Stability Methods in Non-stationary Systems

Stability Methods in Non-stationary Systems.—A different approach, established in chemical reactor analysis, was proposed by Sherrington and B. F. Gray. They define the conditions under which the local stability (i.e. stability at a particular instant in time) may be guaranteed, and extend them over the entire time interval by the application of the methods of non-linear stability analysis. The 

Gray and M. E. Sherrington, (a) Combustion and Flame, 1972,19,43S (6) ibid, p. 44S. J. S. LaSalle and S. Lefschatz, Stability by Liapunov s Second Method with Applications , Academic Ih-ess, New York, 1961. 

(a) Theory and Application of Liapunov s Direct Method , Prentice-Hall, London, 1963 (h) Stability of Motion , Springer-Verlag, Berlin, 1967. 

Fignre 9 Identification of the sensitivity of the integral curve (0 iKO tvith the stability of the perturbed solution. Curve (a) represents asymptotic decay of the perturbed trajectory. Curves (b) and (c) represent perturbation in the sensitive reghn where the new trajectory may remain close to the null solution (b) or diverge fiam it (c) 

The direct method of Liapimov relies on defining a function V(S,0 whidi is a direct measure of the perturbations. We wish to ensure that i and C remain small. For this we require K to be real-valued and positive definite. If P 0 the perturbations will be decreasing or constant the stronger condition 1 0 ensures that perturbations are always decreasing and that the equilibrium state is asymptotically 

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