Two-Point Problem for Systems Satisfying the Controllability Condition with Lie Brackets of the Second Order
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We study a two-point control problem for systems linear in control. The class of problems under consideration satisfies a controllability condition with Lie brackets up to the second order, inclusively. To solve the problem, we use trigonometric polynomials whose coefficients are computed by expanding the solutions into the Volterra series. The proposed method allows one to reduce the two-point control problem to a system of algebraic equations. It is shown that this algebraic system has (locally) at least one real solution. The proposed method for the construction of control functions is illustrated by several examples.
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- 2.R. W. Brockett, “Control theory and singular Riemannian geometry,” in: P. J. Hilton and G. S. Young (editors), New Directions of Applied Mathematics, Springer, New York (1981), pp. 11–27.Google Scholar
- 6.T. Chumachenko and A. Zuyev, “Application of the return method to the steering of nonlinear systems,” in: K. Kozłowski (editor), Robot Motion and Control, Springer, Berlin (2009), pp. 83–91.Google Scholar
- 10.F. Jean, Control of Nonholonomic Systems: from sub-Riemannian Geometry to Motion Planning, Springer, Cham (2014).Google Scholar
- 12.G. Lafferriere and H. J. Sussmann, “Motion planning for controllable systems without drift,” in: Proc. IEEE Internat. Conf. on Robotics and Automation, Sacramento (1991), IEEE (1991), pp. 1148–1153.Google Scholar
- 14.R. M. Murray and S. S. Sastry, “Steering nonholonomic systems using sinusoids,” in: Proc. 29th IEEE CDC, Honolulu, HI (1990), pp. 2097–2101.Google Scholar
- 17.R. Yang, P. S. Krishnaprasad, and W. Dayawansa, “Optimal control of a rigid body with two oscillators,” in: Mech. Day, Fields Inst. Comm., Vol. 7, Amer. Math. Soc., Providence, RI (1996), pp. 233–260.Google Scholar
- 18.V. I. Zubov, Lectures on the Control Theory [in Russian], Nauka, Moscow (1975).Google Scholar