Sparsity-promoting optimal control for a class of distributed systems

We consider a linear quadratic optimal control problem with an additional penalty on the number of communication links in the distributed controller. We reformulate this combinatorial optimization problem as a sequence of weighted l₁ problems, where the weighted l₁ norm approximates the counting of the communication links. We identify a class of systems for which the weighted l₁ problem can be formulated as a semideflnite program and therefore its solution can be computed efficiently. Application of the developed algorithm to the optimal control of vehicular formations reveals communication topologies that become sparser as the price of inter-vehicular communications is increased.