Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations

Transactions of the American Mathematical Society

Volumn 357, Issue 6, 2005, Pages 2187-2203

  Goebel, Rafal ^a,b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b NONE   (United States)

Author keywords

Concave convex Hamiltonian; Convex value function; Dual problem; Infinite horizon; Optimal control; Stationary Hamilton Jacobi equation

Indexed keywords

EID: 20144376235     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-05-03817-1     Document Type: Article

References (26)

1. M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhauser, 1997. MR1484411 (99e:49001)
2. L.M. Benveniste and J.A. Scheinkman. Duality theory for dynamic optimization models of economics: the continuous time case. J. Economic Theory, 27:1-19, 1982. MR0662928 (84j:90022)
3. G. Di Blasio. Optimal control with infinite horizon for distributed parameter systems with constrained controls. SIAM J. Control Opt., 29(4):909-925, 1991. MR1111667 (92c:49014)
4. P. Cannarsa and G. Da Prato. Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations. SIAM J. Control Opt., 27(4):861-875, 1989. MR1001924 (90j:49019)
5. D.A. Carlson, A. Haurie, and A. Leizarowitz. Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Springer-Verlag, 1991.
6. F.H. Clarke. Optimization and Nonsmooth Analysis. Wiley, 1983. MR0709590 (85m:49002)
7. H. Frankowska. Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation. Appl. Math. Optim, 19:291-311, 1989. MR0974188 (90b:49043)
8. G.N. Galbraith. Extended Hamilton-Jacobi characterization of value functions in optimal control. SIAM J. Control Optim., 39(1):281-305, 2001. MR1780920 (2001j:49042)
9. R. Goebel. Convex optimal control problems with smooth Hamiltonians. SIAM J. Control Optim. in press.
10. R. Goebel. Planar generalized Hamiltonian systems with large saddle sets. J. Nonlinear Convex Anal., 3(3):365-380, 2002. MR1947105 (2003k:49041)
11. R. Goebel. Stationary Hamilton-Jacobi equations for convex control problems - uniqueness and duality of solutions. In Optimal Control, Stabilization, and Nonsmooth Analysis, Lecture Notes in Control and Information Sciences, Vol. 301, Springer, Berlin, 2004, pp. 313-322. MR2079692
12. R. Goebel. Stabilizing a linear systems with saturation through optimal control. In Proceedings of the 43rd IEEE Conference on Decision and Control, Bahamas, 2004.
13. R. Goebel and R.T. Rockafellar. Generalized conjugacy in Hamilton-Jacobi theory for fully convex Lagrangians. Journal of Convex Analysis, 9(1):463-473, 2002. MR1970567 (2004b:49063)
14. R. Goebel, A. Teel, T. Hu, and Z. Lin. Dissipativity for dual linear differential inclusions through conjugate storage functions. In Proceedings of the 43rd IEEE Conference on Decision and Control, Bahamas, 2004.
15. V. Iftode. Variational solutions of stationary Hamilton-Jacobi equations. Italian J. Pure Appl. Math., 5:117-123, 1999. MR1736005 (2001d:49046)
16. H. Kwakernaak and R. Sivan. Linear Optimal Control Systems. Wiley-Interscience, 1972. MR0406607 (53:10394)
17. A. Leizarowitz. Existence of overtaking optimal trajectories for problems with convex integrands. Mathematics of Operations Research, 10(3):450-461, 1985. MR0798390 (87f:49002)
18. F. Da Lio. On the Bellman equation for infinite horizon problems with unbounded cost functional. Appl. Math. Optim., 41:171-197, 2000. MR1731417 (20011:49050)
19. R.T. Rockafellar. Conjugate convex functions in optimal control and the calculus of variations. J. Math. Anal. Appl., 32:174-222, 1970. MR0266020 (42:929)
20. R.T. Rockafellar. Convex Analysis. Princeton University Press, 1970. MR0274683 (43:445)
21. R.T. Rockafellar. Existence and duality theorems for convex problems of Bolza. Trans. Amer. Math. Soc., 159:1-40, 1971. MR0282283 (43:7995)
22. R.T. Rockafellar. Saddle points of Hamiltonian systems in convex problems of Lagrange. Journal of Optimization Theorey and Applications, 12(4), 1973. MR0358516 (50:10977)
23. R.T. Rockafellar. Hamiltonian trajectories and duality in the optimal control of linear systems with convex costs. SIAM J. Control Optim., 27(5):1007-1025, 1989. MR1009335 (90i:49017)
24. R.T. Rockafellar and R. J-B Wets. Variational Analysis. Springer, 1998. MR1491362 (98m:49001)
25. R.T. Rockafellar and P.R. Wolenski. Convexity in Hamilton-Jacobi theory, 1: Dynamics and duality. SIAM J. Control Optim., 39(5):1323-1350, 2000. MR1825581 (2002b:49053)
26. E.D. Sontag, H.J. Sussmann, and Y.D. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Automat. Control, 39(12):2411-2424, 1994. MR1337566 (96g:93060)