Lyapunov functions defined by Hölder p-norms for linear time-variant systems

For linear time-variant systems x(t) = A(t)×(t), we consider Lyapunov function candidates of the form V
_p
(x,t) = ∥H(t)x∥
_p
, with 1≤p≤∞, defined by matrix-valued functions, H(t): R
₊
→ R
^n×n
, continuously differentiable and non-singular. We prove that the traditional framework based on quadratic Lyapunov functions represents a particular case (i.e. p=2 ) of a more general scenario operating in similar terms for all Hölder p-norms. We propose a unified theory where a Lyapunov function candidate V
_p
(x,t) is characterized by the properties of a solution H(t) of a matrix differential inequality (or, equivalently, matrix differential equation). For any p, 1≤p≤∞, such a differential inequality (equation) is expressed in terms of the matrix measure corresponding to the Hölder p-norm and generalizes the role played by the classical Lyapunov differential inequality (equation) well-known for p=2. Our approach allows the construction of four distinct types of Lyapunov functions, providing criteria for testing stability, uniform stability, asymptotic stability and exponential stability. Finally we discuss the diagonal-type Lyapunov functions and the associated differential inequalities (equations) that are easier to handle because of the diagonal form of H(t).