Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control

Abstract

The aim of this work is to study, from an intrinsic and geometric point of view, second-order constrained variational problems on Lie algebroids, that is, optimization problems defined by a cost function which depends on higher-order derivatives of admissible curves on a Lie algebroid. Extending the classical Skinner and Rusk formalism for the mechanics in the context of Lie algebroids, for second-order constrained mechanical systems, we derive the corresponding dynamical equations. We find a symplectic Lie subalgebroid where, under some mild regularity conditions, the second-order constrained variational problem, seen as a presymplectic Hamiltonian system, has a unique solution. We study the relationship of this formalism with the second-order constrained Euler-Poincaré and Lagrange-Poincaré equations, among others. Our study is applied to the optimal control of mechanical systems.

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

“…Remark 6. Note that the regularity condition given in Proposition 1 represents a first order discretization of the regularity condition for first order vakonomic systems presented, for instance, in [3] (Section 7.3, Equation 7.3.5), [1], [19], [29], [30] and [37]. In general such a condition for continuous time systems is expressed as a 2 × 2 block-matrix A B C D where A corresponds to the matrix giving the classical hyper-regularity condition for the equivalence between Lagrangian and Hamiltonian formalism in mechanics by means of the Legendre transform (this corresponds with the first two entries in the first row of the matrix in Proposition 1).…”

Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

“…The case of k = 2 prolongations was intensively studied by Colombo and his collaborators (see [1,8] and the references therein). In his formalism the second prolongation E [2] is treated as a subset in the prolongation of E along the projection τ : E → M (see Example 2.30), i.e., E [2] ⊂ T E E. Now the initial problem on E [2] can be treated as a constrained problem on T E E, the latter being a first-order algebroid.…”

“…In fact, we show that the latter works fine also for a more general class of higher pre-algebroids, thus further generalization of our theory is possible. We refer to a recent publication [8] and the references therein for numerous concrete interesting examples of higher-order variational problems.…”

The warped product of holomorphic Lie algebroids

“…The connection between the two geometrical settings was analyzed in [20]. Recent applications for Lie algebroids were given in optimal control, interpolation problems, trajectory planning any many more, see for instance [24,9] and the numerous references therein.…”