Using Lyapunov Vectors and Dichotomy to Solve Hyper-Sensitive Optimal Control Problems

The dichotomic basis method for solving completely hyper-sensitive optimal control problems is modified by using Lyapunov exponents and vectors. It is shown that the asymptotic Lyapunov vectors form dichotomic transformations that decouple the unstable dynamics from the stable dynamics. For numerical implementation, finite-time Lyapunov vectors are used to approximate the asymptotic Lyapunov vectors and to construct an approximate dichotomic basis. A reinitialization process is introduced to decrease the error accumulation. The new basis identifies the stable and unstable directions more accurately than the eigenvectors of the Jacobian matrix

[1]  A. M. Lyapunov.  The general problem of the stability of motion , 1992 .

[2]  S. Lam,et al.  The CSP method for simplifying kinetics , 1994 .

[3]  S. Bharadwaj,et al.  Timescale Analysis for Nonlinear Dynamical Systems , 2003 .

[4]  Wiesel.  Continuous time algorithm for Lyapunov exponents. I. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[6]  Ufuk Topcu,et al.  Two-Timescale Nonlinear Dynamics and Slow Manifold Determination , 2005 .

[7]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[8]  L. Barreira,et al.  Lyapunov Exponents and Smooth Ergodic Theory , 2002 .

[9]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[10]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[11]  J. Chow.  A class of singularly perturbed, nonlinear, fixed-endpoint control problems , 1979 .

[12]  Anil V. Rao,et al.  Eigenvector approximate dichotomic basis method for solving hyper‐sensitive optimal control problems , 1999 .

[13]  Anil V. Rao,et al.  Dichotomic basis approach to solving hyper-sensitive optimal control problems , 1999, Autom..

[14]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[15]  Brian D. O. Anderson,et al.  Optimal control problems over large time intervals , 1987, Autom..

[16]  George R. Sell,et al.  The spectrum of an invariant submanifold , 1980 .

[17]  P. Kokotovic,et al.  A dichotomy in linear control theory , 1972 .

[18]  K. Mease,et al.  A new invariance property of Lyapunov characteristic directions , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).