%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/VDOL/lowThrust_10
% name: VDOL/lowThrust_10
% [lowThrust optimal control problem (matrix 10 of 13)]
% id: 2714
% date: 2015
% author: B. Senses, A. Rao
% ed: T. Davis
% fields: name title A b id date author ed kind notes aux
% aux: rowname mapping
% kind: optimal control problem
%-------------------------------------------------------------------------------
% notes:
% Optimal control problem, Vehicle Dynamics & Optimization Lab, UF       
% Anil Rao and Begum Senses, University of Florida                       
% http://vdol.mae.ufl.edu                                                
%                                                                        
% This matrix arises from an optimal control problem described below.    
% Each optimal control problem gives rise to a sequence of matrices of   
% different sizes when they are being solved inside GPOPS, an optimal    
% control solver created by Anil Rao, Begum Senses, and others at in VDOL
% lab at the University of Florida.  This is one of the matrices in one  
% of these problems.  The matrix is symmetric indefinite.                
%                                                                        
% Rao, Senses, and Davis have created a graph coarsening strategy        
% that matches pairs of nodes.  The mapping is given for this matrix,    
% where map(i)=k means that node i in the original graph is mapped to    
% node k in the smaller graph.  map(i)=map(j)=k means that both nodes    
% i and j are mapped to the same node k, and thus nodes i and j have     
% been merged.                                                           
%                                                                        
% This matrix consists of a set of nodes (rows/columns) and the          
% names of these rows/cols are given                                     
%                                                                        
% Anil Rao, Begum Sense, and Tim Davis, 2015.                            
%                                                                        
% VDOL/lowThrust                                                         
%                                                                        
% Low-thrust orbit transfer optimal control problem is taken from        
% Ref.~\cite{betts2010practical}. The goal of the low-thrust orbit       
% transfer problem is to determine the state and the control that        
% minimize the fuel consumption during the orbit transfer of a           
% spacecraft that starts from a low-earth orbit and terminates at the    
% geostationary orbit via low-thrust propulsion systems.  The highly     
% nonlinear dynamics of the low-thrust orbit transfer problem is given   
% in modified equinoctial elements (state of the system) and the thrust  
% direction (control of the system).  Furthermore, the low-thrust        
% optimal control problem is a badly scaled problem because of the       
% small thrust-to-initial-mass ratio, that is typically on the order of  
% $O(10^{-4})$, and the long orbit transfer duration. Badly scaling of   
% the problem leads to a lot of delayed pivots. The specified accuracy   
% tolerance of $10^{-8}$ were satisfied after thirteen mesh iterations.  
% As the mesh refinement proceeds, the size of the KKT matrices          
% increases from 584 to 18476.                                           
%                                                                        
% @book{betts2010practical,                                              
%   title={Practical Methods for Optimal Control and Estimation Using    
%      Nonlinear Programming},                                           
%   author={Betts, John T},                                              
%   volume={19},                                                         
%   year={2010},                                                         
%   publisher={SIAM Press},                                              
%   address = {Philadelphia, Pennsylvania},                              
% }                                                                      
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