Linear time-variant systems: Lyapunov functions and invariant sets defined by Hِlder norms

For linear time-variant systems x ˙ ( t ) = A ( t ) x ( t ) , we consider Lyapunov function candidates of the form V p ( x , t ) = | | H ( t ) x | | p , with 1 ≤ p ≤ ∞ , defined by continuously differentiable and non-singular matrix-valued functions, H ( t ) : R + → R n × n . 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 connecting, by necessary and sufficient conditions, the properties of (i) the matrix-valued function H(t), (ii) the Lyapunov function candidate Vp(x,t) and (iii) the time-dependent set X p ( t ) = { x ∈ R n | | | H ( t ) x | | p ≤ e − rt } , with r≥0. This theory allows the construction of four distinct types of Lyapunov functions and, equivalently, four distinct types of sets which are invariant with respect to the system trajectories. Subsequently, we also get criteria for testing stability, uniform stability, asymptotic stability and exponential stability. For all types of Lyapunov functions, the matrix-valued function H(t) is a solution to a matrix differential inequality (or, equivalently, matrix differential equation) expressed in terms of matrix measures corresponding to Hölder p-norms. Such an inequality (or equation) generalizes the role played by the Lyapunov inequality (equation) in the classical case when p=2. Finally, we discuss the diagonal-type Lyapunov functions that are easier to handle (including the generalized Lyapunov inequality) because of the diagonal form of H(t).