  We consider both discrete and continuous control problems constrained by a
fixed budget of some resource, which may be renewed upon entering a preferred
subset of the state space. In the discrete case, we consider both deterministic
and stochastic shortest path problems with full budget resets in all preferred
nodes. In the continuous case, we derive augmented PDEs of optimal control,
which are then solved numerically on the extended state space with a
full/instantaneous budget reset on the preferred subset. We introduce an
iterative algorithm for solving these problems efficiently. The method's
performance is demonstrated on a range of computational examples, including the
optimal path planning with constraints on prolonged visibility by a static
enemy observer.
  In addition, we also develop an algorithm that works on the original state
space to solve a related but simpler problem: finding the subsets of the domain
"reachable-within-the-budget".
  This manuscript is an extended version of the paper accepted for publication
by SIAM J. on Control and Optimization. In the journal version, Section 3 and
the Appendix were omitted due to space limitations.
