Abstract

The paper establishes properties of the type of (strong) metric regularity of the set-valued map associated with the system of necessary optimality conditions for optimal control problems that are affine with respect to the control (shortly, affine problems). It is shown that for such problems it is reasonable to extend the standard notions of metric regularity by involving two metrics in the image space of the map. This is done by introducing (following an earlier paper by the first and the third named author) the concept of (strong) bi-metric regularity in a general space setting. Lyusternik-Graves-type theorems are proved for (strongly) bi-metrically regular maps, which claim stability of these regularity properties with respect to "appropriately small" perturbations. Based on that, it is proved that in the case of a map associated with affine optimal control problems, the strong bi-metric regularity is invariant with respect to linearization. This result is complemented with a sufficient condition for strong bi-metric regularity for linear-quadratic affine optimal control problems, which applies to the "linearization" of a nonlinear affine problem. Thus the same conditions are also sufficient for strong bi-metric regularity in the nonlinear affine problem.

Author information

Contact details are reproduced from the original publication and may be historical.

Vladimir M. Veliov

Inst. of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Austria

vladimir.veliov@tuwien.ac.at

Keywords

  • Optimal control
  • affine problems
  • metric regularity
  • solution stability
  • perturbed control problems

Mathematics Subject Classification

49J30, 49K15, 49K40

References

  1. A. A. Agrachev, G. Stefani, P. Zezza: Strong optimality for a bang-bang trajectory, SIAM J. Control Optim. 41(4) (2003) 991–1014.https://doi.org/10.1137/S036301290138866XGoogle Scholar
  2. W. Alt, R. Baier, M. Gerdts, F. Lempio: Approximation of linear control problems with bang-bang solutions, Optimization 62(1) (2013) 9–32.https://doi.org/10.1080/02331934.2011.568619Google Scholar
  3. W. Alt, C. Schneider, M. Seydenschwanz: Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions, Appl. Math. Computation 287(C) (2016) 104–124.https://doi.org/10.1016/j.amc.2016.04.028Google Scholar
  4. A. L. Dontchev, W. W. Hager: Lipschitzian stability in nonlinear control and optimization, SIAM J. Control Optim. 31 (1993) 569–603.https://doi.org/10.1137/0331026Google Scholar
  5. A. L. Dontchev, M. I. Krastanov, V. M. Veliov: On the existence of Lipschitz continuous optimal feedback control in ODE control problems, submitted; available as Research Report 2018-04, ORCOS, TU Wien (2018).Google Scholar
  6. A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd ed., Springer, New York (2014).https://doi.org/10.1007/978-1-4939-1037-3Google Scholar
  7. U. Felgenhauer: On stability of bang-bang type controls, SIAM J. Control Optim. 41(6) (2003) 1843–1867.https://doi.org/10.1137/S0363012901399271Google Scholar
  8. U. Felgenhauer, L. Poggiolini, G. Stefani: Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour, Control and Cybernetics 38 (2009) 1305–1325.Google Scholar
  9. H. Frankowska, M. Quincampoix: Hölder metric regularity of set-valued maps, Math. Program. 132 (2012) 333–354.https://doi.org/10.1007/s10107-010-0401-7Google Scholar
  10. A. D. Ioffe: Variational Analysis of Regular Mappings: Theory and Applications, Springer, Cham (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  11. H. Maurer, N. P. Osmolovskii: Second order sufficient conditions for time-optimal bang-bang control, SIAM J. Control Optim. 42(6) (2004) 2239–2263.https://doi.org/10.1137/S0363012902402578Google Scholar
  12. J. Preininger, T. Scarinci, V. M. Veliov: Metric regularity properties in bang-bang type linear-quadratic optimal control problems, Set-Valued Var. Analysis 27(2) (2019) 381–404.https://doi.org/10.1007/s11228-018-0488-1Google Scholar
  13. M. Quincampoix, V. M. Veliov: Metric regularity and stability of optimal control problems for linear systems, SIAM J. Control Optim. 51(5) (2013) 4118–4137.https://doi.org/10.1137/130914383Google Scholar
  14. S. M. Robinson: Strongly regular generalized equations, Math. Oper. Res. 5 (1980) 43–62.https://doi.org/10.1287/moor.5.1.43Google Scholar

Suggested citation

M. Quincampoix, T. Scarinci, V. M. Veliov. “On the Metric Regularity of Affine Optimal Control Problems.” Journal of Convex Analysis 27 (2020), No. 2, 509–533. https://doi.org/10.68381/jca27027

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.