  Originating in the artificial intelligence literature, optimistic planning
(OP) is an algorithm that generates near-optimal control inputs for generic
nonlinear discrete-time systems whose input set is finite. This technique is
therefore relevant for the near-optimal control of nonlinear switched systems,
for which the switching signal is the control. However, OP exhibits several
limitations, which prevent its application in a standard control context.
First, it requires the stage cost to take values in [0,1], an unnatural
prerequisite as it excludes, for instance, quadratic stage costs. Second, it
requires the cost function to be discounted. Third, it applies for reward
maximization, and not cost minimization. In this paper, we modify OP to
overcome these limitations, and we call the new algorithm OPmin. We then make
stabilizability and detectability assumptions, under which we derive
near-optimality guarantees for OPmin and we show that the obtained bound has
major advantages compared to the bound originally given by OP. In addition, we
prove that a system whose inputs are generated by OPmin in a receding-horizon
fashion exhibits stability properties. As a result, OPmin provides a new tool
for the near-optimal, stable control of nonlinear switched discrete-time
systems for generic cost functions.
