Acta mathematica scientia, Series B >
A MAXIMUM PRINCIPLE APPROACH TO STOCHASTIC H2/H∞ CONTROL WITH RANDOM JUMPS
Received date: 2013-01-05
Revised date: 2014-05-27
Online published: 2015-03-20
Supported by
The first author is supported by the Doctoral foundation of University of Jinan (XBS1213) and the National Natural Science Foundation of China (11101242).
A necessary maximum principle is given for nonzero-sum stochastic differential games with random jumps. The result is applied to solve the H2/H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2/H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps.
Qixia ZHANG , Qiliang SUN . A MAXIMUM PRINCIPLE APPROACH TO STOCHASTIC H2/H∞ CONTROL WITH RANDOM JUMPS[J]. Acta mathematica scientia, Series B, 2015 , 35(2) : 348 -358 . DOI: 10.1016/S0252-9602(15)60006-6
[1] Hinrichsen D, Pritchard A J. Stochastic H∞. SIAM Journal on Control and Optimization, 1998, 36: 1504- 1538
[2] Zames G. Feedback and optimal sensitivity: Model reference transformation, multiplicative seminorms and approximative inverses. IEEE Transactions on Automatic Control, 1981, 26(2): 301-320
[3] Limbeer D J, Anderson B D, Hendel B. A Nash Game Approach to Mixed H2/H∞ Control. IEEE Trans- actions on Automatic Control, 1994, 39(1): 69-82
[4] Zhang W H, Zhang H S, Chen B X. Stochastic H2/H∞ control with (x, u, v)-dependent noise: Finite horizon case. Automatica, 2006, 42(11): 1891-1898
[5] Rishel R. A minimumprinciple for controlled jump processes. Lecture Notes in Economics and Mathematical Systems, 1975, 107: 493-508
[6] An T T T, Øksendal B. A Maximum Principle for Stochastic Differential Games with Partial Information. Journal of Optimization Theory and Applications, 2008, 139(3): 463-483
[7] Boel R K, Varaiya D. Optimal control of jump processes. SIAM Journal on Control and Optimization, 1977, 15: 92-119
[8] Tang S J, Li X J. Necessary condition for optimal control of stochastic systems with random jumps. SIAM Journal on Control and Optimization, 1994, 32: 1447-1475
[9] Wu Z, Yu Z Y. Linear quadratic nonzero-sum differential games with random jumps. Applied Mathematics and Mechanics, English Edition, 2005, 26(8): 1034-1039
[10] Peng S G. Backward stochastic differential equations and applications to optimal control. Applied Mathe- matics and Optimization, 1993, 27: 125-144
[11] Shi J T, Wu Z. A Risk-sensitive Stochastic Maximum Principle for Optimal Control of Jump Diffusions and its Applications. Acta Mathematica Scientia, 2011, 31B(2): 419-433
[12] Wang G C, Yu Z Y. A partial information non-zero sum differential game of backward stochastic differential equations with applications. Automatica, 2012, 48: 342-352
[13] Yong J M, Zhou X Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: Springer- Verlag, 1999
[14] Zhang D T. Backward Linear-Quadratic Stochastic Optimal Control and Nonzero-sum Differential Game Problem with Random Jumps. Journal of Systems Science and Complexity, 2011, 24(4): 647-662
[15] Peng S G, Wu Z. Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control. SIAM Journal on Control and Optimization, 1999, 37(3): 825-843
[16] Wu Z. Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration. Journal of the Australian Mathematical Society, 2003, 74: 249-266
/
| 〈 |
|
〉 |