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Robust feedback switching control: dynamic programming and viscosity solutions
We consider a robust switching control problem. The controller only observes
the evolution of the state process, and thus uses feedback (closed-loop)
switching strategies, a non standard class of switching controls introduced in
this paper. The adverse player (nature) chooses open-loop controls that
represent the so-called Knightian uncertainty, i.e., misspecifications of the
model. The (half) game switcher versus nature is then formulated as a two-step
(robust) optimization problem. We develop the stochastic Perron method in this
framework, and prove that it produces a viscosity sub and supersolution to a
system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which
envelope the value function. Together with a comparison principle, this
characterizes the value function of the game as the unique viscosity solution
to the HJB equation, and shows as a byproduct the dynamic programming principle
for robust feedback switching control problem.Comment: to appear on SIAM Journal on Control and Optimizatio
Controlled diffusion processes
This article gives an overview of the developments in controlled diffusion
processes, emphasizing key results regarding existence of optimal controls and
their characterization via dynamic programming for a variety of cost criteria
and structural assumptions. Stochastic maximum principle and control under
partial observations (equivalently, control of nonlinear filters) are also
discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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