Uniform Asymptotic Stability of Evolutionary Processes in a Banach Space

This paper contains two major results. The first one is to obtain necessary and sufficient conditions for the uniform asymptotic stability of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. This is the substance of Theorem 1 and its corollaries. The second is to extend the bounded-input, bounded-output criteria of O. Perron and R. Bellman to evolutionary processes in a Banach space. This is done in Theorems 6 through 8. In the special case of Hilbert spaces it is shown that uniform asymptotic stability of the evolutionary processes considered is equivalent to the existence of positive Hermitian bilinear functionals whose derivatives along trajectories give rise to negative bilinear Hermitian functionals. This is the analogue of the usual Lyapunov theory for linear differential equations defined in a finite Euclidean space.