  This paper considers the infinite horizon optimal control problem for
nonlinear systems. Under the condition of nonlinear controllability of the
system to any terminal set containing the origin and forward invariance of the
terminal set, we establish a regularized solution approach consisting of a
``finite free final time" optimal transfer problem to the terminal set which
renders the set globally asymptotically stable. Further, we show that the
approximations converge to the optimal infinite horizon cost as the size of the
terminal set decreases to zero. We also perform the analysis for the discounted
problem and show that the terminal set is asymptotically stable only for a
subset of the state space and not globally. The theory is empirically evaluated
on various nonholonomic robotic systems to show that the cost of our
approximate problem converges and the transfer time into the terminal set is
dependent on the initial state of the system, necessitating the free final time
formulation.
