# American Institute of Mathematical Sciences

June  2019, 9(2): 223-255. doi: 10.3934/mcrf.2019012

## Construction of the minimum time function for linear systems via higher-order set-valued methods

 1 Universität Bayreuth, Mathematisches Institut, 95440 Bayreuth, Germany 2 Otto von Guericke University Magdeburg, Department of Mathematics, Universitätsplatz 2, 39106 Magdeburg, Germany

* Corresponding author: Thuy T. T. Le

Received  December 2016 Revised  March 2018 Published  November 2018

Fund Project: The second author is supported by a PhD fellowship for foreign students at the Università di Padova funded by Fondazione CARIPARO. This paper was developed while the second author was visiting the Department of Mathematics of the University of Bayreuth.

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., if only Hölder continuity holds) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Hölder continuous or even discontinuous.

Citation: Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012
##### References:

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##### References:
Part of the triangulation
Minimum time functions for Example 5.1 with different control sets
Minimum time function for Example 5.1 with U = [−1, 1]2, $\mathcal{S} = \left\{ 0 \right\}$
Minimum time function for Example 5.2a) with target set {0} resp. B0.05(0)
Minimum time functions for Example 5.2b)
Approximate optimal trajectories for Example 5.2a) resp. b)
Minimum time functions for Examples 5.3 and 5.4
Euler and Heun's iterates, minimum time function for Example 5.5 resp
Reachable sets and minimum time functions for Example 5.6
Reachable sets with various end times tf for Examples 5.7 and 5.8
Reachable sets with various end times and different target sets for Example 5.9
Reachable sets with various end times and different control sets for Example 5.10
error estimates for Example 5.1 with different control and target sets
 $N_{\mathcal{R}} = N_U$ $U=B_1(0)$, $\mathcal{S}=B_{0.25}(0)$ $U=[-1, 1]^2$, $\mathcal{S}=B_{0.25}(0)$ $U=[-1, 1]^2$, $\mathcal{S}=\left\{ 0 \right\}$ 0.04 50 0.2951 0.2265 0.02 100 0.1862 0.1180 0.01 200 0.1332 0.0122 0.005 400 0.1132 0.0062 0.0025 800 0.0683 0.0062
 $N_{\mathcal{R}} = N_U$ $U=B_1(0)$, $\mathcal{S}=B_{0.25}(0)$ $U=[-1, 1]^2$, $\mathcal{S}=B_{0.25}(0)$ $U=[-1, 1]^2$, $\mathcal{S}=\left\{ 0 \right\}$ 0.04 50 0.2951 0.2265 0.02 100 0.1862 0.1180 0.01 200 0.1332 0.0122 0.005 400 0.1132 0.0062 0.0025 800 0.0683 0.0062
Error estimates for Ex. 5.2 a) for combination methods of order 1 and 2
 h $N_{\mathcal{R}}$ Euler scheme & Riemann sum Heun's scheme & trapezoid rule 0.04 50 0.2951 0.2265 0.02 100 0.1862 0.1180 0.01 200 0.1332 0.0122 0.005 400 0.1132 0.0062 0.0025 800 0.0683 0.0062
 h $N_{\mathcal{R}}$ Euler scheme & Riemann sum Heun's scheme & trapezoid rule 0.04 50 0.2951 0.2265 0.02 100 0.1862 0.1180 0.01 200 0.1332 0.0122 0.005 400 0.1132 0.0062 0.0025 800 0.0683 0.0062
Error estimates for Ex. 5.2 a) for Runge-Kutta meth. of order 1 and 2
 h $N_{\mathcal{R}}$ set-valued Euler method set-valued Heun method 0.04 50 0.2330 0.2265 0.02 100 0.1681 0.1180 0.01 200 0.1149 0.0122 0.005 400 0.0753 0.0062 0.0025 800 0.0318 0.0062
 h $N_{\mathcal{R}}$ set-valued Euler method set-valued Heun method 0.04 50 0.2330 0.2265 0.02 100 0.1681 0.1180 0.01 200 0.1149 0.0122 0.005 400 0.0753 0.0062 0.0025 800 0.0318 0.0062
Error estimates for Example 5.3 for methods of order 1 and 2
 h Euler scheme & Riemann sum Heun's scheme & trapezoid rule 0.05 0.170 0.1153 0.025 0.095 0.0470 0.0125 0.0599 0.0133 0.00625 0.0285 0.0032
 h Euler scheme & Riemann sum Heun's scheme & trapezoid rule 0.05 0.170 0.1153 0.025 0.095 0.0470 0.0125 0.0599 0.0133 0.00625 0.0285 0.0032
Error estimates for Example 5.5 with set-valued methods of order 1 and 2
 h $N_{\mathcal{R}}$ set-valued Euler scheme set-valued Heun's scheme 0.5 50 0.0848 0.1461 0.1 100 0.0060 0.0076 0.05 200 0.0015 0.0020 0.025 400 0.00042 0.000502 0.0125 800 0.000108 0.000126
 h $N_{\mathcal{R}}$ set-valued Euler scheme set-valued Heun's scheme 0.5 50 0.0848 0.1461 0.1 100 0.0060 0.0076 0.05 200 0.0015 0.0020 0.025 400 0.00042 0.000502 0.0125 800 0.000108 0.000126
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