In this paper which is closely related to the previous paper  we specify general theory developed there. We study the structure of Jacobi fields in the case of an analytic system and piecewise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piecewise analytic in this case but may have jump discontinuities. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighbourhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps.
This is a preview of subscription content, log in to check access.
Buy single article
Instant unlimited access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Agrachev A. 1998. Feedback?invariant optimal control theory and differential geometry, II. Jacobi curves for singular extremals. J. Dynamical and Control Systems, pp 583?604,.
Agrachev A. 2004. Nonlinear and optimal control theory, chapter Geometry of optimal control problems and Hamiltonian systems, pp 1–59, Springer.
Agrachev A, Beschastnyi I. Jacobi fields in optimal control: Morse and Maslov indices. preprint.
Agrachev A, Beschastnyi I. Symplectic geometry of constrained optimization. Regular and Chaotic Dynamics 2017;22:750–770.
Agrachev A, Gamkrelidze R. 1990. Nonlinear controllability and optimal control, chapter Symplectic geometry for optimal control, pp 263–277, CRC Press Book.
Agrachev A, Gamkrelidze R. 1997. Feedback–?invariant optimal control theory and differential geometry, I. Regular extremals. J. Dynamical and Control Systems, pp 343?-389.
Agrachev A, Gamkrelidze R. 1998. Geometry of feedback and optimal control, chapter Symplectic methods in optimization and control, pp 19–77, CRC Press Book.
Agrachev A, Rizzi L, Silveira P. On conjugate times of LQ optimal control problems. J Dynamical and Control Systems 2015;21:625–641.
Agrachev A, Sachkov Y. 2004. Control theory from a geometric point of view. Springer.
Agrachev A, Steffani G, Zezza P. Strong optimality of a bang-bang trajectory. SIAM J on Control and Optimization 2002;41:981–1014.
Arnold V. The Sturm theorems and symplectic geometry. Funct Anal Appl 1985; 19:251–259.
Bonnard B, Chyba M. 2003. Singular trajectories and their role in control theory. Springer.
Caillau J-B, Fejoz J, Orieux M, Roussarie R. Singularities of min time affine control systems. preprint, hal-01718345, version 1.
Cappell S, Lee R, Miller E. On the Maslov index. Comm Pure Appl Math 1994;47:121–186.
Coddington E, Levinson N. 1984. Theory of ordinary differential equations. Krieger pub co.
de Gosson M. 2000. Symplectic geometry and quantum mechanics. Birkhauser.
Guillemin V, Sternberg S. 1977. Geometric asymptotics. American Mathematical Society.
Horn R, Johnson C. 2012. Matrix analysis. Cambridge University Press, 2 edn.
Morse M. Singular quadratic functionals. Math Ann 1973;201:60–76.
Morse M, Leighton W. Singular quadratic functionals. Trans Amer Math Soc 1936;36:252–286.
Olver F, Maximon L. 2010. NIST handbook Of mathematical functions, chapter Bessel functions, pp 215–286, Cambridge University Press.
Osmolovskii N, Maurer H. 2016. Advances in mathematical modeling optimization and optimal control, chapter Second-order optimality conditions for broken extremals and bang-bang controls, pp 147–201, Springer.
Reid W. 1972. Riccati differential equations. Academic Press.
Shättler H, Ledzewicz U. 2012. Geometric optimal control. Springer.
Shayman M. Phase portrait of the matrix Riccati equation. SIAM J Control and Optimization 1986;24(1):1–65.
Sussmann H, Liu W. 1995. Shortest paths for sub-Riemannian metrics on rank-two distributions.
Sussmann HJ. 1986. Algebraic and geometric methods in nonlinear control theory, chapter Envelopes, conjugate points, and optimal bang-bang extremals, pp 325–346. D Reidel Publishing company.
Teschl G. 2012. Ordinary differential equations and dynamical systems. American Mathematical Society.
Wagner V. The geometrical theory of the simplest n-dimensional singular problem of the calculus of variations. Rec Math [Mat Sbornik] N S 1947;63:321–364.
We would like to thank Luca Rizzi for pointing out an error in the initial formulation of the oscillation theorem and Jean-Baptiste Caillau, Francesco Rossi and the anonymous referee for helpful discussions and remarks that helped to improve the article.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Agrachev, A., Beschastnyi, I. Jacobi Fields in Optimal Control: One-dimensional Variations. J Dyn Control Syst (2020) doi:10.1007/s10883-019-09467-0
- Optimal control
- Second-order conditions
- Jacobi equations
- Singular systems