The Euler Method for Linear Control Systems Revisited

Haunschmied, J.L.; Pietrus, Alain; Veliov, Vladimir M.

doi:10.1007/978-3-662-43880-0_9

Cited by 14 publications

(17 citation statements)

References 5 publications

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“…The results of order k + 1 from Lemma 1.3 and Theorem 1.4 are strongly connected to the findings in [19] and [32]. In [32] the authors derived stability results for Mayer-type problems with the help of metric regularity and smoothness assumptions on the problem parameters.…”

Section: Remarkmentioning

confidence: 66%

“…Under weaker assumptions on the structure of the switching function we derive a condition of order k + 1, which will play an important role in the analysis of convergence for discrete regularization approaches. This condition is closely connected to the controllability index in [19,32] (see Remark 3). We make the following assumption on bang-bang regularity (cf.…”

Section: X(t) T W (T)x(t) + X(t) T S(t)u(t) + W(t) T X(t) + R (T) T Umentioning

confidence: 89%

“…Using the stability results from [32] it was shown in [19] that the direct Euler discretization for Mayer-type problems converges with order O(h 1/k ) w.r.t. the controllability index k and the mesh size h. Under the Assumptions (A1) and (A2) k we will show convergence of order O(h 1/(k+1) ) for a discrete regularization approach applied to linear-quadratic problems of type (OS) with bang-bang controls.…”

Section: Remarkmentioning

confidence: 98%

See 2 more Smart Citations

Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

Seydenschwanz

2015

Comput Optim Appl

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We analyze a combined regularization-discretization approach for a class of linear-quadratic optimal control problems. By choosing the regularization parameter α with respect to the mesh size h of the discretization we approximate the optimal bangbang control. Under weaker assumptions on the structure of the switching function we generalize existing convergence results and prove error estimates of order O(h 1/(k+1) ) with respect to the controllability index k.

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Section: Remarkmentioning

confidence: 66%

Section: X(t) T W (T)x(t) + X(t) T S(t)u(t) + W(t) T X(t) + R (T) T Umentioning

confidence: 89%

Section: Remarkmentioning

confidence: 98%

See 1 more Smart Citation

Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

Seydenschwanz

2015

Comput Optim Appl

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“…We recall that the Euler method has already been profoundly investigated in the case where bang-bang controls appear (e.g. [1,2,13,18,26]). As mentioned in the introduction, in doing this we use an idea that originates from [15,28] and was implemented in [23] in the case of Mayer's problems.…”

Section: Discretization Schemementioning

confidence: 99%

“…Results on the stability of solutions with respect to disturbances were also recently obtained, see [4,12,14,25] and the bibliography therein. Based on these results, error estimates for the accuracy of the Euler discretization scheme applied to various classes of affine optimal control problems were obtained in [1,2,13,18,26,27]. The error estimates are at most of first order with respect to the discretization step, which is natural in view of the discontinuity of the optimal control.…”

Section: Introductionmentioning

confidence: 99%

Higher-order numerical scheme for linear quadratic problems with bang–bang controls

Scarinci

Veliov

2017

Comput Optim Appl

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This paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang-bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler's scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances.Keywords Optimal control · Numerical methods · Bang-bang control · Linear-quadratic optimal control problems · Time-discretization methods Mathematics Subject Classification

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On the regularity and stability of the Mayer‐type convex optimal control problems

Djendel

Zhang

Zhou

2022

Asian Journal of Control

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This paper is devoted to sufficient condition for Strong Metric sub‐Regularity (SMsR for short) of the set‐valued mapping corresponding to the local description of Pontryagin maximum principle for the Mayer‐type optimal control problems with convexity condition of the Hamiltonian and functional. In particular, stability property of optimal control for the Mayer‐type problem has been established for the occasion of a polyhedral control set and entirely bang‐bang solution structure. Moreover, based on the sufficiency of SMsR and stability property of optimal control, we give the approximate errors of Euler discretization methods utilized to such problems.

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The Euler Method for Linear Control Systems Revisited

Cited by 14 publications

References 5 publications

Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

Higher-order numerical scheme for linear quadratic problems with bang–bang controls

On the regularity and stability of the Mayer‐type convex optimal control problems

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