Fiscal Year	2015
Division	Graduate School of Engineering & Applied Science
Department	Applied Mathematics
Investigator(s)	Kang, Wei Wilcox, Lucas C.
Sponsor	Defense Advanced Research Projects Agency (DoD)
Summary	We propose to develop an innovative numerical method, which combines causality free algorithms with sparse grids, to solve HJB equations of dimensions d< 10, which are considered as very high dimensions for PDEs. In a recent paper, the PIs solved the HJB equation with six state variables for the optimal attitude control of satellite systems equipped with control momentum wheels. This result is encouraging in our effort of mitigating the curse of dimensionality. Different from existing approaches, a causality free method on sparse grids has several advantages, including: (1) the algorithm has perfect parallelism; (2) the grid size is significantly smaller than that of dense grids; (3) there is no spatial error at grid points. The goal of the project is to take the advantage of these numerical properties to solve HJB equations from applications that are interesting to DoD such as UAVs and AUVs; to develop a theoretical foundation of error estimation; and to carry out experimentations to test the capability of the real-time feedback based on the HJB solution. We expect to solve some HJB equations with four to ten variables that are currently considered to be, in general, not numerically solvable.
Keywords
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