Second-order constrained variational problems on Lie algebroids: Applications to optimal control

Leonardo Colombo
2017 Journal of Geometric Mechanics (JGM)  
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
more » ... ations. 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. Contents 2010 Mathematics Subject Classification. Primary: 70H25; Secondary: 70H30, 70H50, 37J15, 58K05, 70H03, 37K05. Key words and phrases. Lie algebroids, optimal control, higher-order mechanics, higher-order variational problems. 1 2 LEONARDO COLOMBO 4.2. Optimal control problems of underactuated mechanical systems on Lie algebroids 36 Acknowledgments 42 REFERENCES 42 SECOND-ORDER PROBLEMS ON LIE ALGEBROIDS 3 mechanics from the point of view of Lie algebroids obtained through the application of a constrained variational principle. They have developed a constraint algorithm for presymplectic Lie algebroids generalizing the well know constraint algorithm of Gotay, Nester and Hinds [38, 39] and they have also established the Skinner and Rusk formalism on a Lie algebroids. Some of the results given in this work are an extension of this framework in the context of constrained second-order systems. In this work we choose a framework to study mechanics based in the Skinner-Rusk formalism, which combines simultaneously some features of the Lagrangian and Hamiltonian classical formalisms to study the dynamics associated with optimal control problems as in [4] . The idea of this formulation was to obtain a common description for both regular and singular dynamics, obtaining simultaneously the Hamiltonian and Lagrangian formulations of the dynamics. Over the years, however, Skinner and Rusk's framework was extended in many directions: It was originally developed for first-order autonomous mechanical systems [74] , and later generalized to non-autonomous dynamical systems [2, 25, 72] , control systems [4] and, more recently to classical field theories [12, 29, 75] . Briefly, in this formulation, one starts with a differentiable manifold Q as the configuration space, and the Whitney sum T Q ⊕ T * Q as the evolution space (with canonical projections π 1 : T Q ⊕ T * Q −→ T Q and π 2 : T Q ⊕ T * Q −→ T * Q). Define on T Q ⊕ T * Q the presymplectic 2-form Ω = π * 2 ω Q , where ω Q is the canonical symplectic form on T * Q, and note that the rank of the presymplectic form is everywhere equal to 2n. If the dynamical system under consideration admits a Lagrangian description, with Lagrangian L ∈ C ∞ (T Q), one can obtain a (presymplectic)-Hamiltonian representation on T Q ⊕ T * Q given by the presymplectic 2-form Ω and the Hamiltonian function H = π 1 , π 2 − π * 1 L , where ·, · denotes the natural pairing between vectors and covectors on Q. In this Hamiltonian system the dynamics is given by vector fields X, which are solutions to the Hamiltonian equation iX Ω = dH. If L is regular, then there exists a unique vector field X solution to the previous equation, which is tangent to the graph of the Legendre map. In the singular case, it is necessary to develop a constraint algorithm in order to find a submanifold (if it exists) where there exists a well-defined vector field solution. Recently, higher-order variational problems have been studied for their importance in applications to aeronautics, robotics, computer-aided design, air traffic control, trajectory planning, and in general, problems of interpolation and approximation of curves on Riemannian manifolds [6, 11, 41, 48, 52, 63, 64, 66] . There are variational principles which involve higher-order derivatives by Gay Balmaz et.al., [30, 31, 32] , (see also [50] ) since from it one can obtain the equations of motion for Lagrangians where the configuration space is a higher-order tangent bundle. More recently, there have been an interest in the study of the geometrical structures associated with higher-order variational problems with the aim of a deepest understanding of those objects [20, 23, 71, 61, 44, 45, 46] as well as the relationship between higher-order mechanics and graded bundles [8, 9, 10] . Optimal control on Lie algebroids has been a subject of study among the last years by extending the Pontryagin maximum principle to control systems onLie algebroids as it has been shown in [1], [58], [37] , [68] . The first two references given before are based on the geometry of prolongations of Lie algebroids (in sense of Higgins and Mackenzie [53]) over the vector bundle projection of a dual bundle; the same approach as we use in this work. The first reference focuses on kinematic
doi:10.3934/jgm.2017001 fatcat:3w36skz57rbjrjeej2ejhiefpq