×

zbMATH — the first resource for mathematics

Second-order sufficient conditions for state-constrained optimal control problems. (English) Zbl 1059.49027
Summary: Second-order sufficient optimality conditions (SSC) are derived for an optimal control problem subject to mixed control-state and pure state constraints of order one. The proof is based on a Hamilton-Jacobi inequality and it exploits the regularity of the control function as well as the associated Lagrange multipliers. The obtained SSC involve the strict Legendre-Clebsch condition and the solvability of an auxiliary Riccati equation. They are weakened by taking into account the strongly active constraints.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
49L99 Hamilton-Jacobi theories
Software:
BNDSCO
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Pickenhain, S., and Tammer, K., Sufficient Conditions for Local Optimality in Multidimensional Control Problems with State Restrictions, Zeitschrift für Analysis und ihre Anwendungen, Vol. 10, pp. 397-405, 1991 · Zbl 0767.49016
[2] Maurer, H., and Pickenhain, S., Second-Order Sufficient Conditions for Control Problems with Mixed Control-State Constraints, Journal of Optimization Theory and Applications, Vol. 86, pp. 649-667, 1995. · Zbl 0874.49020
[3] Hartl, R. F., Sethi, S. P., and Vickson, R. G., A Survey of the Maximum Principle for Optimal Control Problems with State Constraints, SIAM Review, Vol. 37, pp. 181-218, 1995. · Zbl 0832.49013
[4] Malanowski, K., On Normality of Lagrange Multipliers for State-Constrained Optimal Control Problems, Optimization, Vol. 52, pp. 75-91, 2003. · Zbl 1057.49004
[5] Zeidan, V., The Riccati Equation for Optimal Control Problems with Mixed State-Control Problems: Necessity and Sufficiency, SIAM Journal on Control and Optimization, Vol. 32, pp. 1297-1321, 1994. · Zbl 0810.49024
[6] Malanowski, K., and Maurer, H., Sensitivity Analysis for Parametric Problems with Control-State Constraints, Computational Optimization and Applications, Vol. 5, pp. 253-283, 1996. · Zbl 0864.49020
[7] Hager, W. W., Lipschitz Continuity for Constrained Processes, SIAM Journal on Control and Optimization, Vol. 17, pp. 321-338, 1979. · Zbl 0426.90083
[8] Neustadt, L. W., Optimization: A Theory of Necessary Conditions, Princeton University Press, Princeton, New Jersey, 1976. · Zbl 0353.49003
[9] Fiacco, A. V., and Mccormick, G. P., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York, NY, 1968. · Zbl 0193.18805
[10] Maurer, H., and Zowe, J., First and Second-Order Necessary and Sufficient Optimality Conditions for Infinite-Dimensional Programming Problems, Mathematical Programming, Vol. 16, pp. 98-110, 1979. · Zbl 0398.90109
[11] Haynsworth, E. V., Determination of the Inertia of a Partitioned Hermitian Matrix, Linear Algebra and Applications, Vol. 1, pp. 73-81, 1968. · Zbl 0155.06304
[12] Maurer, H., First and Second-Order Sufficient Optimality Conditions in Mathematical Programming and Optimal Control, Mathematical Programming Study, Vol. 14, pp. 163-177, 1981. · Zbl 0448.90069
[13] Malanowski, K., and Maurer, H., Sensitivity Analysis for State?Constrained Optimal Control Problems, Discrete and Continuous Dynamical Systems, Vol. 4, pp. 241-272, 1998. · Zbl 0952.49022
[14] Augustin, D., and Maurer, H., Computational Sensitivity Analysis for State-Constrained Control Problems, Annals of Operations Research, Vol. 101, pp. 75-99, 2001. · Zbl 1005.49022
[15] Augustin, D., and Maurer, H., Second-Order Sufficient Conditions and Sensitivity Analysis for the Optimal Control of a Container Crane under State Constraints, Optimization, Vol. 49, pp. 351-368, 2001. · Zbl 1005.49023
[16] Kawasaki, H., and Zeidan, V., Conjugate Points for Variational Problems with Equality State Constraints, SIAM Journal on Control and Optimization, Vol. 39, pp. 433-456, 2000. · Zbl 0972.49012
[17] Oberle, H. J., and Grimm, W., BNDSCO: A Program for the Numerical Solution of Optimal Control Problems, Report 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
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.