11 research outputs found
Symmetry Reduction of Optimal Control Systems and Principal Connections
This paper explores the role of symmetries and reduction in nonlinear control
and optimal control systems. The focus of the paper is to give a geometric
framework of symmetry reduction of optimal control systems as well as to show
how to obtain explicit expressions of the reduced system by exploiting the
geometry. In particular, we show how to obtain a principal connection to be
used in the reduction for various choices of symmetry groups, as opposed to
assuming such a principal connection is given or choosing a particular symmetry
group to simplify the setting. Our result synthesizes some previous works on
symmetry reduction of nonlinear control and optimal control systems. Affine and
kinematic optimal control systems are of particular interest: We explicitly
work out the details for such systems and also show a few examples of symmetry
reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure
Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups
We study the reduction by symmetry for optimality conditions in optimal
control problems of left-invariant affine multi-agent control systems, with
partial symmetry breaking cost functions for continuous-time and discrete-time
systems. We recast the optimal control problem as a constrained variational
problem with a partial symmetry breaking Lagrangian and obtain the reduced
optimality conditions from a reduced variational principle via symmetry
reduction techniques in both settings, continuous-time, and discrete-time. We
apply the results to a collision and obstacle avoidance problem for multiple
vehicles evolving on in the presence of a static obstacle.Comment: 20 page
Symmetry reduction and recovery of trajectories of optimal control problems via measure relaxations
We address the problem of symmetry reduction of optimal control problems
under the action of a finite group from a measure relaxation viewpoint. We
propose a method based on the moment-SOS aka Lasserre hierarchy which allows
one to significantly reduce the computation time and memory requirements
compared to the case without symmetry reduction. We show that the recovery of
optimal trajectories boils down to solving a symmetric parametric polynomial
system. Then we illustrate our method on the symmetric integrator and the
time-optimal inversion of qubits.Comment: 38 pages, 23 figure