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In this paper we advocate for Isaacs' method for the solution of differentialgames to be applied to the solution of optimal control problems. To make theargument, the vehicle employed is Pontryagin's canonical optimal controlexample, which entails a double integrator plant. However, rather thancontrolling the state to the origin, we require the end state to reach aterminal set that contains the origin in its interior. Indeed, in practice, itis required to control to a prescribed tolerance rather than reach a desiredend state; constraining the end state to a terminal manifold of co-dimensionn-1 renders the optimal control problem easier to solve. The global solution ofthe optimal control problem is obtained and the synthesized optimal control lawis in state feedback form. In this respect, two target sets are considered: asmooth circular target and a square target with corners. Closed-loopstate-feedback control laws are synthesized that drive the double integratorplant from an arbitrary initial state to the target set in minimum time. Thisis 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 usingthe necessary conditions for optimality provided by Pontryagin's MaximumPrinciple (PMP). In this paper, the case is made for Isaacs' method for thesolution of differential games to be applied to the solution of optimal controlproblems by way of the juxtaposition of the PMP and DP methods.

The Synthesis of Optimal Control Laws Using Isaacs' Method for the Solution of Differential Games