Quadratic order conditions for bang-singular extremals

Numerical Algebra, Control and Optimization

Volumn 2, Issue 3, 2012, Pages 511-546

  Soledad Aronna, M ^a,b   Frederic Bonnans J ^b   Dmitruk, Andrei V ^c   Lotito, Pablo A ^d  

^a FCEIA UNR   (Argentina)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^c MOSCOW STATE UNIVERSITY   (Russian Federation)

^d UNIVERSIDAD NACIONAL DEL CENTRO DE LA PROVINCIA DE BUENOS AIRES   (Argentina)

Author keywords

Bang singular solution; Control constraint; Optimal control; Second order condition; Singular arc

Indexed keywords

EID: 84892574173     PISSN: 21553289     EISSN: 21553297     Source Type: Journal    
DOI: 10.3934/naco.2012.2.511     Document Type: Article

References (60)

1. A. A. Agrachev and R. V. Gamkrelidze, Second order optimality principle for a time-optimal problem, Math. USSR, Sbornik, 100, (1976).
2. A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint, " volume 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004.
3. A. A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), 991-1014.
4. V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control, " Nauka, Moscow, 1979, in Russian.
5. M. S. Aronna, J. F. Bonnans and P. Martinon, A shooting algorithm for problems with singular arcs, INRIA Research Rapport Nr. 7763, October 2011, arXiv:1206.0839.
6. B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 929-956 (electronic).
7. U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds, " Springer-Verlag, Berlin, 2004.
8. A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control, " Hemisphere Publishing Corp. Washington, D. C., 1975.
9. Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.
10. Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095.
11. R. Cominetti and J.-Penot, Tangent sets of order one and two to the positive cones of some functional spaces, Applied Mathematics and Optimization, 36 (1997), 291-312.
12. A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, 18 (1977).
13. A. V. Dmitruk, Quadratic conditions for a Pontryagin minimum in an optimal control problems, linear in the control, with a constraint on the control, Dokl. Akad. Nauk SSSR, 28 (1983), 364-368.
14. A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with ineèualities, Math. Notes, 35 (1984), 427-435.
15. A. V. Dmitruk, Quadratic order conditions for a Pontryagin minimum in an optimal control problem linear in the control, Math. USSR Izvestiya, 28 (1987), 275-303.
16. A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306.
17. A. V. Dmitruk and K. K. Shishov, Analysis of a èuadratic functional with a partly singular Legendre condition, Moscow University Comput. Math. and Cybernetics, 34 (2010), 16-25.
18. L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79 (1957), 497-516.
19. A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems with constraints, USSR Comp. Math. and Math. Phys., 5 (1965), 1-80.
20. N. Dunford and J. Schwartz, "Linear Operators, Vol I, " Interscience, New York, 1958.
21. U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control Optim., 41 (2003), 1843-1867 (electronic).
22. U. Felgenhauer, Optimality and sensitivity for semilinear bang-bang type optimal control problems, Int. J. Appl. Math. Comput. Sci., 14 (2004), 447-454.
23. U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state eèuation, Control Cybernet., 34 (2005), 763-785.
24. R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, J. SIAM Control, 10 (1972), 127-168.
25. P. Gajardo, H. Ramírez C. and A. Rapaport, Minimal time seèuential batch reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856.
26. B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, J. SIAM Control, 4 (1966), 716-731.
27. B. S. Goh, The second variation for the singular Bolza problem, J. SIAM Control, 4 (1966), 309-325.
28. M. R. Hestenes, Applications of the theory of èuadratic forms in Hilbert space to the calculus of variations, Pacific J. Math., 1 (1951), 525-581.
29. A. Hoffman, On approximate solutions of systems of linear ineèualities, Journal of Research of the National Bureau of Standards, Section B, Mathematical Sciences, 49 (1952), 263-265.
30. D. H. Jacobson and J. L. Speyer, Necessary and sufficient conditions for optimality for singular control problems: A limit approach, J. Math. Anal. Appl., 34 (1971), 239-266.
31. D. H. Jacobson, M. M. Lele and J. L. Speyer, New necessary conditions of optimality for control problems with state-variable ineèuality constraints, Journal of Mathematical Analysis and Applications, 35 (1971), 255-284.
32. H. J. Kelley, A second variation test for singular extremals, AIAA Journal, 2 (1964), 1380-1382.
33. H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals, in "Topics in Optimization", Academic Press, New York, (1967), 63-101.
34. R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 1439-1444.
35. A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. on Control, 15 (1977), 256-293.
36. S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces, in "Applied Mathematics and Optimization", Springer, New York, (1979), 49-62.
37. U. Ledzewicz and H. Schättler, Multi-input optimal control problems for combined tumor antiangiogenic and radiotherapy treatments, Journal of Optimization Theory and Applications, published as 'online first' [to appear 2012].
38. E. S. Levitin, A. A. Milyutin and N. P. Osmolovskiǐ, Higher order conditions for local minima in problems with constraints, Uspekhi Mat. Nauk, 33 (1978), 85-148, 272.
39. A. A. Markov, Some examples of the solution of a special kind of problem on greatest and least èuantities, Soobshch. Karkovsk. Mat. Obshch., 1 (1887), 250-276, (in Russian).
40. H. Maurer, Numerical solution of singular control problems using multiple shooting technièues, J. of Optimization Theory and Applications, 18 (1976), 235-257.
41. H. Maurer and N. P. Osmolovskii, Second order optimality conditions for bang-bang control problems, Control and Cybernetics, 32 (2003), 555-584.
42. H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control, SIAM J. Control Optim., 42 (2003), 2239-2263.
43. A. A. Milyutin, On èuadratic conditions for an extremum in smooth problems with a finitedimensional range, Methods of the Theory of Extremal Problems in Economics, (1981), 138-177.
44. A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control, " American Mathematical Society, 1998.
45. H. G. Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636.
46. N. P. Osmolovskii, Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations, J. Math. Sci. (N. Y.), 123 (2004), 3987-4122.
47. L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem, SIAM J. Control Optimization, 49 (2011), 140-161.
48. L. Poggiolini and G. Stefani, On second order sufficient conditions for a bang-singular arc, Proceedings of science-SISSA, (2005).
49. L. Poggiolini and G. Stefani, Minimum time optimality of a partially singular arc: second order conditions, In "Lagrangian and Hamiltonian methods for nonlinear control 2006" (Lecture Notes in Control and Inform. Sci.), Springer, Berlin, 366 (2007), 281-291.
50. L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-singular extremal in the minimum time problem, Control Cybernet., 37 (2008), 469-490.
51. R. T. Rockafellar, "Convex Analysis, " Princeton University Press, Princeton, New Jersey, 1970.
52. W. Rudin, "Real and Complex Analysis, " Mc Graw-Hill, New York, 1987.
53. A. V. Sarychev, Firstand second-order sufficient optimality conditions for bang-bang controls, SIAM J. Control Optim., 3 (1997), 565-588.
54. H. Schättler, A local feedback synthesis of time-optimal stabilizing controls in dimension three, Math. Control Signals Systems, 4 (1991), 293-313.
55. H. Schättler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs, Forum Math., 5 (1993), 203-241.
56. P. Souères and J. P. Laumond, Shortest paths synthesis for a car-like robot, IEEE Trans. Automat. Control, 41 (1996), 672-688.
57. H. J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), 1145-1162.
58. H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the C1 nonsingular case, SIAM J. Control Optim., 25 (1987), 433-465.
59. H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optim., 25 (1987), 868-904.
60. H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: A worked out example of the use of geometric technièues in nonlinear optimal control, Rutgers Center for Systems and Control Technical Report 91-10, 1991.