Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

1.3    Notations and Preliminaries

In this section, we introduce notations and some important definitions that will be used frequently in the book.

1.3.1    Notation

The notation will be standard most of the times except where stated otherwise. Moreover, N will denote the set of natural numbers, while Z will denote the set of integers. Similarly,ℜ, ℜⁿ will denote respectively, the real line and the n-dimensional real vector space, t ∈ ℜ will denote the time parameter.

$\mathcal{X}$, M, N,… will denote differentiable-manifolds of dimension n which are locally Euclidean and $TM={\cup }_{x\in M}{T}_{x}M,T^{\star }M={\cup }_{x\in M}{T}_x^{\star }M$ will denote respectively the tangent and cotangent bundles of M with dimensions 2n. Moreover, π and π^⋆ will denote the natural projections T M → M and T