A verification theorem for nonlinear systems with nonlinear ℒ2-gain

The conventional (linear) notion of finite ℒ
₂
-gain has been extensively studied and applied in the analysis and control of both linear and nonlinear dynamical systems over the last few decades. The connection between this finite gain property and dissipation is well-known, and is fundamental to providing a verification mechanism for the property via solution of a related Hamilton-Jacobi-Bellman equation. Motivated by an interest in broadening the applicability of finite ℒ
₂
-gain in the analysis and control of nonlinear systems, this paper presents a generalized verification mechanism that permits nonlinear gain functions to be incorporated in the notion of finite ℒ
₂
-gain.