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

  • * Corresponding author: Thuy T. T. Le

    * Corresponding author: Thuy T. T. Le

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

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  • 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.

    Mathematics Subject Classification: 49N60, 93B03 (49N05, 49M25, 52A27).


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  • Figure 1.  Part of the triangulation

    Figure 2.  Minimum time functions for Example 5.1 with different control sets

    Figure 3.  Minimum time function for Example 5.1 with U = [−1, 1]2, $\mathcal{S} = \left\{ 0 \right\}$

    Figure 4.  Minimum time function for Example 5.2a) with target set {0} resp. B0.05(0)

    Figure 5.  Minimum time functions for Example 5.2b)

    Figure 6.  Approximate optimal trajectories for Example 5.2a) resp. b)

    Figure 7.  Minimum time functions for Examples 5.3 and 5.4

    Figure 8.  Euler and Heun's iterates, minimum time function for Example 5.5 resp

    Figure 9.  Reachable sets and minimum time functions for Example 5.6

    Figure 10.  Reachable sets with various end times tf for Examples 5.7 and 5.8

    Figure 11.  Reachable sets with various end times and different target sets for Example 5.9

    Figure 12.  Reachable sets with various end times and different control sets for Example 5.10

    Table 1.  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
     | Show Table
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    Table 2.  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
     | Show Table
    DownLoad: CSV

    Table 3.  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
     | Show Table
    DownLoad: CSV

    Table 4.  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
     | Show Table
    DownLoad: CSV

    Table 5.  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
     | Show Table
    DownLoad: CSV
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