Abstract.The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.Mathematics Subject Classification. 93E20, 60H10.
International audienceThough it is a trivial fact that the observability of a linear state space system is conserved by output feedback, it requires a rigorous proof to generalize this result to uniform complete observability, which is defined with the observability Gramian. The purpose of this paper is to present such a proof. Some issues in existing results are also discussed. The uniform complete observability of closed loop systems is useful for the analysis of some adaptive systems and of the Kalman filter
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