Remsing, C. C.2024-07-122024-07-122009Remsing, C. C. (2009). Stability and optimal control. Maltepe Üniversitesi. s. 126.9.78605E+12https://www.maltepe.edu.tr/Content/Media/CkEditor/03012019014112056-AbstractBookICMS2009Istanbul.pdf#page=331https://hdl.handle.net/20.500.12415/2264We consider the problem of minimizing a quadratic cost functional J = 1 2 R T 0 ¡ c1u 2 1 + · · · + c`u 2 ` ¢ dt over the trajectories of a left-invariant control system ? evolving on a matrix Lie group G, which is affine in controls. The final time T > 0 is fixed and there are no restrictions on the values of the control variables. Each such invariant optimal control problem defines the appropriate Hamiltonian H on the dual g ? of the Lie algebra of G through the Pontryagin’s Maximum Principle. The integral curves of the corresponding Hamiltonian vector field H~ (with respect to the minus Lie-Poisson structure on g ? ) are called extremal curves. In this paper we are concerned with regular extremal curves. When the Lie algebra g admits a non-degenerate invariant bilinear form h·, ·i : g × g ? R, the Hamilton equations take a more familiar form. This is always possible if g is semisimple. Lyapunov stability of Hamiltonian equilibria is investigated by using the energy-Casimir method. Explicit computations are done in the special case of the rotation group SO (3).enCC0 1.0 Universalinfo:eu-repo/semantics/openAccessConference Object127126