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