Mathematics – Optimization and Control
Scientific paper
2012-02-19
Mathematics
Optimization and Control
arXiv admin note: text overlap with arXiv:0801.4285 by other author
Scientific paper
In this paper, we study the optimal control system driven by stochastic differential equations (SDEs) of mean- field type, in which the control variable has two components, the first being absolutely continuous and the second singular. On the other hand, the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the the cost functional is also of mean field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Using the approach developed by Bahlali [Bahlali, S.: Necessary and sufficient optimality conditions for relaxed and strict control problems. SIAM J. Control Optim. 2(4), 2078--2095 (2008)], we establish a necessary condition of optimality for two models. The first concerns the relaxed controls, which are measure-valued processes in which an optimal solution exists. The second is a particular case of the first and relates to strict control problems. Our stochastic maximum principle uses only the first-order adjoint equation.
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