Geometric constrained variational calculus. II: The second variation (Part I)

References

• 1. E. Massa, D. Bruno, G. Luria and E. Pagani, Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061. Link, Web of Science, Google Scholar
• 2. D. Bruno, G. Luria and E. Pagani, On the gauge structure of the calculus of variations with constraints, Int. J. Geom. Methods Mod. Phys. 08 (2011) 1723–1746. Link, Web of Science, Google Scholar
• 3. L. Cesari, Optimization Theory and Applications (Springer, New York, 1983). Crossref, Google Scholar
• 4. V. M. Alekscseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control (Contemporary Soviet Mathematics, 1987). Crossref, Google Scholar
• 5. H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples (Springer, New York, 2010). Google Scholar
• 6. A. A. Agrachev, G. Stefani and P. Zezza, An invariant second variation in optimal control, Int. J. Control 71 (1998) 689–715. Crossref, Web of Science, Google Scholar
• 7. O. Bolza, Lectures on the Calculus of Variations (The University of Chicago Press, Chicago, 1904). Google Scholar
• 8. L. Tonelli, Fondamenti di Calcolo delle Variazioni I e II (Nicola Zanichelli, Bologna, 1921). Google Scholar
• 9. C. Caratheodory, Calculus of Variations and Partial Differential Equations of First Order (Chelsea Publishing Company, New York, 1982). Originally published as Variationsrechnung und Partielle Differentialgleichungen erster Ordnung by B. G. Teubner, Berlin, 1935. Google Scholar
• 10. G. A. Bliss, Lectures on the Calculus of the Variations (The University of Chicago Press, Chicago, 1946). Google Scholar
• 11. M. R. Hestenes, Calculus of Variations and Optimal Control Theory (Wiley, New York, 1966). Google Scholar
• 12. I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ, 1963). Google Scholar
• 13. H. Sagan, Introduction to the Calculus of Variations (McGraw-Hill Book Company, New York, 1969). Google Scholar
• 14. M. Morse, Variational Analysis (Wiley, New York, 1973). Google Scholar
• 15. C. Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, 1949). Reprinted by Dover Publications (1970). Google Scholar
• 16. M. Giaquinta and S. Hildebrandt, Calculus of Variations I, II (Springer, Berlin, 1996). Google Scholar
• 17. A. A. Agrachev and Yu. L. Sachov, Control Theory From the Geometric Viewpoint (Springer, Berlin, 2004). Crossref, Google Scholar
• 18. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Process (Interscience, New York, 1962). Google Scholar
• 19. W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri Poincaré 27 (1977) 101–114. Web of Science, Google Scholar
• 20. W. M. Tulczyjew, A symplectic formulation of particle dynamics, in Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics, Vol. 570 (Springer, 1977), pp. 57–463. Crossref, Google Scholar
• 21. W. M. Tulczyjew, Symmetries and constants of motion for dynamics in implicit form, Ann. Inst. Henri Poincaré 57 (1992) 146–166. Web of Science, Google Scholar
• 22. J. Radon, Über die Oszillationstheoreme der konjugierten Punkte biem Probleme von Lagrange, Sitzungsberichte Münchner 1927 (1927) 243–257. Google Scholar
• 23. J. Radon, Zum Problem von Lagrange, Abh. Math. Sem. Univ. Hamburg 6 (1928) 273–299. Crossref, Google Scholar
• 24. W. H. Reid, A matrix differential equation of Riccati type, Amer. J. Math. 68 (1946) 237–246. Crossref, Web of Science, Google Scholar