# American Institute of Mathematical Sciences

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September  2015, 35(9): 4293-4322. doi: 10.3934/dcds.2015.35.4293

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

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

Received  April 2014 Revised  October 2014 Published  April 2015

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.
Citation: Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293
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##### References:
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