Approximate finite-horizon optimal control with policy iteration

In this paper, the policy iteration algorithm for the finite-horizon optimal control of continuous time systems is addressed. The finite-horizon optimal control with input constraints is formulated in the Hamilton-Jacobi-Bellman (HJB) equation by using a suitable nonquadratic function. The value function of the HJB equation is obtained by solving a sequence of cost functions satisfying the generalized HJB (GHJB) equations with policy iteration. The convergence of the policy iteration algorithm is proved and the admissibility of each iterative policy is discussed. Using the least squares method with neural networks (NN) approximation of the cost function, the approximate solution of the GHJB equation converges uniformly to that of the HJB equation. A numerical example is given to illustrate the result.