Convergence Analysis of Value Iteration Adaptive Dynamic Programming for Continuous-Time Nonlinear Systems

This article is concerned with the convergence property and error bounds analysis of value iteration (VI) adaptive dynamic programming for continuous-time (CT) nonlinear systems. The size relationship between the total value function and the single integral step cost is described by assuming a contraction assumption. Then, the convergence property of VI is proved while the initial condition is an arbitrary positive semidefinite function. Moreover, the accumulated effects of approximation errors generated in each iteration are taken into consideration while using approximators to implement the algorithm. Based on the contraction assumption, the error bounds condition is proposed, which ensures the approximated iterative results converge to a neighborhood of the optimum, and the relation between the optimal solution and approximated iterative results is also derived. To make the contraction assumption more concrete, an estimation way is proposed to derive a conservative value of the assumption. Finally, three simulation cases are given to validate the theoretical results.