An a priori estimate for discrete approximations in nonlinear optimal control

SIAM Journal on Control and Optimization

Volumn 34, Issue 4, 1996, Pages 1315-1328

  Dontchev, Asen L ^a  

^a Mathematical Reviews   (United States)

Author keywords

Discrete approximations; Error estimates; Optimal control

Indexed keywords

APPROXIMATION THEORY; CONSTRAINT THEORY; ERRORS; INTEGRATION; NONLINEAR CONTROL SYSTEMS; OPTIMIZATION;

DISCRETE APPROXIMATIONS; ERROR ESTIMATES; NONLINEAR OPTIMAL CONTROL PROBLEM; RIEMANN INTEGRABILITY;

OPTIMAL CONTROL SYSTEMS;

EID: 0030194741     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S036301299426948X     Document Type: Article

References (67)

1. W. ALT, On the approximation of infinite optimization problems with an aplication to optimal control problems, Appl. Math. Optim., 12 (1984), pp. 15-27.
2. E. R. AVAKOV AND F. P. VASIL'EV, Difference approximation of a maximin problem of optimal control with phase constraints, Vestnik Moscov. Univ. Ser. XV Vychisl. Mat. Kibernet., 2 (1982), pp. 11-17. (In Russian.)
3. M. BARDI AND M. FALCONE, An approximation scheme for the minimum time function, SIAM J. Control Optim., 28 (1990), pp. 950-965.
4. J. M. BORWEIN AND A. S. LEWIS, Duality relationship for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), pp. 325-338.
5. B. M. BUDAK, E. M. BERKOVICH, AND E. N. SOLOV'EVA, Difference approximations in optimal control problems, SIAM J. Control, 7 (1969), pp. 18-31. (Originally published in Russian in Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1968, no. 2, pp. 41-55.)
6. B. M. BUDAK, E. M. BERKOVICH, AND E. N. SOLOV'EVA, Difference approximations in optimal control problems, SIAM J. Control, 7 (1969), pp. 18-31. (Originally published in Russian in Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1968, no. 2, pp. 41-55.)
7. _, The convergence of difference approximations in optimal control, Zh. Vychisl. Mat. i Mat. Fiz., 9 (1969), pp. 533-547. (In Russian.)
8. B. M. BUDAK AND F. P. VASIL'EV, Some Computational Aspects of Optimal Control Problems, Moscow University Press, Moscow, 1975. (In Russian.)
9. I. CAPUZZO DOLCETIA, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim., 10 (1983), pp. 367-377.
10. I. CAPUZZO DOLCETIA AND M. FALCONE, Discrete dynamic programming and viscosity solutions of the Bellman equation, Ann. Inst. H. Poincaré Anal. Non-Lineaire, 6 (1989), suppl., pp. 161-183.
11. M. G. CRANDALL AND P. L. LIONS, Two approximations of solutions of Hamilton-Jacobi equations, Math. of Comp., 43 (1984), pp. 1-19.
12. J. CULLUM, Discrete approximations to continuous optimal control problems, SIAM J. Control, 7 (1969), pp. 32-49.
13. _, An explicit procedure for discretizing continuous, optimal control problems, J. Optim. Theory Appl., 8 (1971), pp. 15-34.
14. _, Finite-dimensional approximations of state constrained continuous optimal control problems, SIAM J. Control, 10 (1972), pp. 649-670.
15. J. W. DANIEL, On the approximate minimization of functionals, Math. Comp., 23 (1969), pp. 573-581.
16. _, On the convergence of a numerical method in optimal control, J. Optim. Theory Appl., 4 (1969), pp. 330-342.
17. _, The Approximate Minimization of Functionals, Wiley-Interscience, New York 1983.
18. A. L. DONTCHEV, Error estimates for a discrete approximation to constrained control problems, SIAM J. Numer. Anal., 18 (1981), pp. 500-514.
19. _, Perturbations, Approximations and Sensitivity Analysis of Optimal Control Systems, Lecture Notes in Control and Inform. Sci. 52, Springer, New York, 1983.
20. _, Duality methods in constrained best interpolation, Mathematica Balkanica, 1 (1987), pp. 96-105.
21. _, Best interpolation in a strip, J. Approx. Theory, 73 (1993), pp. 334-342.
22. A. L. DONTCHEV AND W. W. HAGER, Lipschitz stability in nonlinear control and optimization, SIAM J. Control Optim., 31 (1993), pp. 569-603.
23. A. L. DONTCHEV, W. W. HAGER, A. B. POORE, AND B. YANG, Optimality, stability and convergence in nonlinear control, Appl. Math. Optim., 31 (1995), pp. 297-326.
24. A. L. DONTCHEV AND W. W. HAGER, Euler approximation to the feasible set, Numer. Funct. Anal. Optim., 15 (1994), pp. 245-261.
25. A. L. DONTCHEV AND F. LEMPIO, Difference methods for differential inclusions - a survey, SIAM Rev., 34 (1992), pp. 263-294.
26. A. L. DONTCHEV AND T. ZOLEZZI, Well-Posed Optimization Problems, Lecture Notes in Math. 1543, Springer, New York, 1993.
27. J. C. DUNN, Diagonally modified conditional gradient method for input constrained optimal control problems, SIAM J. Control Optim., 24 (1986), pp. 1177-1191.
28. J. C. DUNN AND T. TIAN, Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions, SIAM J. Control Optim., 30 (1992), pp. 1361-1348.
29. Y. M. ERMOL'EV, V. P. GULENKO, AND T. I. TSARENKO, Finite element methods in optimal control, Naukova Dumka, Kiev, 1978. (In Russian.)
30. M. FALCONE, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim., 15 (1987), pp. 1-13. Corrigenda: Appl. Math. Optim., 23 (1991), pp. 213-214.
31. M. FALCONE, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim., 15 (1987), pp. 1-13. Corrigenda: Appl. Math. Optim., 23 (1991), pp. 213-214.
32. R. P. FEDORENKO, Approximate Solution of Optimal Control Problems, Nauka, Moscow, 1971. (In Russian.)
33. R. GONZALES AND E. ROFMAN, On deterministic control problems: An approximation procedure for the optimal cost, SIAM J. Control Optim., 23 (1985), Part I: The stationary problem, pp. 242-265, Part II: The nonstationary case, pp. 267-285.
34. R. GONZALES AND E. ROFMAN, On deterministic control problems: An approximation procedure for the optimal cost, SIAM J. Control Optim., 23 (1985), Part I: The stationary problem, pp. 242-265, Part II: The nonstationary case, pp. 267-285.
35. R. GONZALES AND E. ROFMAN, On deterministic control problems: An approximation procedure for the optimal cost, SIAM J. Control Optim., 23 (1985), Part I: The stationary problem, pp. 242-265, Part II: The nonstationary case, pp. 267-285.
36. W. W. HAGER, The Ritz-Trefftz method for state and control constained optimal control problems, SIAM J. Numer. Anal., 12 (1975), pp. 854-867.
37. _, Rate of convergence for discrete approximations to unconstrained control problems, SIAM J. Numer. Anal., 13 (1976), pp. 449-471.
38. _, Convex control and dual approximations, Control Cybernet., 8 (1979), part 1: pp. 5-12, part 2: pp. 321-338.
39. _, Convex control and dual approximations, Control Cybernet., 8 (1979), part 1: pp. 5-12, part 2: pp. 321-338.
40. _, Convex control and dual approximations, Control Cybernet., 8 (1979), part 1: pp. 5-12, part 2: pp. 321-338.
41. _, Multiplier method for nonlinear optimal control, SIAM J. Numer. Anal., 27 (1990), pp. 1061-1080.
42. W. W. HAGER AND G. D. IANCULESCU, Dual approximations in optimal control, SIAM J. Control Optim., 22 (1984), pp. 423-465.
43. C. T. KELLEY AND E. W. SACHS, Mesh independence of the gradient projection method for optimal control problems, SIAM J. Control Optim., 30 (1992), pp. 477-493.
44. A. A. LEVIKOV, Error estimate for solution of the linear optimal control problem, Avtomat. i Telemekh., 3 (1982), pp. 71-78. (In Russian.)
45. K. C. P. MACHIELSEN, Numerical Solution of Optimal Control Problems with State Constraints by Sequential Quadratic Programming in Function Space, Dissertation, Technische Hogeschool Eindhoven, Eindhoven, 1987.
46. K. MALANOWSKI, On convergence of finite-difference approximations to control and state constrained optimal control problems, Archiwum Aut. i Telem., 24 (1979), pp. 319-337.
47. _, On convergence of finite-difference approximations to optimal control problems for systems with control appearing linearly, Archiwum Aut. i Telem., 24 (1979), pp. 155-171.
48. _, Convergence of approximations vs. regularity of solutions for convex control-constrained optimal control problems, Appl. Math. Optim., 8 (1981), pp. 69-95.
49. B. MOBDUKHOVICH, On difference approximations of optimal control systems, Appl. Math. Mech., 42 (1978), pp. 452-461.
50. _, Approximation Method in Problems of Optimization and Control, Nauka, Moscow, 1988. (In Russian.)
51. _, Discrete Approximations and Refined Euler-Lagrange Conditions for Nonconvex Differential Inclusions, IMA preprint series 1115, March 1993.
52. H. J. PESCH, A practical guide to the solution of real-life optimal control problems, Control Cybernet., 23 (1994), pp. 7-60.
53. E. POLAK, Computational Methods in Optimization: A Unified Approach, Academic Press, New York, 1971.
54. _, A historical survey of computations methods in optimal control, SIAM Rev., 15 (1973), pp. 553-548.
55. _, On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems, Math. Programming, 62 (1993), pp. 385-414.
56. E. POLAK AND LIMIN HE, Rate-preserving strategies for semi-infinite programming and optimal control, SIAM J. Control Optim., 30 (1992), pp. 543-572.
57. E. POLAK, T. H. YANG, AND D. Q. MAYNE, A method of centers based on barrier functions for solving optimal control problems with continuous state and control constraints, SIAM J. Control Optim., 31 (1993), pp. 159-179.
58. G. W. REDDIEN, Collocation at Gauss points as a discretization in optimal control, SIAM J. Control Optim., 17 (1979), pp. 298-316.
59. K. L. TEO, C. J. GOH AND K. H. WANG, A Unified Computational Approach to Optimal Control Problems, Longman Sci. Tech., Harlow, 1991.
60. F. P. VASIL'EV, Methods for Solving Extremum Problems, Nauka, Moscow, 1981. (In Russian.)
61. V. V. VASIN, Discrete approximation and stability in extremal problems, Zh. Vychisl. Mat. i Mat. Fiz., 22 (1982), pp. 824-839.
62. V. M. VELIOV, Second order discrete approximations to strongly convex differential inclusions, Systems Control Lett., 13 (1989), pp. 263-269.
63. _, Second order discrete approximations to linear differential inclusions, SIAM J. Numer. Anal., 29 (1992), pp. 439-451.
64. _, Best approximations of control/uncertain differential systems by means of discrete-time systems, IIASA WP-91-45, November 1991.
65. J. WARGA, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), pp. 111-145.
66. P. R. WOLENSKI, The exponential formula for the reachable set of a Lipschitz differential inclusion, SIAM J. Control Optim., 28 (1990), pp. 1148-1161.
67. S. E. WRIGHT, Consistency of primal-dual approximations for convex optimal control problems, SIAM J. Control Optim., 33 (1995), pp. 1489-1509.