A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines |
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Authors: | Nithin Govindarajan Cornelis C de Visser Kalmanje Krishnakumar |
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Affiliation: | 1. Delft Center for Systems and Control, Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, 2628 CD Delft, The Netherlands;2. Control and Simulation division, Faculty of Aerospace Engineering, Delft University of Technology, 2600GB Delft, The Netherlands;3. Intelligent Systems Division, NASA Ames Research center, Moffett Field, CA 94035, USA |
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Abstract: | This paper presents a sparse collocation method for solving the time-dependent Hamilton–Jacobi–Bellman (HJB) equation associated with the continuous-time optimal control problem on a fixed, finite time-horizon with integral cost functional. Through casting the problem in a recursive framework using the value-iteration procedure, the value functions of every iteration step is approximated with a time-varying multivariate simplex B-spline on a certain state domain of interest. In the collocation scheme, the time-dependent coefficients of the spline function are further approximated with ordinary univariate B-splines to yield a discretization for the value function fully in terms of piece-wise polynomials. The B-spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems. The proposed algorithm is demonstrated on a pair of benchmark example problems. Simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation. |
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Keywords: | Optimal feedback control Hamilton&ndash Jacobi&ndash Bellman equation Adaptive dynamic programming Splines Collocation method |
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