Solution of the boundary regulator problem

Now let W be the operator of the quadratic form infw J (x 0 , tio; <*>)· Then: Theorem 3 I f xq € Xo then

min J(xo] u) = inf inf J(xo — Duo% txoi <*>) = inf ( ti t i0 W tio

%w x0 - Du0 Uo >.

The operator W is the operator of the quadratic form infw J(£oj o^; w) (com­ pare sect. 2). Our solution of the regulator problem is based on this result: we identify the optimal control for xo G Xo by using the theory of the singular regu­ lator problem applied to the problem described by (10), (11). Then, the solution is extended to all xq G X by a continuity argument.