An approximate solution to the norm optimal control problem

The problem of finding the minimum input norm required to bring a dynamical system from an arbitrary initial state to a given end-point constraint set under specified state constraints, in a finite time interval, is considered. Even when there exists a control input which achieves the constraints, there may not be one with a minimum norm. For input-affine systems, it is shown that there is an admissible bang-bang control input whose norm is arbitrarily close to the minimum-norm. Since bang-bang functions are completely specified by their amplitude and switching times, the optimal norm can be numerically estimated by performing a finite dimensional search.