A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines

doi:10.1016/j.automatica.2014.07.012

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A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines

2014-09-01

PRE

Nithin Govindarajan *

Coen C. de Visser

Kalmanje Krishnakumar

DOI：10.1016/j.automatica.2014.07.012

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Abstract

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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. (C) 2014 Elsevier Ltd. All rights reserved.

Keywords:

Optimal feedback control

Hamilton-Jacobi-Bellman equation

Adaptive dynamic programming

Splines

Collocation method

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