Infinite-dimensional linear programming approach to singular stochastic control

SIAM Journal on Control and Optimization

Volumn 35, Issue 2, 1997, Pages 604-625

  Taksar, Michael I ^a  

^a Stony Brook University   (United States)

Author keywords

Controlled diffusion processes; Primary and dual linear programs; Stochastic control; Stochastic differential equations; Variational inequalities

Indexed keywords

DIFFERENTIAL EQUATIONS; DIFFUSION; MAXIMUM PRINCIPLE; NUMERICAL CONTROL SYSTEMS; OPTIMAL CONTROL SYSTEMS; PROBLEM SOLVING; RANDOM PROCESSES; VARIATIONAL TECHNIQUES;

CONTROLLED DIFFUSION PROCESS; COST FUNCTION; INFINITE DIMENSIONAL LINEAR PROGRAMMING APPROACH; STOCHASTIC CONTROL; VARIATIONAL INEQUALITIES;

LINEAR PROGRAMMING;

EID: 0031098307     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S036301299528685X     Document Type: Article

References (34)

1. E.J. ANDERSON AND P. NASH, Linear Programming in Infinite-Dimensional Spaces, John Wiley, Chichester, 1989.
2. C. DELLACHERIE AND P.-A. MEYER, Probabilites et Potentiel. Theorie des Martingales, Hermann, Paris, 1980.
3. N. DUNFORD AND J.T. SCHWARTZ, Linear Operators. Part I: General Theory, John Wiley, New York, 1988.
4. E.B. DYNKIN, Markov Processes I, II, Springer-Verlag, Berlin, 1965.
5. S.N. ETHIER AND T.G. KURTZ, Markov Processes: Characterization and Convergence, John Wiley, New York, 1986.
6. W.H. FLEMING AND R.W. RISHEL, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin, 1975.
7. W.H. FLEMING AND H.M. SONER, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, Berlin, 1992.
8. W.H. FLEMING AND D. VERMES, Convex duality approach to the optimal control of diffusions, SIAM J. Control Optim., 27 (1989), pp. 1136-1155.
9. W.H. FLEMING AND D. VERMES, Generalized Solutions in the Optimal Control of Diffusions, IMA Vol. Math. Appl. 10, W.H. Fleming and P.L. Lions, eds., Springer-Verlag, New York, 1985, pp. 119-127.
10. M.I. FREIDLIN, Functional Integration and Partial Differential Equations, Princeton University Press, Princeton, NJ, 1985.
11. A. FRIEDMAN, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, 1975.
12. A. FRIEDMAN, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
13. J.M. HARRISON AND M.I. TAKSAR, Instantaneous control of Brownian motion, Math. Oper. Res., 8 (1983), pp. 439-453.
14. D. HERNANDEZ-HERNANDEZ, O. HERNANDEZ-LERMA, AND M.I. TAKSAR, The linear programming approach to deterministic control problems, Applicationes Mathematicae, 14 (1996), pp. 17-33.
15. D. HERNANDEZ-HERNANDEZ AND O. HERNANDEZ-LERMA, Discounted cost Markov decision processes on Borel spaces: The linear programming formulation, J. Math. Anal. Appl., 183 (1994), pp. 335-351.
16. O. HERNANDEZ-LERMA AND J.B. LASSERRE, Linear programming and average optimality of Markov control processes on Borel spaces - unbounded cost, SIAM J. Control Optim., 32 (1994), pp. 480-500.
17. R.M. LEWIS AND R.B. VINTER, Relaxation of optimal control problems to equivalent convex programs, J. Math. Anal. Appl., 74 (1980), pp. 475-493.
18. I. KARATZAS, A class of singular stochastic control problems, Adv. in Appl. Probab., 22 (1983), pp. 225-254.
19. I. KARATZAS AND S.E. SHREVE, Connection between optimal stopping and singular stochastic control I, II, SIAM J. Control Optim., 22 (1984), pp. 856-877; 23 (1985), pp. 433-541.
20. I. KARATZAS AND S.E. SHREVE, Connection between optimal stopping and singular stochastic control I, II, SIAM J. Control Optim., 22 (1984), pp. 856-877; 23 (1985), pp. 433-541.
21. J.L. KELLEY, General Topology, Van Nostrand, New York, 1955.
22. O.A. LADYZHENSKAIA AND N.N. URALTSEVA, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
23. P.L. LIONS AND A.S. SZNITMAN, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 3 (1984), pp. 511-537.
24. R.S. LIPTSER AND A.N. SHIRYAYEV, Theory of Martingales, Kluwer Academic Publ., Dordrecht, the Netherlands, 1989.
25. J.L. MENALDI, M. ROBIN, AND M.I. TAKSAR, Singular ergodic control of multidimensional Gaussian process, Math. Control Signals Systems, 5 (1992), pp. 93-114.
26. J.L. MENALDI AND M.I. TAKSAR, Optimal correction problem of a multidimensional stochastic system, Automatica, 25 (1989), pp. 223-232.
27. A.P. ROBERTSON AND W. ROBERTSON, Topological Vector Spaces, Cambridge University Press, Cambridge, UK, 1964.
28. M.I. TAKSAR, Average optimal singular control and a related stopping problem, Math. Oper. Res., 10 (1985), pp. 63-81.
29. M.I. TAKSAR, Singular control in a multidimensional space with control costs proportional to displacement, in Proc. International Conference on Optimization, Singapore, 1987, pp. 314-324.
30. M.I. TAKSAR, Convex solutions to variational inequalities and multidimensional singular control, in The Dynkin Festschrift. Markov Processes and Their Applications, M. Freidlin, ed., Birkhäuser, Boston, Baden, Berlin, 1994, pp. 371-386.
31. D. VERMES, Optimal control of piecewise deterministic Markov processes, Stochastics, 14 (1985), pp. 165-207.
32. R.B. VINTER AND R.M. LEWIS, The equivalence of strong and weak formulations for certain problems in optimal control, SIAM J. Control Optim., 16 (1978), pp. 546-570.
33. L. WEIN, Optimal control of a two-station Brownian network, Math. Oper. Res., 15 (1990), pp. 215-242.
34. L. WEIN, Scheduling networks of queues: Heavy traffic analysis of a multistation network with controllable inputs, Oper. Res., 40 (1992), pp. 312-334.