  A new fundamental solution semigroup for operator differential Riccati
equations is developed. This fundamental solution semigroup is constructed via
an auxiliary finite horizon optimal control problem whose value functional
growth with respect to time horizon is determined by a particular solution of
the operator differential Riccati equation of interest. By exploiting
semiconvexity of this value functional, and the attendant max-plus linearity
and semigroup properties of the associated dynamic programming evolution
operator, a semigroup of max-plus integral operators is constructed in a dual
space defined via the Legendre-Fenchel transform. It is demonstrated that this
semigroup of max-plus integral operators can be used to propagate all solutions
of the operator differential Riccati equation that are initialized from a
specified class of initial conditions. As this semigroup of max-plus integral
operators can be identified with a semigroup of quadratic kernels, an explicit
recipe for the aforementioned solution propagation is also rendered possible.
