On necessary and sufficient conditions for near-optimal singular stochastic controls

In this paper we discuss the necessary and sufficient conditions for near-optimal singular stochastic controls for the systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland’s variational principle and some delicate estimates of the state and adjoint processes. It is well known that optimal singular controls may fail to exist even in simple cases. This justifies the use of near-optimal singular controls, which exist under minimal conditions and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to choose suitable ones, that are convenient for implementation. This result is a generalization of Zhou’s stochastic maximum principle for near-optimality to singular control problem.

[1]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[2]  Xun Yu Zhou,et al.  Deterministic Near-Optimal Controls. Part II: Dynamic Programming and Viscosity Solution Approach , 1996, Math. Oper. Res..

[3]  Xun Yu Zhou,et al.  Deterministic near-optimal control, part 1: Necessary and sufficient conditions for near-optimality , 1995 .

[4]  U. Haussmann,et al.  Singular Optimal Stochastic Controls II: Dynamic Programming , 1995 .

[5]  I. Ekeland.  On the variational principle , 1974 .

[6]  L. Bittner.  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishechenko, The Mathematical Theory of Optimal Processes. VIII + 360 S. New York/London 1962. John Wiley & Sons. Preis 90/– , 1963 .

[7]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[8]  Brahim Mezerdi,et al.  Necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs , 2009, Syst. Control. Lett..

[9]  B. Mezerdi,et al.  A General Stochastic Maximum Principle for Singular Control Problems , 2005 .

[10]  Farid Chighoub,et al.  Near optimality conditions in stochastic control of jump diffusion processes , 2011, Syst. Control. Lett..

[11]  Nonlinear kinetic theory of vehicular traffic , 1974 .

[12]  K. Teo,et al.  Near-Optimal Controls of a Class of Volterra Integral Systems , 1999 .

[13]  Xun Yu Zhou.  Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality , 1998 .

[14]  L. Alvarez.  A class of solvable singular stochastic control problems , 1999 .

[15]  A. Chala,et al.  The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients , 2005 .

[16]  Boris M. Miller,et al.  Singular Stochastic Control Problems , 2004, SIAM J. Control. Optim..

[17]  E. Blum,et al.  The Mathematical Theory of Optimal Processes. , 1963 .

[18]  Luis H. R. Alvarez,et al.  Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems , 2000, SIAM J. Control. Optim..

[19]  U. Haussmann,et al.  Singular Optimal Stochastic Controls I: Existence , 1995 .

[20]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[21]  S. Peng.  A general stochastic maximum principle for optimal control problems , 1990 .

[22]  Boris M. Miller,et al.  Maximum Principle for Singular Stochastic Control Problems , 2006, SIAM J. Control. Optim..

[23]  I. Karatzas,et al.  Finite-Fuel Singular Control With Discretionary Stopping , 2000 .

[24]  Yuh-tai Ju.  SINGULAR OPTIMAL CONTROL , 1980 .

[25]  U. Haussmann,et al.  The stochastic maximum principle for a singular control problem , 1994 .

[26]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .