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In the present paper we obtain a closed-form solution for the class of continuous-time algebraic Riccati equations (ARE) with vanishing state weight, whenever the unstable eigenvalues are distinct. The AREs in such a class solve a minimum energy control problem. The obtained closed-form solution gives insight on issues such as loss of controllability and it might also prove comparable in terms of numerical precision over current solving algorithms. Keywords: Algebraic Riccati equation; Closed-form solution; Controllability; Minimum energy control; Signal-to-noise ratio.