Some Applications of Quasi-Velocities in Optimal Control

Abstract

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently [20]. We also provide several examples to illustrate our construction.

Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

“…α , (P α ) (1) , (P α ) (2) } and using the expression of Tulczyjew's isomorphism and the condition y α = z α , equation (7), (8) and (9) are equivalents to…”

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

“…Thus, if (e, e , ve) ∈ T τ E E; then ρ1(e, e , ve) = (e, ve) ∈ TeE, and τ (1) E (e, e , ve) = e ∈ E. Next, we introduce two canonical operations that we have on a Lie algebroid E. The first one is obtained using the Lie algebroid structure of E and the second one is a consequence of E being a vector bundle. On one hand, if f ∈ C ∞ (M ) we will denote by f c the complete lift to E of f defined by f c (e) = ρ(e)(f ) for all e ∈ E. Let X be a section of E then there exists a unique vector field X c on E, the complete lift of X, satisfying the two following conditions:…”

“…If (x i , y A ) are local fibred coordinates on E, (ρ i A , C C AB ) are the corresponding local structure functions on E and {e (1) A , e…”

“…If (x i , pA) are the local coordinates on E * associated with the local basis {e A } of Γ( E * ), then (x i , pA, y a ) are local coordinates on W0 and we may consider the local basis { e (1) A , ( e A ) (2) , e , ( e A ) (2) , where (ě, e * ) ∈ W0 and ν(ě, e * ) = x. If ([[·, ·]] ν , ρ ν ) is the Lie algebroid structure on T ν E, we have that…”

“…As we know from subsection (2.2), the basis of sections {eA} of E induces a local basis of the sections of T τ E E given by e (1) A (e) = e, eA(τE(e)), ρ i…”