Lagrangian and Legendre test functions are derived. One formulation allows for a weak enforcement of boundary conditions, which imposes end conditions only up to the accuracy of the numerical approximation itself. Additionally, the multi-scale formulation can reduce the dimension of multi-scale optimal control problems, those in which the states and controls evolve on different timescales. Finally, numerical examples are shown to demonstrate the versatile nature of Galerkin optimal control.
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Giraldo, F.X. (2000);The weak Lagrange-Galerkin finite element method for the two-dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized ...
Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method, Draft Wilcox, Lucas C.; Stadler, Georg; Bui-Thanh, Tan; Ghattas, Omar (2013);This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. ...
Giraldo, F.X. (1998);The purpose of this paper is to introduce a new method formed by fusing the Lagrange-Galerkin and spectral element methods. The Lagrange-Galerkin method traces the characteristic curves of the solution and, consequently, ...