Geometric reduction in optimal control theory with symmetries

Reports on Mathematical Physics, 2004

In this paper we study general symmetries for optimal control problems making use of the geometric formulation proposed in . This framework allows us to reduce the number of equations associated with optimal control problems with symmetry and compare the solutions of the original system with the solutions of the reduced one. The reconstruction of the optimal controls starting from the reduced problem is also explored.

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

1986 25th IEEE Conference on Decision and Control, 1986

Based on E51 we discuss the use of symmetries i n solving optimal control problems.

International Journal of Geometric Methods in Modern Physics, 2011

A new relation among a class of optimal control systems and Lagrangian systems with symmetry is discussed. It will be shown that a family of solutions of optimal control systems whose control equation are obtained by means of a group action are in correspondence with the solutions of a mechanical Lagrangian system with symmetry. This result also explains the equivalence of the class of Lagrangian systems with symmetry and optimal control problems discussed in [1, 2]. The explicit realization of this correspondence is obtained by a judicious use of Clebsch variables and Lin constraints, a technique originally developed to provide simple realizations of Lagrangian systems with symmetry. It is noteworthy to point out that this correspondence exchanges the role of state and control variables for control systems with the configuration and Clebsch variables for the corresponding Lagrangian system. These results are illustrated with various simple applications.

Journal of Geometry and Physics, 2012

In this paper, we first study the Poisson reductions of controlled Hamiltonian (CH) system and symmetric CH system by controllability distributions. These reductions are the extension of Poisson reductions by distribution for Poisson manifolds to that for phase spaces of CH systems with external force and control. We give Poisson reducible conditions of CH system by controllability distribution, and prove that the Poisson reducible property for CH systems leaves invariant under the CH-equivalence. Moreover, we study the Poisson reduction of symmetric CH system by G-invariant controllability distribution. Next, we consider the singular Poisson reduction and SPR-CH-equivalence for CH system with symmetry, and prove the singular Poisson reduction theorem of CH system. We also study the relationship between Poisson reduction for singular Poisson reducible CH systems by G-invariant controllability distribution and that for associated reduced CH system by reduced controllability distribution. At last, some examples are given to state the theoretical results.

1998

This is the second article in the series that began in . Jacobi curves were defined, computed, and studied in that paper for regular extremals of smooth control systems. Here we do the same for singular extremals. The last section contains a feedback classification and normal forms of generic single-input affine in control systems on a 3-dimensional manifold.

Applied Mathematics E-Notes, 2003

We extend the second Noether theorem to optimal control problems which are invariant under symmetries depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. As far as we consider a semi-invariance notion, and the transformation group may also depend on the control variables, the result is new even in the classical context of the calculus of variations.

2003

Control theory is a young branch of mathematics that has developed mostly in the realm of engineering problems. It is splitted in two major branches; control theory of problems described by partial difierential equa-tions where control are exercized either by boundary terms and/or inhomoge-neous terms and where the objective functionals are mostly quadratic forms; and control theory of problems described by parameter dependent ordinary difierential equations. In this case it is more frequent to deal with non-linear systems and non-quadratic objective functionals [49]. In spite that control theory can be consider part of the general theory of difierential equations, the problems that inspires it and some of the results obtained so far, have configured a theory with a strong and definite personality that is already of-fering interesting returns to its ancestors. For instance, the geometrization of nonlinear a卤ne-input control theory problems by introducing Lie-geometrical methods into...

Arxiv preprint arXiv: …, 2010

Abstract: We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed ...

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).