  We investigate conditions of optimality for an infinite horizon control
problem and consider their correspondence with the value function. Assuming
Lipschitz continuity of the value function, we prove that sensitivity relations
plus the normal form version of the Pontryagin Maximum Principle is a necessary
and sufficient condition for the optimality criteria that correspond to this
value function. Different criteria of optimality under different asymptotic
constraints may be used, including almost strong and classical optimality
proposed by D.Bogusz. Special attention is devoted to weakly agreeable
criteria. We also obtain the conditions on control system (like
controllability) that guarantee the Lipschitz continuity of the value function,
without any other asymptotic conditions besides finiteness of the value
function. Some examples are discussed. In particular, it was shown that the
same control, regarded as agreeable optimal and overtaking optimal control, can
correspond to different (everywhere) value functions.
