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

Colombo, Giovanni; Le, Thuy Thi Thien

doi:10.3934/dcds.2015.35.4293

Cited by 5 publications

(17 citation statements)

References 19 publications

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“…For a given x ∈ R n , the problem of computing approximately the minimum time T (x) to reach S by following the dynamics (1) is deeply investigated in literature. It was usually obtained by solving the associated discrete Hamilton-Jacobi-Bellman equation (HJB), see, for instance, [13,25,19,28]. Neglecting the space discretization we obtain an approximation of T (x).…”

Section: By Definitionmentioning

confidence: 99%

“…We consider either the small ball B 0.05 (0) or the origin as target set S. Then the minimum time function is 1 2 -Hölder continuous for the first choice of S see [34,19] is only absolutely continuous with respect to τ for some directions l ∈ S 1 with l 1 = 0. Hence, we can expect that the convergence order for the set-valued quadrature method is at most 2.…”

Section: Robert Baier and Thuy T T Lementioning

confidence: 99%

“…for some positive constant C. The inequality (31) shows that the error estimate is improved in comparison with the one obtained in [19] and does not assume explicitly the regularity of optimal solutions as in [14]. One possibility to define the modulus of continuity satisfying the required property of non-decrease in Theorem 3.3 is as follows:…”

Section: 3mentioning

confidence: 99%

“…Minimum time functions are usually computed by solving the Hamilton-Jacobi-Bellman (HJB) equations and by the dynamic programming principle, see e.g. [16,17,13,14,15,19,28]. In this approach, the minimal requirement on the regularity of T (x) is the continuity, see e.g.…”

mentioning

confidence: 99%

“…In this approach, the minimal requirement on the regularity of T (x) is the continuity, see e.g. [13,19,28]. The solution of a HJB equation with suitable boundary conditions gives immediately -after a transformation -the minimum time function and its level sets provide a description of the reachable sets.…”

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confidence: 99%

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

Baier¹,

Le²

2019

Mathematical Control &Amp; Related Fields

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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., Hölder continuity) 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.2010 Mathematics Subject Classification. 49N60, 93B03(49N05, 49M25, 52A27).

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Section: By Definitionmentioning

confidence: 99%

Section: Robert Baier and Thuy T T Lementioning

confidence: 99%

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Baier¹,

Le²

2019

Mathematical Control &Amp; Related Fields

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A double-sided dynamic programming approach to the minimum time problem and its numerical approximation

Grüne

2017

Applied Numerical Mathematics

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Small-time local attainability for a class of control systems with state constraints

Thuy

Marigonda

2017

ESAIM: COCV

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In this paper we consider the problem of small time local attainability (STLA) for nonlinear finite-dimensional time-continuous control systems in presence of state constraints. More precisely, given a nonlinear control system subjected to state constraints and a closed set S, we provide sufficient conditions to steer to S every point of a suitable neighborhood of S along admissible trajectories of the system, respecting the constraints, and giving also an upper estimate of the minimum time needed for each point to reach the target. Methods of nonsmooth analysis are used.

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

Cited by 5 publications

References 19 publications

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

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

A double-sided dynamic programming approach to the minimum time problem and its numerical approximation

Small-time local attainability for a class of control systems with state constraints

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