Approximate finite-horizon optimal control without PDE´s

The problem of controlling the state of a system, from a given initial condition, during a fixed time interval minimizing at the same time a criterion of optimality is commonly referred to as finite-horizon optimal control problem. It is well-known that one of the standard solutions to the finite-horizon optimal control problem relies upon the solution of the Hamilton-Jacobi-Bellman (HJB) partial differential equation, which may be difficult or impossible to obtain in closed-form. Herein we propose a methodology to avoid the explicit solution of such HJB pde for input-affine nonlinear systems by means of a dynamic extension. This results in a dynamic time-varying state feedback yielding an approximate solution to the finite-horizon optimal control problem.