  We consider the uplink transmission between a multi-antenna ground station
and an unmanned aerial vehicle (UAV) swarm. The UAVs are assumed as intelligent
agents, which can explore their optimal three dimensional (3-D) deployment to
maximize the channel capacity of the multiple input multiple output (MIMO)
system. Specifically, considering the limitations of each UAV in accessing the
global information of the network, we focus on a decentralized control strategy
by noting that each UAV in the swarm can only utilize the local information to
achieve the optimal 3-D deployment. In this case, the optimization problem can
be divided into several optimization sub-problems with respect to the rank
function. Due to the non-convex nature of the rank function and the fact that
the optimization sub-problems are coupled, the original problem is NP-hard and,
thus, cannot be solved with standard convex optimization solvers.
Interestingly, we can relax the constraint condition of each sub-problem and
solve the optimization problem by a formulated UAVs channel capacity
maximization game. We analyze such game according to the designed reward
function and the potential function. Then, we discuss the existence of the pure
Nash equilibrium in the game. To achieve the best Nash equilibrium of the MIMO
system, we develop a decentralized learning algorithm, namely decentralized
UAVs channel capacity learning. The details of the algorithm are provided, and
then, the convergence, the effectiveness and the computational complexity are
analyzed, respectively. Moreover, we give some insightful remarks based on the
proofs and the theoretical analysis. Also, extensive simulations illustrate
that the developed learning algorithm can achieve a high MIMO channel capacity
by optimizing the 3-D UAV swarm deployment with the local information.
