  In this paper, we propose a generalized successive approximation method
(SAM), called invariantly admissible policy iteration (PI), for finding the
solution to a class of input-affine nonlinear optimal control problems by
iterations. Unlike the existing SAM, the proposed method updates the domain of
the next policy and value function for admissibility (and invariance). In the
existing SAM, the admissibility of the generated policies are guaranteed under
the two implicit assumptions regarding Lyapunov's theorem and invariance, both
of which are presented and discussed in this paper and are generally not true.
On the contrary, the proposed invariantly admissible PI guarantees the
admissibility in a more refined manner, without such assumptions. The
admissibility and invariance of the updated region, with respect to the
corresponding policies, are mathematically prove under the specific invariant
admissible update rule. We also provide monotonic decreasing and uniform
convergence properties of the sequence of value functions under certain
conditions. Finally, numerical simulations are presented to illustrate the
proposed PI method and its effectiveness.
