Stability Theory for Strong and Mild Solutions

Abstract

If Open image in new window is a finite-dimensional Hilbert space, then Lyapunov proved that the mild solution {u x (t)} of the Cauchy problem \(\dot{u} (t) = Au(t)\), with the initial condition Open image in new window satisfies the following condition (a) Open image in new window , with \(r,\;c_{0}>0\), for t>0, if and only if, there exists a positive definite matrix R satisfying two conditions: (i) Open image in new window , Open image in new window , \(c_{1}\; c_{2}>0\), and (ii) A ∗ R+RA=−I. The infinite dimensional analogue of this theorem is given by Datko. However, in this case the operator R does not satisfy the lower bound in condition (i). Lyapunov uses this lower bound crucially in going from linear to non-linear case. The solution in non-linear case will satisfy condition (a), termed exponential stability. When the operator A generates a pseudo-contraction semigroup, we produce an operator R satisfying both conditions (i) and (ii) in a deterministic equation. At this stage, we discuss stochastic equations and demonstrate how to study exponential stability for mild solutions of non-linear equations. We cannot use the Ito formula for mild solutions of stochastic differential equations, thus we have to approximate them using Yosida approximation, which produces a sequence of strong solutions. We use this approximation to obtain a sufficient condition for exponential stability in the mean square sense (m.s.s.) when the Lyapunov function Ψ exists. However, in order to use this approximation, we need to assume that Ψ is locally bounded. In the linear case we produce such a Lyapunov function obtaining necessary and sufficient conditions for the exponential stability in the m.s.s. of the solution. Then the first order approximation is used to study the exponential stability in the m.s.s. of mild solutions to non-linear equations following the ideas of Lyapunov. We provide the stochastic analogue of Datko’s theorem in the Appendix.