Positive Definiteness of Generalized Homogeneous Functions

Identification of positive definiteness of functions is crucial in control theory. However for generalized homogeneous functions, there does not exist an effective method to identify the positive definiteness. In this paper, we consider Lipschitz continuous generalized homogeneous functions. For the functions, we propose a new method to identify the positive definiteness of the functions. Moreover, we apply our proposed method to an optimal homogeneous finite-time control problem. We confirm the effectiveness of the proposed method through the example.