  We consider stochastic motion planning in single-source single-destination
robotic relay networks, under a cooperative beamforming framework. Assuming
that the communication medium constitutes a spatiotemporal stochastic field, we
propose a 2-stage stochastic programming formulation of the problem of
specifying the positions of the relays, such that the expected reciprocal of
their total beamforming power is maximized. Stochastic decision making is made
on the basis of random causal CSI. Recognizing the intractability of the
original problem, we propose a lower bound relaxation, resulting to a
nontrivial optimization problem with respect to the relay locations, which is
equivalent to a small set of simple, tractable subproblems. Our formulation
results in spatial controllers with a predictive character; at each time slot,
the new relay positions should be such that the expected power reciprocal at
the next time slot is maximized. Quite interestingly, the optimal control
policy to the relaxed problem is purely selective; under a certain sense, only
the best relay should move.
