International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 245-254
doi:10.1155/S0161171293000286

Nonsmooth analysis approach to Isaac's equation

Abstract

We study Isaacs' equation (∗)wt(t,x)+H(t,x,wx(t,x))=0 (H is a highly nonlinear function) whose “natural” solution is a value W(t,x) of a suitable differential game. It has been felt that even though Wx(t,x) may be a discontinuous function or it may not exist everywhere, W(t,x) is a solution of (∗) in some generalized sense. Several attempts have been made to overcome this difficulty, including viscosity solution approaches, where the continuity of a prospective solution or even slightly less than that is required rather than the existence of the gradient Wx(t,x). Using ideas from a very recent paper of Subbotin, we offer here an approach which, requiring literally no regularity assumptions from prospective solutions of (∗), provides existence results. To prove the uniqueness of solutions to (∗), we make some lower- and upper-semicontinuity assumptions on a terminal set Γ. We conclude with providing a close relationship of the results presented on Isaacs' equation with a differential games theory.