A class of differential Riccati equations (DREs) is considered whereby the evolution of any solution can be identified with the propagation of a value function of a corresponding optimal control problem arising in L2-gain analysis. By exploiting the semigroup properties inherited from the attendant dynamic programming principle, a max-plus primal space fundamental solution semigroup of max-plus linear max-plus integral operators is developed that encapsulates all such value function propagations. Using this semigroup, a new one-parameter fundamental solution semigroup of matrices is developed for the aforementioned class of DREs. It is demonstrated that this new semigroup can be used to compute particular solutions of these DREs, and to characterize finite escape times (should they exist) in a relatively simple way compared with that provided by the standard symplectic fundamental solution semigroup.