# Second order analysis of control-affine problems with scalar state constraint

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## Abstract

In this article we establish new second order necessary and sufficient optimality conditions for a class of control-affine problems with a scalar control and a scalar state constraint. These optimality conditions extend to the constrained state framework the Goh transform, which is the classical tool for obtaining an extension of the Legendre condition.

## Mathematics Subject Classification

49K15 49K27## Notes

### Acknowledgments

We wish to thank the anonymous referees for their bibliographical advices. This work was partially supported by the European Union under the 7th Framework Pro-gramme FP7-PEOPLE-2010-ITN Grant Agreement Number 264735-SADCO. The last stage of this research took place while the first author was holding a postdoctoral position at IMPA, Rio de Janeiro, with CAPES-Brazil funding.

## References

- 1.Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)MATHGoogle Scholar
- 2.Aftalion, A., Bonnans, J.: Optimization of running strategies based on anaerobic energy and variations of velocity. SIAM J. Appl. Math.
**74**(5), 1615–1636 (2014)MathSciNetMATHCrossRefGoogle Scholar - 3.Agrachev, A., Sachkov, Y.: Control theory from the geometric viewpoint. In: Encyclopaedia of Mathematical Sciences, 87, Control Theory and Optimization, II. Springer, Berlin (2004)Google Scholar
- 4.Agrachev, A.A., Stefani, G., Zezza, P.L.: Strong optimality for a bang–bang trajectory. SIAM J. Control Optim.
**41**, 991–1014 (2002)MathSciNetMATHCrossRefGoogle Scholar - 5.Aronna, M.S.: Singular Solutions in Optimal Control: Second Order Conditions and a Shooting Algorithm. In: Technical report, ArXiv (2013). (Published online as arXiv:1210.7425, submitted)
- 6.Aronna, M.S., Bonnans, J.F., Dmitruk, A.V., Lotito, P.A.: Quadratic order conditions for bang-singular extremals. Numer. Algebra Control Optim.
**2**(3), 511–546 (2012)MathSciNetMATHCrossRefGoogle Scholar - 7.Arutyunov, A.V.: On necessary conditions for optimality in a problem with phase constraints. Dokl. Akad. Nauk SSSR
**280**(5), 1033–1037 (1985)MathSciNetMATHGoogle Scholar - 8.Arutyunov, A.V.: Optimality Conditions: Abnormal and degenerate problems, Volume 526 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (2000). Translated from the Russian by S. A. VakhrameevGoogle Scholar
- 9.Bonnans, J.F., Hermant, A.: Revisiting the analysis of optimal control problems with several state constraints. Control Cybern.
**38**(4A), 1021–1052 (2009)MathSciNetMATHGoogle Scholar - 10.Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)MATHCrossRefGoogle Scholar
- 11.Bonnard, B., Faubourg, L., Launay, G., Trélat, E.: Optimal control with state constraints and the space shuttle re-entry problem. J. Dyn. Control Syst.
**9**(2), 155–199 (2003)MathSciNetMATHCrossRefGoogle Scholar - 12.Cominetti, R.: Metric regularity, tangent sets and second order optimality conditions. J. Appl. Math. Optim.
**21**, 265–287 (1990)MathSciNetMATHCrossRefGoogle Scholar - 13.de Pinho, M.R., Ferreira, M.M., Ledzewicz, U., Schaettler, H.: A model for cancer chemotherapy with state–space constraints. Nonlinear Anal. Theory Methods Appl.
**63**(5), e2591–e2602 (2005)MATHCrossRefGoogle Scholar - 14.Ekeland, I., Temam, R.: Convex Analysis and Variational Problems, Volume 1 of Studies in Mathematics and its Applications. North-Holland, Amsterdam, 1976. French edition: Analyse convexe et problèmes variationnels. Dunod, Paris (1974)Google Scholar
- 15.Felgenhauer, U.: Optimality and sensitivity for semilinear bang–bang type optimal control problems. Int. J. Appl. Math. Comput. Sci.
**14**(4), 447–454 (2004)MathSciNetMATHGoogle Scholar - 16.Frankowska, H., Tonon, D.: Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints. SIAM J. Control Optim.
**51**(5), 3814–3843 (2013)MathSciNetMATHCrossRefGoogle Scholar - 17.Gabasov, R., Kirillova, F.M.: High-order necessary conditions for optimality. J. SIAM Control
**10**, 127–168 (1972)MathSciNetMATHCrossRefGoogle Scholar - 18.Goh, B.S.: Necessary conditions for singular extremals involving multiple control variables. J. SIAM Control
**4**, 716–731 (1966)MathSciNetMATHCrossRefGoogle Scholar - 19.Goh, B.S., Leitmann, G., Vincent, T.L.: Optimal control of a prey–predator system. Math. Biosci.
**19**, 263–286 (1974)MathSciNetMATHCrossRefGoogle Scholar - 20.Graichen, K., Petit, N.: Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions. In: 17th World Congress of The International Federation of Automatic Control, Volume Proceedings of the 2008 IFAC World Congress, pp. 14301–14306, Seoul (2008). IFACGoogle Scholar
- 21.Hestenes, M.R.: Applications of the theory of quadratic forms in Hilbert space to the calculus of variations. Pac. J. Math.
**1**(4), 525–581 (1951)MathSciNetMATHCrossRefGoogle Scholar - 22.Hoffman, A.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand Sect. B Math. Sci.
**49**, 263–265 (1952)MathSciNetCrossRefGoogle Scholar - 23.Jacobson, D.H., Speyer, J.L.: Necessary and sufficient conditions for optimality for singular control problems: a limit approach. J. Math. Anal. Appl.
**34**, 239–266 (1971)MathSciNetMATHCrossRefGoogle Scholar - 24.Karamzin, D.Y.: Necessary conditions for an extremum in a control problem with phase constraints. Zh. Vychisl. Mat. Mat. Fiz.
**47**(7), 1123–1150 (2007)MathSciNetGoogle Scholar - 25.Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Program.
**41**, 73–96 (1988)MathSciNetMATHCrossRefGoogle Scholar - 26.Kelley, H.J.: A second variation test for singular extremals. AIAA J.
**2**, 1380–1382 (1964)MathSciNetMATHCrossRefGoogle Scholar - 27.Lyusternik, L.: Conditional extrema of functions. Math. USSR-Sb
**41**, 390–440 (1934)Google Scholar - 28.Malanowski, K., Maurer, H.: Sensitivity analysis for parametric control problems with control-state constraints. Comput. Optim. Appl.
**5**(3), 253–283 (1996)MathSciNetMATHCrossRefGoogle Scholar - 29.Maurer, H.: On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control Optim.
**15**(3), 345–362 (1977)MathSciNetMATHCrossRefGoogle Scholar - 30.Maurer, H., Kim, J.-H.R., Vossen, G.: On a state-constrained control problem in optimal production and maintenance. In: Deissenberg, C., Hartl, R.F. (eds.) Optimal Control and Dynamic Games, Applications in Finance, Management Science and Economics, vol. 7, pp. 289–308. Springer, Berlin (2005)Google Scholar
- 31.Maurer, H., Osmolovskii, N.P.: Second order optimality conditions for bang–bang control problems. Control Cybern.
**32**, 555–584 (2003)MATHGoogle Scholar - 32.McDanell, J.P., Powers, W.F.: Necessary conditions joining optimal singular and nonsingular subarcs. SIAM J. Control
**9**, 161–173 (1971)MathSciNetMATHCrossRefGoogle Scholar - 33.Milyutin, A.A., Osmolovskii, N.N.: Calculus of Variations and Optimal Control. American Mathematical Society, Providence (1998)MATHGoogle Scholar
- 34.Osmolovskii, N.P., Maurer, H.: Equivalence of second order optimality conditions for bang-bang control problems. I. Main results. Control Cybern.
**34**(3), 927–950 (2005)MathSciNetMATHGoogle Scholar - 35.Osmolovskii, N.P., Maurer, H.: Applications to Regular and Bang–Bang Control, Volume 24 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2012). Second-order necessary and sufficient optimality conditions in calculus of variations and optimal controlGoogle Scholar
- 36.Poggiolini, L., Spadini, M.: Strong local optimality for a bang–bang trajectory in a Mayer problem. SIAM J. Control Optim.
**49**, 140–161 (2011)MathSciNetMATHCrossRefGoogle Scholar - 37.Poggiolini, L., Stefani, G.: Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst.
**17**(4), 469–514 (2011)MathSciNetMATHCrossRefGoogle Scholar - 38.Rampazzo, F., Vinter, R.: Degenerate optimal control problems with state constraints. SIAM J. Control Optim.
**39**(4), 989–1007 (2000). (electronic)MathSciNetMATHCrossRefGoogle Scholar - 39.Robinson, S.M.: First order conditions for general nonlinear optimization. SIAM J. Appl. Math.
**30**, 597–607 (1976)MathSciNetMATHCrossRefGoogle Scholar - 40.Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHCrossRefGoogle Scholar
- 41.Russak, I.B.: Second-order necessary conditions for general problems with state inequality constraints. J. Optim. Theory Appl.
**17**(112), 43–92 (1975)MathSciNetMATHCrossRefGoogle Scholar - 42.Russak, I.B.: Second order necessary conditions for problems with state inequality constraints. SIAM J. Control
**13**, 372–388 (1975)MathSciNetMATHCrossRefGoogle Scholar - 43.Schattler, H.: A local field of extremals near boundary arc-interior arc junctions. In: 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ’05. pp. 945–950 (2005)Google Scholar
- 44.Schättler, H.: Local fields of extremals for optimal control problems with state constraints of relative degree 1. J. Dyn. Control Syst.
**12**(4), 563–599 (2006)MathSciNetMATHCrossRefGoogle Scholar - 45.Seywald, H., Cliff, E.M.: Goddard problem in presence of a dynamic pressure limit. J. Guid. Control Dyn.
**16**(4), 776–781 (1993)MATHCrossRefGoogle Scholar

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