Discounted Near-Optimal Control of Affine Systems via a Progressive Cost Evolution Formulation

In this essay, we develop a combined value iteration (CVI) framework to address discounted optimal control problems for discrete-time affine nonlinear systems. First, generated by novel value iteration (NVI), the admissibility is investigated for the iterative control policy. Note that the relaxation factor leads to the adjustable convergence speed. Second, the constraint condition for the discount factor is established to guarantee the admissibility of the iterative control policy derived from stabilizing value iteration (SVI). In addition, the monotonicity is discussed for the iterative cost function sequence. More importantly, CVI is constructed based on NVI and SVI. Third, produced by CVI, the system stability under the evolving control sequence is ensured by the introduction of the attraction region. In the end, a numerical example is involved to confirm the related theoretical results.