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IMA Journal of Mathematical Control and Information Advance Access originally published online on March 2, 2009
IMA Journal of Mathematical Control and Information 2009 26(1):95-104; doi:10.1093/imamci/dnn001
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A new algorithm for finding numerical solutions of optimal feedback control

Bao-Zhu Guo{dagger}

Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, People's Republic of China and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

Bing Sun

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People's Republic of China and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

{dagger} Email: bzguo{at}iss.ac.cn

Received on March 15, 2007; Revision received October 17, 2007. A new algorithm for finding numerical solutions of optimal feedback control based on dynamic programming is developed. The algorithm is based on two observations: (1) the value function of the optimal control problem considered is the viscosity solution of the associated Hamilton–Jacobi–Bellman (HJB) equation and (2) the appearance of the gradient of the value function in the HJB equation is in the form of directional derivative. The algorithm proposes a discretization method for seeking optimal control–trajectory pairs based on a finite-difference scheme in time through solving the HJB equation and state equation. We apply the algorithm to a simple optimal control problem, which can be solved analytically. The consistence of the numerical solution obtained to its analytical counterpart indicates the effectiveness of the algorithm.

Keywords: optimal feedback control; viscosity solution; dynamic programming; numerical solution; exponential stability.


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