Quadratures of Pontryagin extremals for optimal control problems (original) (raw)

Optimal control and integrability on Lie groups

Any left-invariant optimal control problem (with quadratic cost) can be lifted, via the celebrated Maximum Principle, to a Hamiltonian system on the dual of the Lie algebra of the underlying state space G. The (minus) Lie-Poisson structure on the dual space g * is used to describe the (normal) extremal curves. As an illustration, a typical left-invariant optimal control problem on the rotation group SO (3) is investigated. The reduced Hamilton equations associated with an extremal curve are derived and then explicitly integrated by Jacobi elliptic functions.

Integrability and optimal control

This paper considers left-invariant control affine systems evolving on matrix Lie groups. Any left-invariant optimal control problem (with quadratic cost) can be lifted, via the celebrated Maximum Principle, to a Hamiltonian system on the dual of the Lie algebra of the underlying state space G. The (minus) Lie-Poisson structure on the dual space g * is used to describe the (normal) extremal curves. Complete integrability of (reduced) Hamiltonian dynamical systems is discussed briefly. Some observations concerning Casimir functions and the case of semisimple (matrix) Lie groups are made. As an application, a drift-free left-invariant optimal control problem on the rotation group SO (3) is investigated. The reduced Hamilton equations associated with an extremal curve are derived in a simple and elegant manner. Finally, these equations are explicitly integrated by Jacobi elliptic functions.

Conserved quantities along the Pontryagin extremals of quasiinvariant optimal control problems

Proc. 10th Mediterranean Conference on Control and …, 2002

We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether theorem are provided, in the direction which enlarges the scope of its application. We formulate a more general version of Noether's theorem for optimal control problems, which incorporates the possibility to consider a family of transformations depending on several parameters and, what is more important, to deal with quasi-invariant and not necessarily invariant optimal control problems. We trust that this latter extension provides new possibilities and we illustrate it with several examples, not covered by the previous known optimal control versions of Noether's theorem.

Integrating Hamiltonian systems defined on the Lie groups SO (4) and SO (1, 3)

2007

In this paper we study constrained optimal control problems on semi-simple Lie groups. These constrained optimal control problems include Riemannian, sub-Riemannian, elastic and mechanical problems. We begin by lifting these problems, through the Maximum Principle, to their associated Hamiltonian formalism. As the base manifold is a Lie group G the cotangent bundle is realized as the direct product G × g * where g * is the dual of the Lie algebra g of G. The solutions to these Hamiltonian vector fields l ∈ g * , are called extremal curves and the projections g(t) ∈ G are the corresponding optimal solutions. The main contribution of this paper is a method for deriving explicit expressions relating the extremal curves l ∈ g * to the optimal solutions g(t) ∈ G for the special cases of the Lie groups SO(4) and SO(1, 3). This method uses the double cover property of these Lie groups to decouple them into lower dimensional systems. These lower dimensional systems are then solved in terms of the extremals using a coordinate representation and the systems dynamic constraints. This illustrates that the optimal solutions g(t) ∈ G are explicitly dependent on the extremal curves.

Symbolic computation of variational symmetries in optimal control

arXiv preprint math/0604072, 2006

We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether's first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2, 3, 5, 8).

Optimal Control and higher-order mechanics for systems with symmetries

2012

An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems and we devote special attention to the particular case of underactuated mechanical systems

Quasi-invariant optimal control problems

Portugaliæ Mathematica, 2004

We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether theorem are provided, in the direction which enlarges the scope of its application. We formulate a more general version of Noether's theorem for optimal control problems, which incorporates the possibility to consider a family of transformations depending on several parameters and, what is more important, to deal with quasi-invariant and not necessarily invariant optimal control problems. We trust that this latter extension provides new possibilities and we illustrate it with several examples, not covered by the previous known optimal control versions of Noether's theorem.

Geometry of Optimal Control Problems and Hamiltonian Systems

Lecture Notes in Mathematics, 2008

These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 hours course. The goal was to give an idea of the general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian Geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita connection, Riemannian curvature and related topics. I tried to make the presentation as light as possible: gave more details in smooth regular situations and referred to the literature in more complicated cases. There is an evidence that the results described in the notes and treated in technical papers we refer to are just parts of a united beautiful subject to be discovered on the crossroads of Differential Geometry, Dynamical Systems, and Optimal Control Theory. I will be happy if the course and the notes encourage some young ambitious researchers to take part in the discovery and exploration of this subject. Contents I Lagrange multipliers' geometry 3

Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

Sbornik: Mathematics, 2011

On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis.