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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On near optimal control of systems with slow observables. / Gaitsgory, Vladimir; Rossomakhine, Sergey.
In: SIAM Journal on Control and Optimization, Vol. 55, No. 3, 2017, p. 1398-1428.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 -