{
  "id": "1701.06973",
  "version": "v1",
  "published": "2017-01-24T16:40:03.000Z",
  "updated": "2017-01-24T16:40:03.000Z",
  "title": "Optimal Control Problems with Symmetry Breaking Cost Functions",
  "authors": [
    "Anthony Bloch",
    "Leonardo Colombo",
    "Rohit Gupta",
    "Tomoki Ohsawa"
  ],
  "comment": "Paper submitted to a journal on August 2016. Comments welcome",
  "categories": [
    "math.OC",
    "cs.SY",
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "We investigate symmetry reduction of optimal control problems for left-invariant control systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous-time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler--Poincar\\'e equations from a variational principle. By applying a Legendre transformation to it, we recover the Lie-Poisson equations obtained by A. D. Borum [Master's Thesis, University of Illinois at Urbana-Champaign, 2015] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie-Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-24T16:40:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70G45",
      "70H03",
      "70H05",
      "37J15",
      "49J15"
    ],
    "keywords": [
      "optimal control problem",
      "variational principle",
      "lie-poisson equations",
      "partial symmetry breaking cost functions",
      "partial symmetry breaking lagrangian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}