This paper presents a unified and extended approach to $ L_2 $-gain analysis of smooth nonlinear systems using Hamilton-Jacobi equations and their relation to invariant manifolds of Hamiltonian vector fields. It also develops a nonlinear analog of the state-space approach to linear $ H_\infty $ control, specifically the state feedback $ H_\infty $ optimal control problem. The paper discusses the relationship between $ H_\infty $ control of nonlinear systems and their linearized counterparts. It shows that if the $ H_\infty $ control problem for the linearized system is solvable, then locally a solution to the nonlinear $ H_\infty $ control problem exists. The paper also explores the connection between the $ L_2 $-gain of a nonlinear system and its linearized version, and presents results on the existence of smooth solutions to Hamilton-Jacobi equations and their implications for control theory. It further discusses the relationship between the $ L_2 $-gain and the stability of the closed-loop system, and provides conditions under which the closed-loop system is asymptotically stable. The paper concludes with a discussion of the nonlinear state feedback $ H_\infty $ optimal control problem and its relation to the $ L_2 $-gain analysis of nonlinear systems.This paper presents a unified and extended approach to $ L_2 $-gain analysis of smooth nonlinear systems using Hamilton-Jacobi equations and their relation to invariant manifolds of Hamiltonian vector fields. It also develops a nonlinear analog of the state-space approach to linear $ H_\infty $ control, specifically the state feedback $ H_\infty $ optimal control problem. The paper discusses the relationship between $ H_\infty $ control of nonlinear systems and their linearized counterparts. It shows that if the $ H_\infty $ control problem for the linearized system is solvable, then locally a solution to the nonlinear $ H_\infty $ control problem exists. The paper also explores the connection between the $ L_2 $-gain of a nonlinear system and its linearized version, and presents results on the existence of smooth solutions to Hamilton-Jacobi equations and their implications for control theory. It further discusses the relationship between the $ L_2 $-gain and the stability of the closed-loop system, and provides conditions under which the closed-loop system is asymptotically stable. The paper concludes with a discussion of the nonlinear state feedback $ H_\infty $ optimal control problem and its relation to the $ L_2 $-gain analysis of nonlinear systems.