An approximation approach to optimal control problems

Jinghao Zhu^{∗}  Zhiqiang Zou^{†}

June 27, 2002

Abstract

For nonlinear analytic systems there have been many works on control problems since the eighties last century. In [1], Lie series were used in studying the controllablity of the nonlinear analytic systems. In this paper we use Lie series to construct difference algorithm in obtaining the discrete solutions of optimizations on reachable sets of nonlinear analytic systems. Taking advantage of the uniform convergence of Lie series on the interval of ”small” time , we seek the integral of the optimal control function on each interval of ”small” time. We shall see that the optimal control takes effect to the problem concerned by these integral dates. We can also construct the optimal control by these dates to meet some given requirments. The algorithm is given to introduce the process including pre-estimation and correction to approximate the solutions of the optimization problems.

Consider the following nonlinear control system

ẋ = f(x) + G(x)u, x(0) = x_{0} ∈ X ⊂ R^{n}, t ∈ [0, T],

where f : R^{n} ⟶ R^{n} and G : R^{n} ⟶ R^{n} × R^{m} are real analytic mappings and where the admissible controls u(t), taking values in some compact set U ⊂ R^{m}, are Riemann integrable. Let Q(x) be a locally Lipschitz function. We pose the following optimization problem: Find an admissible û(.), such that

Q(x̂(T)) = min_{u∈U}{Q(x_{u}(T))}.  (1.2)

Keywords: Optimization, Optimal control, Discrete solutions, Lie sereis.

References

1. H.J.Sussmann, Lie brackets and local controllability: A sufficient condition for scalar-input systems, Siam J. Control and Optimization Vol. 21, No. 5, September 1983 , Macmillan Publishing Company, New York 1988

∗Department of Applied Mathematics,Tongji University, Shanghai, China, jinghaok@online.sh.cn †Department of Applied Mathematics, Tongji University, Shanghai, China,