Contents I. Introduction
THE norm has played an important role in the study and analysis of robust optimal control theory since its original formulation in an input-output setting by Zames, [16]. Earlier solution techniques involved operator-theoretic methods. State space solutions were rigorously derived in [4] for the linear system case that required solving several associated Riccati equations. Later, more insight into the problem was given after the linear control problem was posed as a zero-sum two-person differential game by Başar [2]. The nonlinear counterpart of the control theory was developed by Van der Schaft [13]. He utilized the notion of dissipativity introduced by Willems [14], [15] and formulated the control theory into a nonlinear gain optimal control problem. He made use of the fact that the norm in the frequency domain is nothing but the -induced norm from the input time-function to the output-time function for initial zero state. The -gain optimal control problem requires solving a Hamilton-Jacobi equation, namely the Hamilton-Jacobi-Isaacs (HJI) equation. Conditions for the existence of smooth solutions of the Hamilton-Jacobi equation were studied through invariant manifolds of Hamiltonian vector fields and the relation with the Hamiltonian matrices of the corresponding Riccati equation for the linearized problem, [13]. Later some of these conditions were relaxed by Isidori and Astolfi [6], into critical and noncritical cases.
