Nonlinear Optimal Control Theory

Hamilton-Jacobi Theory 367

Remark 12.9.2. Assumption 12.9.1(i) implies Assumption 12.2.1-r (iii). As-

sumption 12 .9.1 and Theorems 12.5.3 and 12.6.2 imply tha t the value function

W is the unique viscosity solution of the Hamilton-J acobi equation (1 2.5.1)

with boundary condition V (T, x) = g(x).

Remark 12.9.3. In the definition of viscosity solution the test functions were

taken to be of class C

(1)

, as in the definition given by Crandall and Lions.

An examination o f the proofs in Sections 12.5 and 12.6 shows that we did not

use the continuity of the partial derivatives of the test functions. All we used

was the existence of partial derivatives. Therefore, fo r our purposes we could

have restricted ourselves to test functions that pos sess partial ...