Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T07:04:32.692Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  22 June 2011

Nikolai P. Osmolovskii
Nikolai P. Osmolovskii*
Affiliation:
Systems Research Institute, ul. Newelska 6, 01-447 Warszawa, Poland Politechnika Radomska, ul. Malczewskiego 20A, 26-600 Radom, Poland University of Natural Sciences and Humanities in Siedlce, ul. 3 Maja 54, 08-110 Siedlce, Poland. [email protected]
Get access

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.

Keywords

Pontryagin’s principlecritical conequadratic formsecond order sufficient conditionquadratic growthHoffman’s error bound
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 452 - 482
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bonnans, J.F. and Hermant, A., Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 561–598. Google Scholar
Bonnans, J.F. and Osmolovskii, N.P., Second-order analysis of optimal control problems with control and initial-final state constraints. J. Convex Anal. 17 (2010) 885–913. Google Scholar
J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimal Control Problems. Springer, New York (2000).
Dmitruk, A.V., Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in control. I. Deciphering theorem. Izv. Akad. Nauk SSSR 50 (1986) 284–312. Google Scholar
Dmitruk, A.V., Quadratic conditions for a Pontryagin minimum in an optimal control problem linear in control. II. Theorem on weakening inequality constraints. Izv. Akad. Nauk SSSR 51 (1987) 812–832. Google Scholar
Dubovitski, A.Ya. and Milyutin, A.A., 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. Google Scholar
Hoffman, A.J., On approximate solutions of systems of linear inequalities. J. Res. Nat’l Bur. Standarts 49 (1952) 263–265. Google Scholar
Levitin, E.S., Milyutin, A.A. and Osmolovskii, N.P., Higher-order local minimum conditions in problems with constraints. Uspekhi Mat. Nauk 33 (1978) 85–148; English translation in Russian Math. Surveys 33 (1978) 97–168. Google Scholar
Malanowski, K., Stability and sensitivity of solutions to nonlinear optimal control problems. Appl. Math. Optim. 32 (1994) 111–141. Google Scholar
K. Malanowski, Sensitivity analysis for parametric control problems with control–state constraints. Dissertationes Mathematicae CCCXCIV. Polska Akademia Nauk, Instytut Matematyczny, Warszawa (2001) 1–51.
Maurer, H., First and second order sufficient optimality conditions in mathematical programming and optimal control. Mathematical Programming Study 14 (1981) 163–177. Google Scholar
Maurer, H. and Pickenhain, S., Second order sufficient conditions for optimal control problems with mixed control-state constraints. J. Optim. Theory Appl. 86 (1995) 649–667. Google Scholar
A.A. Milyutin, Maximum Principle in the General Optimal Control Problem. Fizmatlit, Moscow (2001) [in Russian].
Milyutin, A.A. and Osmolovskii, N.P., 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. Google Scholar
Milyutin, A.A. and Osmolovskii, N.P., 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. Google Scholar
Milyutin, A.A. and Osmolovskii, N.P., 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. Google Scholar
A.A. Milyutin and N.P. Osmolovskii, Calculus of Variations and Optimal Control, Translations of Mathematical Monographs 180. American Mathematical Society, Providence (1998).
Osmolovskii, N.P., On a system of linear inequalities on a convex set. Usp. Mat. Nauk. 32 (1977) 223–224 [in Russian]. Google Scholar
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].
N.P. Osmolovskii, Theory of higher order conditions in optimal control. Ph.D. thesis, Moscow (1988) [in Russian].
Osmolovskii, N.P., Quadratic optimality conditions for broken extremals in the general problem of calculus of variations. J. Math. Sci. 123 (2004) 3987–4122. Google Scholar
Osmolovskii, N.P., Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints. J. Math. Science 173 (2011) 1–106. Google Scholar
Zeidan, V., Extended Jacobi sufficiency criterion for optimal control. SIAM J. Control. Optim. 22 (1984) 294–301. Google Scholar
Zeidan, V., The Riccati equation for optimal control problems with mixed state-control constraints : necessity and sufficiency. SIAM J. Control Optim. 32 (1994) 1297–1321. Google Scholar