Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints

Nikolai P. Osmolovskii

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 452-482
  • ISSN: 1292-8119

Abstract

top
Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions.

How to cite

top

Osmolovskii, Nikolai P.. "Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 452-482. <http://eudml.org/doc/221921>.

@article{Osmolovskii2012,
abstract = {Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions. },
author = {Osmolovskii, Nikolai P.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Pontryagin’s principle; critical cone; quadratic form; second order sufficient condition; quadratic growth; Hoffman’s error bound; Pontryagin's principle; Hoffman's error bound},
language = {eng},
month = {7},
number = {2},
pages = {452-482},
publisher = {EDP Sciences},
title = {Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints},
url = {http://eudml.org/doc/221921},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Osmolovskii, Nikolai P.
TI - Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/7//
PB - EDP Sciences
VL - 18
IS - 2
SP - 452
EP - 482
AB - Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions.
LA - eng
KW - Pontryagin’s principle; critical cone; quadratic form; second order sufficient condition; quadratic growth; Hoffman’s error bound; Pontryagin's principle; Hoffman's error bound
UR - http://eudml.org/doc/221921
ER -

References

top
  1. J.F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. Henri Poincaré, Anal. Non Linéaire26 (2009) 561–598.  
  2. J.F. Bonnans and N.P. Osmolovskii, Second-order analysis of optimal control problems with control and initial-final state constraints. J. Convex Anal.17 (2010) 885–913.  
  3. J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimal Control Problems. Springer, New York (2000).  
  4. A.V. Dmitruk, Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in control. I. Deciphering theorem. Izv. Akad. Nauk SSSR50 (1986) 284–312.  
  5. A.V. Dmitruk, Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in control. II. Theorem on weakening inequality constraints. Izv. Akad. Nauk SSSR51 (1987) 812–832.  
  6. A.Ya. Dubovitski and A.A. Milyutin, Extremum problems in the presence of restrictions. Zh. Vychislit. Mat. i Mat. Fiz.5 (1965) 395–453; English translation in U.S.S.R. Comput. Math. Math. Phys.5 (1965) 1–80.  
  7. A.J. Hoffman, On approximate solutions of systems of linear inequalities. J. Res. Nat’l Bur. Standarts49 (1952) 263–265.  
  8. E.S. Levitin, A.A. Milyutin and N.P. Osmolovskii, Higher-order local minimum conditions in problems with constraints. Uspekhi Mat. Nauk33 (1978) 85–148; English translation in Russian Math. Surveys33 (1978) 97–168.  
  9. K. Malanowski, Stability and sensitivity of solutions to nonlinear optimal control problems. Appl. Math. Optim.32 (1994) 111–141.  
  10. K. Malanowski, Sensitivity analysis for parametric control problems with control–state constraints. Dissertationes MathematicaeCCCXCIV. Polska Akademia Nauk, Instytut Matematyczny, Warszawa (2001) 1–51.  
  11. H. Maurer, First and second order sufficient optimality conditions in mathematical programming and optimal control. Mathematical Programming Study14 (1981) 163–177.  
  12. H. Maurer and S. Pickenhain, Second order sufficient conditions for optimal control problems with mixed control-state constraints. J. Optim. Theory Appl.86 (1995) 649–667.  
  13. A.A. Milyutin, Maximum Principle in the General Optimal Control Problem. Fizmatlit, Moscow (2001) [in Russian].  
  14. A.A. Milyutin and N.P. Osmolovskii, High-order conditions for a minimum on a set of sequences in the abstract problem with inequality constraints. Comput. Math. Model.4 (1993) 393–400.  
  15. A.A. Milyutin and N.P. Osmolovskii, High-order conditions for a minimum on a set of sequences in the abstract problem with inequality and equality constraints. Comput. Math. Model.4 (1993) 401–409.  
  16. A.A. Milyutin and N.P. Osmolovskii, High-order conditions with respect to a subsystem of constraints in the abstract minimization problem on a set of sequences. Comput. Math. Model.4 (1993) 410–418.  
  17. A.A. Milyutin and N.P. Osmolovskii, Calculus of Variations and Optimal Control, Translations of Mathematical Monographs180. American Mathematical Society, Providence (1998).  
  18. N.P. Osmolovskii, On a system of linear inequalities on a convex set. Usp. Mat. Nauk. 32 (1977) 223–224[in Russian].  
  19. N.P. Osmolovskii, Higher-Order Necessary and Sufficient Conditions in Optimal Control. Parts 1 and 2, Manuscript deposited in VINITI April 1, No. 2190-B and No. 2191-B (1986) [in Russian].  
  20. N.P. Osmolovskii, Theory of higher order conditions in optimal control. Ph.D. thesis, Moscow (1988) [in Russian].  
  21. N.P. Osmolovskii, Quadratic optimality conditions for broken extremals in the general problem of calculus of variations. J. Math. Sci.123 (2004) 3987–4122.  
  22. N.P. Osmolovskii, Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Science173 (2011) 1–106.  
  23. V. Zeidan, Extended Jacobi sufficiency criterion for optimal control. SIAM J. Control. Optim.22 (1984) 294–301.  
  24. V. Zeidan, The Riccati equation for optimal control problems with mixed state-control constraints : necessity and sufficiency. SIAM J. Control Optim.32 (1994) 1297–1321.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.