Efficient Algorithms for Solving Hamilton-Jacobi-Bellman Equations

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. More precisely, optimal controls problems involve a dynamic system with input quantities, called controls, and some quantity, called cost , to be minimized. An optimal control is a set of differential equations describing the paths of the control variables that optimise the cost. Finding solutions to problems of this nature involves a significantly high degree of difficulty in terms of cost and power compared with the related task of solving optimal open-loop control problems. Moreover, stability is a major problem in the feedback control problem, which may tend to overcorrect errors that can cause oscillations of constant or changing amplitude. A feedback control problem essentially depends on both state and time variables, and so its determination by numerical schemes has one serious drawback, it is the so called curse of dimensionality. Therefore, efficient numerical methods are needed for the accurate determination of optimal feedback controls.