On the relation between partially observed, stochastic optimal control and deterministic infinite dimensional optimal control

M.H.A. Davis, G. Burstein
[1992] Proceedings of the 31st IEEE Conference on Decision and Control  
We explore in this paper the relation between partially observed stochastic optimal control and deterministic infinite dimensional control.The former is formulated as the control problem for the Zakai stochastic partial differential equation with adapted controls which is reduced to a family of pathwise deterministic infinite dimensional control problems for the robust Zakai random partial differential equation with possibly anticipating controls.This is done using the robustifying gauge
more » ... ifying gauge transformation and by introducing the nonanticipativity of the control processes as an equality constraint via a Lagrange multiplier stochastic process. An explicit formula is obtained for this in terms of the acljoint process of the maximum principle for infinite cliniensioiial optinial control [l] applied to the control problem of the robust Zakai equation. This Lagrange multiplier has an interpretation as a price system for small violations of the constraintin this case small anticipative perturbations of the nonanticipative controls.This is a generalization of our results proved in [4],(5] for the stochastic control problem with complete observations where this was reduced to pathwise deterministic (finite dimensional) optimal c,ontrol using the decomposition formula for the flow of stochastic differential equations. We presmt. these results at the beginning of this paper. Iiixifi.,l.(x.~io(t,w))qO( t,x)(v. -uo(t.,))dtdx 2 0 for all We get the usual vanishing $1 the U -gradient of the Hainiltoiiinn if the optimal control is interior to %.It is also h w i i in [3,p.190] that J J v E 91 a.\. (4.7)
doi:10.1109/cdc.1992.371360 fatcat:ojaehh52sjbp3iohgrtjgbsifu