  The aim of this paper is to address optimality of stochastic control
strategies via dynamic programming subject to total variation distance
ambiguity on the conditional distribution of the controlled process. We
formulate the stochastic control problem using minimax theory, in which the
control minimizes the pay-off while the conditional distribution, from the
total variation distance set, maximizes it.
  First, we investigate the maximization of a linear functional on the space of
probability measures on abstract spaces, among those probability measures which
are within a total variation distance from a nominal probability measure, and
then we give the maximizing probability measure in closed form. Second, we
utilize the solution of the maximization to solve minimax stochastic control
with deterministic control strategies, under a Markovian and a non-Markovian
assumption, on the conditional distributions of the controlled process. The
results of this part include: 1) Minimax optimization subject to total
variation distance ambiguity constraint; 2) new dynamic programming recursions,
which involve the oscillator seminorm of the value function, in addition to the
standard terms; 3) new infinite horizon discounted dynamic programming
equation, the associated contractive property, and a new policy iteration
algorithm. Finally, we provide illustrative examples for both the finite and
infinite horizon cases. For the infinite horizon case we invoke the new policy
iteration algorithm to compute the optimal strategies.
