Stabilization by means of periodic output feedback

We consider linear time-invariant continuous-time systems x˙(t)=Ax(t)+bu(t), y(t)=cx(t) with 2-dimensional state x∈ℛ ², scalar input u∈ℛ, and scalar output y∈ℛ. The matrices A,b and c are constant and of appropriate dimension. We discuss the problem of making the above linear system exponentially stable by means of a static time-varying output feedback u(t)=k(t)y(t). Easily verifiable necessary and sufficient conditions for this problem to be solvable are presented. Moreover, the proof of the sufficiency part is constructive; that is, it supplies the required feedback gain k(t). The paper thus solves an open problem posed by R. Brockett (1998) for the particular case of scalar input scalar output second-order systems. We assume throughout the paper that b≠(0 0)^T and c≠(0 0)