On finite-time stabilization of evolution equations: A homogeneous approach

Abstract

Generalized monotone dilation in a Banach space is introduced. Classical theorems on existence and uniqueness of solutions of nonlinear evolution equations are revised. A universal homogeneous feedback control for a finite-time stabilization of linear evolution equation in a Hilbert space is designed using homogeneity concept. The design scheme is demonstrated for distributed finite-time control of heat and wave equations.

Sliding mode control design using canonical homogeneous norm

“…Next, a change of coordinates, which transforms an asymptotically stable generalized homogeneous system to a quadratically stable, one is presented. In both cases, the so‐called canonical homogeneous norm is utilized for the construction of the corresponding coordinate transformation. The combination of these 2 results allows the necessary and sufficient stability condition to be presented in terms of existence of the quadratic‐like Lyapunov function V ( x )= x ⊤ Ξ( x ) P Ξ( x ) x , where P = P ⊤ ≻0 and the nonsingular matrix Ξ is constant along any ray from the origin (ie, Ξ( e s x )=Ξ( x ) for x ≠0, ) and for x ≠0.…”

Quadratic-like stability of nonlinear homogeneous systems

“…In [28] such a homogeneous norm was called canonical since it is induced by the canonical norm · in R n and x d = x = 1 on the unit sphere S. Obviously,…”

“…s ∂s ∂u s=ln u d the formula (3) can be derived using Implicit Function Theorem [30] applied to the equation d(−s)u = 1 (see, also [28]). …”

“…Next, a change of coordinate, which transforms asymptotically stable generalized homogeneous system to a quadratically stable one, is presented. In both cases the so-called canonical homogeneous norm [28] is utilized for construction of the corresponding coordinate transformation. Combination of these two results allows the necessary and sufficient stability condition to be presented in terms of existence of the quadratic-like Lyapunov function V (x) = x Ξ(x)P Ξ(x)x, where P = P > 0 and the nonsingular matrix Ξ is constant along any ray from the origin (i.e.…”

Fast Control Systems: Nonlinear Approach

“…In [34] this homogeneous norm was called canonical since it is induced by the canonical norm · in R n and x d = x = 1 on the unit sphere S. Obviously…”