Regularity properties of optimal trajectories of single-input control systems in dimension three
- 23 Downloads
Let \(\dot q = f(q) + ug(q)\) be a smooth control system on a three-dimensional manifold. Given a point q0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.
KeywordsManifold Switching Time Optimal Trajectory Regularity Property Independence Condition
Unable to display preview. Download preview PDF.
- 1.A. A. Agrachev, “On regularity properties of extremal controls,” J. Dyn. Control Systems, 1, No. 3, 319–324 (1995).Google Scholar
- 2.A. A. Agrachev and R. V. Gamkrelidze, “A second-order optimality principle for a time-optimal problem,” Math. USSR Sb., 29, No. 4, 547–576 (1976).Google Scholar
- 3.A. A. Agrachev and R. V. Gamkrelidze, “Symplectic geometry for optimal control,” in: Nonlinear Controllability and Optimal Control (H. J. Sussmann, Ed.), Pure Appl. Math., 133, Marcel Dekker (1990).Google Scholar
- 4.A. A. Agrachev and M. Sigalotti, “On the local structure of optimal trajectories in ℝ3,” SIAM J. Control and Optimization (in press).Google Scholar
- 5.H. J. Kelley, R. E. Kopp, and H. Gardner Moyer, “Singular extremals,” in: Topics in Optimization, Academic Press, New York (1967), pp. 63–101.Google Scholar
- 6.I. A. K. Kupka, “The ubiquity of Fuller’s phenomenon,” in: Nonlinear Controllability and Optimal Control (H. J. Sussmann, Ed.), Pure Appl. Math., 133, Marcel Dekker (1990).Google Scholar
- 7.L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes, Wiley, New York (1962).Google Scholar
- 8.H. Schättler, “Regularity properties of optimal trajectories: Recently developed techniques,” in: Nonlinear Controllability and Optimal Control (H. J. Sussmann, Ed.), Pure Appl. Math., 133, Marcel Dekker (1990).Google Scholar
- 9.H. J. Sussmann, “Time-optimal control in the plane,” in: Feedback Control of Linear and Nonlinear Systems, Lect. Notes Control Information Sci., 39, Springer-Verlag, Berlin (1985), pp. 244–260.Google Scholar
- 10.H. J. Sussmann, “A weak regularity for real analytic optimal control problems,” Bibl. Rev. Mat. Iberoam., 2, No. 3, 307–317 (1986).Google Scholar
- 12.M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering, Systems and Control: Foundations and Applications, Birkhäuser, Boston (1994).Google Scholar