dorsal/arxiv
View SchemaPath integrals and symmetry breaking for optimal control theory
| Authors | H. J. Kappen |
|---|---|
| Categories | |
| ArXiv ID | physics/0505066 |
| URL | https://arxiv.org/abs/physics/0505066 |
| DOI | 10.1088/1742-5468/2005/11/P11011 |
Abstract
This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of stochastic control problems the non-linear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton-Jacobi equation to the Schr\"odinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process, that can be computed by stochastic integration or by the evaluation of a path integral. It is shown, how in the deterministic limit the PMP formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as MC sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in number of examples. Examples are given that show the qualitative difference between stochastic and deterministic control and the occurrence of symmetry breaking as a function of the noise.
{
"annotation_id": "069559a9-5847-4dc2-a0b8-2b26bf28de90",
"date_created": "2026-03-02T18:00:57.047000Z",
"date_modified": "2026-03-02T18:00:57.047000Z",
"file_hash": "9edbac588cfc0a9489f903d327f0027ad2a83d0b267aa57815167a8e7820ebd8",
"private": false,
"record": {
"abstract": "This paper considers linear-quadratic control of a non-linear dynamical\nsystem subject to arbitrary cost. I show that for this class of stochastic\ncontrol problems the non-linear Hamilton-Jacobi-Bellman equation can be\ntransformed into a linear equation. The transformation is similar to the\ntransformation used to relate the classical Hamilton-Jacobi equation to the\nSchr\\\"odinger equation. As a result of the linearity, the usual backward\ncomputation can be replaced by a forward diffusion process, that can be\ncomputed by stochastic integration or by the evaluation of a path integral. It\nis shown, how in the deterministic limit the PMP formalism is recovered. The\nsignificance of the path integral approach is that it forms the basis for a\nnumber of efficient computational methods, such as MC sampling, the Laplace\napproximation and the variational approximation. We show the effectiveness of\nthe first two methods in number of examples. Examples are given that show the\nqualitative difference between stochastic and deterministic control and the\noccurrence of symmetry breaking as a function of the noise.",
"arxiv_id": "physics/0505066",
"authors": [
"H. J. Kappen"
],
"categories": [
"physics.gen-ph",
"physics.comp-ph"
],
"doi": "10.1088/1742-5468/2005/11/P11011",
"title": "Path integrals and symmetry breaking for optimal control theory",
"url": "https://arxiv.org/abs/physics/0505066"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "5e3a3b45-b3ac-4007-9281-79da74c99caa",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}