On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*
Journal of Global Optimization
Semismooth Newton Methods for Time-Optimal Control for a Class of ODEs
SIAM Journal on Control and Optimization
Inexact Restoration for Runge-Kutta Discretization of Optimal Control Problems
SIAM Journal on Numerical Analysis
Approximations with error estimates for optimal control problems for linear systems
LSSC'05 Proceedings of the 5th international conference on Large-Scale Scientific Computing
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We study second order sufficient optimality conditions (SSC) for optimal control problems with control appearing linearly. Specifically, time-optimal bang-bang controls will be investigated. In [N. P. Osmolovskii, Sov. Phys. Dokl., 33 (1988), pp. 883--885; Theory of Higher Order Conditions in Optimal Control, Doctor of Sci. thesis, Moscow, 1988 (in Russian); Russian J. Math. Phys., 2 (1995), pp. 487--516; {\it Russian J. Math. Phys.}, 5 (1997), pp. 373--388; Proceedings of the Conference "Calculus of Variations and Optimal Control," Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 198--216; A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, Transl. Math. Monogr. 180, AMS, Providence, RI, 1998], SSC have been developed in terms of the positive definiteness of a quadratic form on a critical cone or subspace. No systematical numerical methods for verifying SSC are to be found in these papers. In the present paper, we study explicit representations of the critical subspace. This leads to an easily implementable test for SSC in the case of a bang-bang control with one or two switching points. In general, we show that the quadratic form can be simplified by a transformation that uses a solution to a linear matrix differential equation. Particular conditions even allow us to convert the quadratic form to perfect squares. Three numerical examples demonstrate the numerical viability of the proposed tests for SSC.