  Adaptive optimal control using value iteration (VI) initiated from a
stabilizing policy is theoretically analyzed in various aspects including the
continuity of the result, the stability of the system operated using any
single/constant resulting control policy, the stability of the system operated
using the evolving/time-varying control policy, the convergence of the
algorithm, and the optimality of the limit function. Afterwards, the effect of
presence of approximation errors in the involved function approximation
processes is incorporated and another set of results for boundedness of the
approximate VI as well as stability of the system operated under the results
for both cases of applying a single policy or an evolving policy are derived. A
feature of the presented results is providing estimations of the region of
attraction so that if the initial condition is within the region, the whole
trajectory will remain inside it and hence, the function approximation results
will be reliable.
