Higher order discrete controllability and the approximation of the minimum time function

Giovanni Colombo; Thuy T. T. Le

doi:10.3934/dcds.2015.35.4293

Article Contents

2015, Volume 35, Issue 9: 4293-4322. Doi: 10.3934/dcds.2015.35.4293

This issue Previous Article A semi-Lagrangian scheme for a degenerate second order mean field game system Next Article Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval

Higher order discrete controllability and the approximation of the minimum time function

Giovanni Colombo^1, and
Thuy T. T. Le^1,

1.
Università di Padova, Dipartimento di Matematica, via Trieste 63, 35121 Padova

Received: March 31, 2014

Revised: September 30, 2014

Published: May 2017

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Abstract

We give sufficient conditions to reach a target for a suitable discretization of a control affine nonlinear dynamics. Such conditions involve higher order Lie brackets of the vector fields driving the state and so the discretization method needs to be of a suitably high order as well. As a result, the discrete minimal time function is bounded by a fractional power of the distance to the target of the initial point. This allows to use methods based on Hamilton-Jacobi theory to prove the convergence of the solution of a fully discrete scheme to the (true) minimum time function, together with error estimates. Finally, we design an approximate suboptimal discrete feedback and provide an error estimate for the time to reach the target through the discrete dynamics generated by this feedback. Our results make use of ideas appearing for the first time in [3] and now extensively described in [12]. Numerical examples are presented.

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Mathematics Subject Classification: 49J52, 49M25, 49L20.

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References

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