Continuity of Lyapunov functions and of energy level for a generalized gradient semigroup
The global attractor of a gradient-like semigroup has a Morse decomposition. 
Associated to this Morse decomposition there is a Lyapunov function 
(differentiable along solutions)-defined on the whole phase space- 
which proves relevant information on the structure of the attractor. 
In this paper we prove the continuity of these
Lyapunov functions under perturbation. On the other hand, 
the attractor of a gradient-like semigroup also has an energy 
level decomposition which is again a Morse
decomposition but with a total order between any two components. We claim
that, from a dynamical point of view, this is the optimal decomposition of 
a global attractor; that is, if we start from the finest Morse decomposition, 
the energy level decomposition is the coarsest Morse decomposition that still 
produces a Lyapunov function which gives the same information about 
the structure of the attractor. We also establish sufficient conditions 
which ensure the stability of this kind of decomposition under perturbation. 
In particular, if connections between different isolated invariant sets inside 
the attractor remain under perturbation, we show the
continuity of the energy level Morse decomposition. The class of Morse-Smale
systems illustrates our results.