Closed-form solution for a class of discrete-time algebraic Riccati equations

In the present paper we obtain a closed-form solution for the class of discrete-time algebraic Riccati equations (ARE) with vanishing state weighting, whenever the unstable eigenvalues are distinct. The AREs in such a class solve a minimum energy control problem for a single-input single-output (SISO) system. The obtained closed-form solution gives insight on issues such as loss of controllability and it might also prove competitive in terms of numerical precision over current solving algorithms.