Control-Affine Systems in Low Dimensions: From Small-Time Reachable Sets to Time-Optimal Syntheses

2012 ◽  
pp. 485-568
Author(s):  
Heinz Schättler ◽  
Urszula Ledzewicz
Keyword(s):  
Small Time ◽  
Reachable Sets ◽  
Affine Systems ◽  
Low Dimensions ◽  
Time Optimal

The Structure of Small-Time Reachable Sets in Low Dimensions

1989 ◽  
Vol 27 (1) ◽  
pp. 120-147 ◽  
Author(s):  
Arthur J. Krener ◽  
Heinz Schättler
Keyword(s):  
Small Time ◽  
Reachable Sets ◽  
Low Dimensions

scholarly journals THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS

2020 ◽  
Vol 6 (1) ◽  
pp. 71
Author(s):  
Mikhail I. Gusev
Keyword(s):  
Asymptotic Behavior ◽  
Nonlinear Control ◽  
Sufficient Conditions ◽  
Linearization Method ◽  
Small Time ◽  
Reachable Set ◽  
Reachable Sets ◽  
Time Interval ◽  
Initial State ◽  
Linearized System

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets.  The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero.  The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension.  The results of numerical simulation illustrate the theoretical conclusions of the paper.

scholarly journals On the convexity of the reachable set with respect to a part of coordinates at small time intervals

2021 ◽  
Vol 31 (2) ◽  
pp. 210-225
Author(s):  
I.O. Osipov
Keyword(s):  
Numerical Modeling ◽  
Sufficient Conditions ◽  
Small Time ◽  
The Other ◽  
Reachable Sets ◽  
Third Order ◽  
Time Intervals ◽  
Linearized System ◽  
Asymptotics Of The Eigenvalues ◽  
Convexity Conditions

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

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