Caillau, J.-B.; Cots, O.; Gergaud, J. Differential continuation for regular optimal control problems. (English) Zbl 1248.49025 Optim. Methods Softw. 27, No. 2, 177-196 (2012). Summary: Regular control problems in the sense of the Legendre condition are defined, and second-order necessary and sufficient optimality conditions in this class are reviewed. Adding a scalar homotopy parameter, differential path-following is introduced. The previous sufficient conditions ensure the definiteness and regularity of the path. The role of automatic differentiation for the implementation of this approach is discussed, and two examples excerpted from quantum and space mechanics are detailed. Cited in 1 ReviewCited in 24 Documents MSC: 49K15 Optimality conditions for problems involving ordinary differential equations 49-04 Software, source code, etc. for problems pertaining to calculus of variations and optimal control Keywords:optimal control; second-order conditions; conjugate points; differential homotopy; automatic differentiation Software:ADIFOR; HOMPACK90 PDFBibTeX XMLCite \textit{J. B. Caillau} et al., Optim. Methods Softw. 27, No. 2, 177--196 (2012; Zbl 1248.49025) Full Text: DOI References: [1] Agrachev A. A., Encyclopaedia of Mathematical Sciences 87 (2004) [2] DOI: 10.1137/1.9780898719154 · Zbl 1036.65047 · doi:10.1137/1.9780898719154 [3] DOI: 10.1002/oca.4660090303 · Zbl 0664.93057 · doi:10.1002/oca.4660090303 [4] DOI: 10.2514/1.18196 · doi:10.2514/1.18196 [5] DOI: 10.1002/oca.709 · Zbl 1072.49502 · doi:10.1002/oca.709 [6] DOI: 10.1109/99.537089 · Zbl 05092146 · doi:10.1109/99.537089 [7] DOI: 10.1007/978-3-642-68220-9_8 · doi:10.1007/978-3-642-68220-9_8 [8] Bonnard B., Mathematics and Applications 40 (2003) [9] DOI: 10.1051/cocv:2007012 · Zbl 1123.49014 · doi:10.1051/cocv:2007012 [10] Bonnard, B., Caillau, J.B. and Cots, O. Energy minimization in two-level dissipative quantum control: The integrable case. AIMS Procs. Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications. May, Dresden. to appear · Zbl 1306.81050 [11] DOI: 10.1063/1.3479390 · Zbl 1309.81118 · doi:10.1063/1.3479390 [12] DOI: 10.1051/cocv/2010004 · Zbl 1213.49027 · doi:10.1051/cocv/2010004 [13] Brezis H., Analyse Fonctionnelle. Théorie et Applications (1992) [14] Caillau J.-B., Automatic Differentiation: From Simulation to Optimization pp 109– (2002) [15] DOI: 10.1023/B:JOTA.0000004870.74778.ae · Zbl 1066.70016 · doi:10.1023/B:JOTA.0000004870.74778.ae [16] Do Carmo M. P., Riemannian Geometry (1992) · doi:10.1007/978-1-4757-2201-7 [17] DOI: 10.1287/moor.4.4.390 · Zbl 0422.55001 · doi:10.1287/moor.4.4.390 [18] DOI: 10.1051/cocv:2006003 · Zbl 1113.49032 · doi:10.1051/cocv:2006003 [19] Haberkorn T., AAIA 27 (6) pp 1046– (2004) [20] Hairer E., Solving Ordinary Differential Equations I, Nonstiff Problems (1993) · Zbl 0789.65048 [21] DOI: 10.1007/BF00248267 · Zbl 0864.49020 · doi:10.1007/BF00248267 [22] DOI: 10.1002/oca.794 · doi:10.1002/oca.794 [23] DOI: 10.1023/A:1022627003404 · Zbl 0896.49021 · doi:10.1023/A:1022627003404 [24] Sarychev A. V., Mat. Sb. 41 pp 338– (1982) [25] DOI: 10.1109/TAC.2010.2047742 · Zbl 1368.49029 · doi:10.1109/TAC.2010.2047742 [26] DOI: 10.1002/oca.781 · doi:10.1002/oca.781 [27] DOI: 10.1145/279232.279235 · Zbl 0913.65042 · doi:10.1145/279232.279235 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.