Sparse feedback in linear control systems

We consider a classical problem of linear static state feedback design in the linear system $\dot x=Ax+Bu$ subject to a nonstandard constraint that the control vector $u=Kx$ has as many zero components as possible.
A simple approach to approximate solutions of such kind of nonconvex problems is proposed, which is based on convexification. The problem reduces to the minimization of special matrix norms subject to the constraints in the form of linear matrix inequalities (LMIs).
The approach can be generalized to numerous problems of robust and optimal control that admit a “sparse” reformulation. To the best of our knowledge, both the solution and the problem formulation are new.