### Abstract

The paper deals with a problem of control of a system characterized by the fact that the influence of controls on the dynamics of certain functions of state variables (called observables) is relatively weak, and the rates of change of these observables are much slower than the rates of change of the state variables themselves. The contributions of the paper are twofold. First, the averaged system whose solutions approximate the trajectories of the slow observables is introduced, and it is shown that the optimal value of the problem of optimal control with time discounting criterion considered on the solutions of the system with slow observables (this problem is referred to as perturbed) converges to the optimal value of the corresponding problem of optimal control of the averaged system. Second, a method for constructing an asymptotically optimal control of the perturbed problem on the basis of an optimal solution of the averaged problem is indicated, sufficient and necessary optimality conditions for the averaged problem are stated, and a way for constructing numerically a near optimal solution of the latter is outlined (the construction being illustrated with an example).

Language | English |
---|---|

Pages | 1398-1428 |

Number of pages | 31 |

Journal | SIAM Journal on Control and Optimization |

Volume | 55 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 |

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### Keywords

- control of slow observables
- averaged system
- average control generating families
- occupational measures
- numerical solution

### Cite this

*SIAM Journal on Control and Optimization*,

*55*(3), 1398-1428. https://doi.org/10.1137/15M1047829

}

*SIAM Journal on Control and Optimization*, vol. 55, no. 3, pp. 1398-1428. https://doi.org/10.1137/15M1047829

**On near optimal control of systems with slow observables.** / Gaitsgory, Vladimir; Rossomakhine, Sergey.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - On near optimal control of systems with slow observables

AU - Gaitsgory, Vladimir

AU - Rossomakhine, Sergey

PY - 2017

Y1 - 2017

N2 - The paper deals with a problem of control of a system characterized by the fact that the influence of controls on the dynamics of certain functions of state variables (called observables) is relatively weak, and the rates of change of these observables are much slower than the rates of change of the state variables themselves. The contributions of the paper are twofold. First, the averaged system whose solutions approximate the trajectories of the slow observables is introduced, and it is shown that the optimal value of the problem of optimal control with time discounting criterion considered on the solutions of the system with slow observables (this problem is referred to as perturbed) converges to the optimal value of the corresponding problem of optimal control of the averaged system. Second, a method for constructing an asymptotically optimal control of the perturbed problem on the basis of an optimal solution of the averaged problem is indicated, sufficient and necessary optimality conditions for the averaged problem are stated, and a way for constructing numerically a near optimal solution of the latter is outlined (the construction being illustrated with an example).

AB - The paper deals with a problem of control of a system characterized by the fact that the influence of controls on the dynamics of certain functions of state variables (called observables) is relatively weak, and the rates of change of these observables are much slower than the rates of change of the state variables themselves. The contributions of the paper are twofold. First, the averaged system whose solutions approximate the trajectories of the slow observables is introduced, and it is shown that the optimal value of the problem of optimal control with time discounting criterion considered on the solutions of the system with slow observables (this problem is referred to as perturbed) converges to the optimal value of the corresponding problem of optimal control of the averaged system. Second, a method for constructing an asymptotically optimal control of the perturbed problem on the basis of an optimal solution of the averaged problem is indicated, sufficient and necessary optimality conditions for the averaged problem are stated, and a way for constructing numerically a near optimal solution of the latter is outlined (the construction being illustrated with an example).

KW - control of slow observables

KW - averaged system

KW - average control generating families

KW - occupational measures

KW - numerical solution

UR - http://www.scopus.com/inward/record.url?scp=85021707281&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/DP130104432

UR - http://purl.org/au-research/grants/arc/DP150100618

U2 - 10.1137/15M1047829

DO - 10.1137/15M1047829

M3 - Article

VL - 55

SP - 1398

EP - 1428

JO - SIAM Journal on Control and Optimization

T2 - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 3

ER -