In studies of perturbation dynamics in physical systems, certain specification of the gov-
erning perturbation dynamical system is generally lacking, either because the perturba-
tion system is imperfectly known or because its specification is intrinsically uncertain, 
while a statistical characterization of the perturbation dynamical system is often avail-
able. In this report exact and asymptotically valid equations are derived for the ensemble 
mean and moment dynamics of uncertain systems. These results are used to extend the 
concept of optimal deterministic perturbation of certain systems to uncertain systems. 
Remarkably, the optimal perturbation problem has a simple solution: In uncertain sys-
tems there is a sure initial condition producing the greatest expected second moment 
perturbation growth.

Perturbation growth in uncertain systems associated with fluid flow is examined concentrating on deriving, solving, and interpreting equations governing the ensemble mean covariance. Covariance evolution equations are obtained for fluctuating operators and illustrative physical examples are solved. Stability boundaries are obtained constructively in terms of the amplitude and structure of operator fluctuation required for existence of bounded second-moment statistics in an uncertain system. The forced stable uncertain system is identified as a primary physical realization of second-moment dynamics by using an ergodic assumption to make the physical connection between ensemble statistics of stable stochastically excited systems and observations of time mean quantities. Optimal excitation analysis plays a central role in generalized stability theory and concepts of optimal deterministic and stochastic excitation of certain systems are extended in this work to uncertain systems. Re- markably, the optimal excitation problem has a simple solution in uncertain systems: there is a pure structure producing the greatest expected ensemble perturbation growth when this structure is used as an initial condition, and a pure structure that is most effective in exciting variance when this structure is used to stochastically force the system distributed in time.

Optimal excitation analysis leads to an interpretation of the EOF structure of the covariance both for the case of optimal initial excitation and for the optimal stochastic excitation distributed in time that maintains the statistically steady state. Concepts of pure and mixed states are introduced for interpreting covariances and these ideas are used to illustrate fundamental limitations on inverting covariances for structure in stochastic systems in the event that only the covariance is known.

Perturbation growth in uncertain systems is examined and related to previous work in which linear stability 
concepts were generalized from a perspective based on the nonnormality of the underlying linear operator. In 
this previous work the linear operator, subject to an initial perturbation or a stochastic forcing distributed in 
time, was either fixed or time varying, but in either case the operator was certain. However, in forecast and 
climate studies, complete knowledge of the dynamical system being perturbed is generally lacking; nevertheless, 
it is often the case that statistical properties characterizing the variability of the dynamical system are known. 
In the present work generalized stability theory is extended to such uncertain systems. The limits in which 
fluctuations about the mean of the operator are correlated over time intervals, short and long, compared to the 
timescale of the mean operator are examined and compared with the physically important transitional case of 
operator fluctuation on timescales comparable to the timescales of the mean operator. Exact and asymptotically 
valid equations for transient ensemble mean and moment growth in uncertain systems are derived and solved. 
In addition, exact and asymptotically valid equations for the ensemble mean response of a stable uncertain 
system to deterministic forcing are derived and solved. The ensemble mean response of the forced stable uncertain 
system obtained from this analysis is interpreted under the ergodic assumption as equal to the time mean of the 
state of the uncertain system as recorded by an averaging instrument. Optimal perturbations are obtained for 
the ensemble mean of an uncertain system in the case of harmonic forcing. Finally, it is shown that the remarkable 
systematic increase in asymptotic growth rate with moment in uncertain systems occurs only in the context of 
the ensemble.