By F. Monroy-Perez, B. Bonnard, J. P. Gauthier
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Extra info for Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications
2 VZ Dn . „ Therefore, is in the linear span of t, n and b . Having reduced the solutions to three dimensional problems, the remaining equations can be integrated by quadrature in terms of the geodesic curvature K,(t). 32 We now turn attention to the problem of Dubins. The passage to the extremal curves is straightforward thanks to the Maximum Principle. Nevertheless, there are some minor conceptual differences due to the constraints. The case n = 2 is slightly different because the boundary of u\ < 1 consists of isolated points ± 1 .
Follows that K, remains invariant for any J-frame lifting. 3 The problems of Dubins and Griffiths The problem of Euler-Grifnths consists of minimizing the functional 1 fT -I K2(s)ds, or more generally, any expression 15 L(n(s))ds, Jo for curves j(t) in M that satisfy prescribed tangential directions at given points xi and x2 in M . In particular, Griffiths shows in 3 that for E™ and Sn each solution curve is confined to a three dimensional submanifold, that the resulting Euler-Lagrange equation is integrable by quadrature, and that each solution curve exhibits a remarkable property that K2(t)r(t) = constant.
1) take their analogous form on g x G: dL(t) = [dHm,L(t)]. (3-2) dt In our cases, SO n + i and SO(l,n) are semi-simple, and the Killing form coincides with the trace form (A, B) = — |Trace AB, while on the semi-direct product R" x SO„(E) the trace form is degenerate. In all three cases, the Lie algebra ge splits into the sum p£ © t where t is the Lie subalgebra isomorphic with so n (R), and where p e is the vector space spanned by the matrices ft = 9dH^ ' f 0 — eai — ea2 • • • — ean\ A = as a — 0 \an vanes over / The elements of 6 are antisymmetric (n + l ) x ( n + l) matrices whose first column consists of zero entries.
Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications by F. Monroy-Perez, B. Bonnard, J. P. Gauthier