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We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w *-lower semicontinuous seminorm p on the dual X * with p ≥ sup C such that p 2 is everywhere Fréchet differentiable in X * ; and as a consequence, the space X is a weakly compactly generated space if and only if there exists a continuous and w *-l.s.c. Fréchet smooth (not necessarily equivalent) norm on X * .

This paper presents stability analysis for a hybrid system consisting of a coupled partial differential equation and ah ordinaxy differential equation arising from shear force feedback control of flexible robots. Conditions on the uniformly exponential decay of solutions of the entire system have not been obtained by using the conventional method such as the Liapunov function method. These conditions are made clear in this paper by reformulating the problem and by doing spectral analysis. It is shown that if the feedbuck gains are properly chosen then the entire hybrid system is uniformly exponentialty stable, which is very useful in practical applications.