Stability in the Sense of Bounded Average Power
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An R m -valued sequence (x k ): = (x k : k = 1, 2, ...), e.g. generated recursively by x k = f k (x k - k , U k ), is called 'average pth power bounded' if (1/K) is bounded uniformly in K = 1, 2,.... (The case p = 2 may correspond to 'power' in the physical sense.) This is a notion of stability. Given estimates of the form: f k (x, u) < a x + k conditions are obtained on the coefficient sequence (a k ) and the input estimates e k := k (u k ) which ensure this form of stability for the output (x k ). In particular, a condition (utilized in an application to adaptive control) is obtained which imposes (i) a bound b on (a k ) and a 'sparsity measure' m (K) on #{kK: a k >} as K ( >1) (ii) average pth power boundedness on (e k ), and (iii) a growth condition on (e k ) related to b and m (). This condition is sharp. 1985 Oxford University Press.