Optimal control and geodesic flows
This publication gives two examples of optimal control problems such that the associated hamiltonian dynamics takes the form of two coupled double bracket equations which take the following form (or a modified version) \[ \dot x=[x, [p,x]], \qquad \dot p=[p, [p,x]]. \] \(x\) corresponds to the state on a Lie algebra and \(p\) is the adjoint variable. The first example consists essentially in minimizing a quadratic cost in the control \(u\) subject to \(\dot x=[x,u]\). \(x\) and \(u\) belong to a suitable Lie algebra such that the negative of the Killing form is positive definite. The corresponding bi-invariant norm is used in the cost functional. The second example consists in minimizing a quadratic cost in the controls \(U\), \(V\) subject to \(\dot Q=UQ-QV\) where \(Q\) is a \(p\times q\) complex matrix and \(U\in u(p)\) (resp. \(V\in u(q))\); \(u(n)\) is the Lie algebra of the unitary group. The norm involved in the cost is the standard trace inner product for matrices. Matrices in the Grassmannian \(G_{q,q+p}(\mathbb{C})\) can be constructed from \(Q\) and the adjoint variable \(P\). These two matrices satisfy similar coupled double bracket equations. The coupled double bracket equations given above are seen to be geodesic equation on the adjoint orbit or on the Grassmannian respectively with respect to a suitable metric. The choice of the metric is conditioned by the fact that tangent vectors have a bracket form in both cases. The details of the proof are missing. Particular cases are examined: the geodesic equations on the sphere \(S^n\), the free rigid body dynamics. The equations are justified by computations which are not always written down. The adjoint equation for \(P\) is not correct due to an error in the derivation (cf. formula 28).
scientific article; zbMATH DE number 983244
scientific article