  In this paper we advocate for Isaacs' method for the solution of differential
games to be applied to the solution of optimal control problems. To make the
argument, the vehicle employed is Pontryagin's canonical optimal control
example, which entails a double integrator plant. However, rather than
controlling the state to the origin, we require the end state to reach a
terminal set that contains the origin in its interior. Indeed, in practice, it
is required to control to a prescribed tolerance rather than reach a desired
end state; constraining the end state to a terminal manifold of co-dimension
n-1 renders the optimal control problem easier to solve. The global solution of
the optimal control problem is obtained and the synthesized optimal control law
is in state feedback form. In this respect, two target sets are considered: a
smooth circular target and a square target with corners. Closed-loop
state-feedback control laws are synthesized that drive the double integrator
plant from an arbitrary initial state to the target set in minimum time. This
is accomplished using Isaacs' method for the solution of differential games,
which entails Dynamic Programming (DP), working backward from the Usable Part
(UP) of the target set, as opposed to obtaining the optimal trajectories using
the necessary conditions for optimality provided by Pontryagin's Maximum
Principle (PMP). In this paper, the case is made for Isaacs' method for the
solution of differential games to be applied to the solution of optimal control
problems by way of the juxtaposition of the PMP and DP methods.
