## Second-order necessary conditions for a strong local minimum in a control problem with general control constraints.(English)Zbl 1421.49019

Summary: We establish some second-order necessary optimality conditions for strong local minima in the Mayer type optimal control problem with a general control constraint $$U \subset \mathrm{I\! R}^m$$ and final state constraint described by a finite number of inequalities. In the difference with the main approaches of the existing literature, the second order tangents to $$U$$ and the second order linearization of the control system are used to derive the second-order necessary conditions. This framework allows to replace the control system at hand by a differential inclusion with convex right-hand side. Such a relaxation of control system, and the differential calculus of derivatives of set-valued maps lead to fairly general statements. We illustrate the results by considering the case of control constraints defined by inequalities involving functions with positively or linearly independent gradients of active constraints, but also in the cases where the known approaches do not apply.

### MSC:

 49K15 Optimality conditions for problems involving ordinary differential equations 34H05 Control problems involving ordinary differential equations 49K21 Optimality conditions for problems involving relations other than differential equations
Full Text:

### References:

 [1] Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Berlin (1990) · Zbl 0713.49021 [2] Frankowska, H., The maximum principle for an optimal solution to a differential inclusion with end point constraints, SIAM J. Control Optim., 25, 145-157, (1987) · Zbl 0614.49017 [3] Frankowska, H., High order inverse mapping theorems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, S6, 283-303, (1989) · Zbl 0701.49040 [4] Frankowska, H., Osmolovskii, N.: Second-order necessary optimality conditions for the Mayer problem subject to a general control constraint. In: Analysis and Geometry in Control theory and its Applications, Springer INdAM Ser., 11, Springer, Cham, pp. 171-207 (2015) · Zbl 1329.49033 [5] Frankowska, H.; Hoehener, D.; Tonon, D., A second-order maximum principle in optimal control under state constraints, Serdica Math. J., 39, 233-270, (2013) · Zbl 1324.49017 [6] Hoehener, D., Variational approach to second-order optimality conditions for control problems with pure state constraints, SIAM J. Control Optim., 50, 1139-1173, (2012) · Zbl 1246.49016 [7] Levitin, ES; Milyutin, AA; Osmolovskii, NP, Conditions of high order for a local minimum in problems with constraints, Russ. Math. Surv., 33, 97-168, (1978) · Zbl 0456.49015 [8] Osmolovskii, NP, Necessary quadratic conditions of extremum for discontinuous controls in optimal control problem with mixed constraints, J. Math. Sci., 183, 435-576, (2012) · Zbl 1263.49001 [9] Osmolovskii, NP, Necessary second-order conditions for a weak local minimum in a problem with endpoint and control constraints, J. Math. Anal. Appl., 457, 1613-1633, (2017) · Zbl 1376.49030 [10] Osmolovskii, N.P., Maurer, H.: Applications to regular and bang-bang control. In: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM, Philadelphia (2012) · Zbl 1263.49002 [11] Pontryagin, L.S., Boltyanski, V.G., Gramkrelidze, R.V., Miscenko, E.F.: The Mathematical Theory of Optimal Processes. Pergamon Press, New York (1964)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.