Periodic near optimal control
type: Item
instance of: scholarly article
title: Periodic near optimal control
zbMATH Open document ID: 0963.49024
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-06-28T00:00:00Z
review text: In this very important and useful paper, the finite time optimal control problem of minimizing an average cost functional for nonlinear singularly perturbed systems is considered. The problem of the existence of near optimal trajectories which are periodic is proposed. For this purpose nonlinear control systems with three time scales are investigated and a limiting system for the slowest motion is constructed by an iterated averaging procedure. Main result: The author shows that global tools, invariant control sets, and local tools, Lie algebra ranks of vector fields are presented under singular perturbations. The existence of periodic near optimal trajectories is verified under the condition that the fast subsystem and an unperturbed slow subsystem are controllable and that a standard Lie algebra rank condition for local accessibility holds.
Mathematics Subject Classification ID: 49N20
Mathematics Subject Classification ID: 49K40
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 34H05
zbMATH DE Number: 1506523
zbMATH Keywords: nonlinear control system
zbMATH Keywords: periodic control
zbMATH Keywords: singular perturbation
zbMATH Keywords: near optimality
zbMATH Keywords: averaging
zbMATH Keywords: periodic near optimal trajectories
zbMATH Keywords: Lie algebra rank condition
author: Götz Grammel
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6874
OpenAlex ID: W2031368216
cites work: Invariant measures of differential inclusions applied to singular perturbations
cites work: Singular Perturbation of a Finite Horizon Problem with State-Space Constraints
cites work: Q5341449
cites work: Q3729587
cites work: Infinite time optimal control and periodicity
cites work: Limit behavior and genericity for nonlinear control systems
cites work: Q4003026
cites work: Averaging Method for One-Sided Lipschitz Differential Inclusions with Generalized Solutions
cites work: The averaging method for perturbations of mixing flows
cites work: Suboptimization of Singularly Perturbed Control Systems
cites work: Suboptimal control of singularly perturbed systems and periodic optimization
cites work: A new reduction technique for a class of singularly perturbed optimal control problems
cites work: Controllability of differential inclusions
cites work: Averaging of singularly perturbed systems
cites work: Estimate of periodic suboptimal controls.
cites work: Nonlinear control systems: An introduction
cites work: Recurrence and invariant measures for degenerate diffusions
cites work: Q3791007
cites work: Q4400449
cites work: Convergence of the Value Functions of Discounted Infinite Horizon Optimal Control Problems with Low Discount Rates
DOI: 10.1006/JMAA.2000.6874
Recommended article: Optimal periodic control
Recommended article: Q4353671
Recommended article: Optimal control of almost-periodic motions
Recommended article: Periodic optimal control in Hilbert space
Recommended article: On the optimal control of almost periodic motions
Recommended article: Nonconcavity and proper optimal periodic control
Recommended article: Periodic Optimal Control, Dissipativity and MPC
Recommended article: Optimal Control of Linear Systems with Almost Periodic Inputs
Recommended article: On optimal periodic control and nested optimization problems
description: scientific article; zbMATH DE number 1506523