Evolution Semigroups and their Applications
by
Yuri Latushkin
University of Missouri-Columbia

Evolution semigroups appear in many areas from transfer operators to magnethohydrodynamics. They are used to characterize asymptotic properties (stability, hyperbolicity, exponential dichotomy) of linear differential equations on Banach spaces and infinite dimensional dynamical systems in terms of spectral properties of the semigroup and its generator. We use methods from the theory of strongly continuous semigroups of linear operators, the theory of nonautonomous abstract Cauchy problems on Banach spaces, and the theory of C^*- and Banach algebras.

In the simplest situation the evolution semigroup {E^t}_{t >= 0} is defined on a space of functions from R to a Banach space X by the formula (E^tf)(\tau)=e^tAf(\tau-t), where {e^tA}_{t >= 0} is a given strongly continuous semigroup on X. In this case, the growth bound of the semigroup {e^tA}_{t >= 0} is equal to the spectral bound for the generator \Gamma of the semigroup {E^t}_{t >= 0}. Thus, the generator \Gamma gives a correct version of the Lyapunov Stability Theorem for an unbounded operator A: the spectrum of \Gamma lies in the open left half of the complex plane if and only if the corresponding differential equation [x\dot]=Ax is uniformly exponentially stable.

The intent of this talk is to show how the theory of evolution semigroups can be used to provide a clarifying perspective, and prove new results, on the uniform exponential stability for general linear control systems, [x\dot](t)=A(t)x(t)+B(t)u(t),  y(t)=C(t)x(t),  t >= 0. The operators A(t) are generally unbounded operators on a Banach space X, while the operators B(t) and C(t) may act on Banach spaces, U and Y, respectively. In addressing the general setting, difficulties arise both from the time-varying aspect and from a loss of Hilbert-space properties. This presentation, however, provides some relatively simple operator-theoretic arguments for properties that extend classical theorems of autonomous systems in finite dimensions. The topics covered here include characterizing internal stability of the nominal system in terms of appropriate input-state-output operators and, subsequently, using these properties to obtain new explicit formulas for bounds on the stability radius in terms of L^p- Fourier multipliers.

Date received: April 27, 1999