Stability and optimal control
AuthorRemsing, C. C.
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CitationRemsing, C. C. (2009). Stability and optimal control. Maltepe Üniversitesi. s. 126.
We 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 Σ 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).
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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