Generalized gradient realization and stability analysis of Herglotz-type Birkhoff’s parametric equations in event space

This paper investigates three types of generalized gradient representations for Herglotz-type Birkhoffian systems in event space and their stability. First, the differential equations and corresponding stability criteria for generalized gradient systems characterized by antisymmetric, symmetric negative definite, and semi-negative definite matrices are presented, respectively. Secondly, the conditions under which the parametric equations of the Herglotz-type Birkhoffian systems in event space can be transformed into the aforementioned three types of generalized gradient systems are derived. Building upon this, the stability of solution for the corresponding Herglotz-type Birkhoffian parametric equations is analyzed by utilizing the properties of generalized gradient systems. Finally, the applicability of the obtained results is verified through specific examples.