×

Bang-singular-bang extremals: sufficient optimality conditions. (English) Zbl 1237.49031

Summary: In this paper, we give second-order sufficient conditions for strong local optimality of a bang-singular-bang extremal in the minimum time problem. The conditions are given in terms of some regularity assumptions on the extremal and of the coercivity of the extended second variation associated with the minimum time problem with fixed endpoints on the singular arc. The conditions are close to the necessary conditions in the usual sense; namely, we require strict inequalities where the necessary conditions have nonstrict inequalities.

MSC:

49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49K15 Optimality conditions for problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. A. Agrachev and R. V. Gamkrelidze. Symplectic methods for optimization and control. In: Geometry of feedback and optimal control (B. Jacubczyk and W. Respondek, eds.). Pure Appl. Math., Marcel Dekker, New York (1997), pp. 1–58. · Zbl 0952.49019
[2] A. A. Agrachev and Yu. L. Sachkov. Control theory from the geometric viewpoint. Springer-Verlag (2004). · Zbl 1062.93001
[3] A. A. Agrachev, G. Stefani, and P. Zezza. An invariant second variation in optimal control. Int. J. Control 71 (1998), No. 5, pp. 689–715. · Zbl 0945.49015 · doi:10.1080/002071798221533
[4] _____ , Strong minima in optimal control. Proc. Steklov Inst. Math. 220 (1998), pp. 4–26.
[5] _____ , Strong optimality for a bang-bang trajectory. SIAM J. Control Optim. 41 (2002), No. 4, pp. 991–1014. · Zbl 1020.49021 · doi:10.1137/S036301290138866X
[6] V. I. Arnold. Mathematical methods in classical mechanics. Springer-Verlag, New York (1980).
[7] F. C. Chittaro and G. Stefani, Singular extremals in multi-input timeoptimal problem: a sufficient condition. Control Cybern. 39 (2010). · Zbl 1287.49020
[8] A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems. Sov. Math. Dokl. 18 (1977). · Zbl 0385.49008
[9] A. V. Dmitruk. Quadratic conditions for a Pontryagin minimum in an optimal control problem, linear in the control, with a constraint on the control. Sov. Math. Dokl. 28 (1983). · Zbl 0549.49010
[10] R. Gabasov and F. Kirillova. High order necessary conditions for optimality. SIAM J. Control Optim. 10 (1972), pp. 127–188. · Zbl 0236.49005 · doi:10.1137/0310012
[11] M. Giaquinta and S. Hildebrandt. Calculus of variations. Springer-Verlag, Berlin (1996). · Zbl 0853.49001
[12] B. S. Goh. The second variation for singular Bolza problems. SIAM J. Control Optim. 4 (1966), pp. 309–325. · Zbl 0146.11906 · doi:10.1137/0304026
[13] M. R. Hestenes. Applications of the theory of quadratic forms in Hilbert space to calculus of variations. Pac. J. Math. 1 (1951), pp. 525–581. · Zbl 0045.20806 · doi:10.2140/pjm.1951.1.525
[14] L. Poggiolini and G. Stefani. State-local optimality of a bang-bang trajectory: a Hamiltonian approach. Systems Control Lett. 53 (2004), pp. 269–279. · Zbl 1157.49305 · doi:10.1016/j.sysconle.2004.05.005
[15] _____ , Sufficient optimality conditions for a bang-singular extremal in the minimum time problem. Control Cybern. 37 (2008), No. 2, 469–490. · Zbl 1235.49054
[16] G. Stefani. Minimum-time optimality of a singular arc: second-order sufficient conditions. In: 43th IEEE Conf. on Decision and Control (2004).
[17] _____ , Strong optimality of singular trajectories. In: Geometric control and nonsmooth analysis (F. Ancona, A. Bressan, P. Cannarsa, F. Clarke, and P. R. Wolenski, eds.). Adv. Math. Appl. Sci. 76, World Scientific, Hackensack, New Jersey (2008), pp. 300–326.
[18] G. Stefani and P. Zezza. Constrained regular LQ-control problems. SIAM J. Control Optim. 35 (1997), No. 3, pp. 876–900. · Zbl 0876.49031 · doi:10.1137/S0363012995286848
[19] M. I. Zelikin and V. F. Borisov. Theory of chattering control. Birkhäuser, Boston–Basel–Berlin (1994). · Zbl 0820.70003
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.