Abstract
In this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations. We consider the case of box constraints on the control and final integral constraints on the state. In contrast to sufficient conditions assuring quadratic growth in the weak sense, i.e. when the cost increases at least quadratically for admissible controls uniformly near to the nominal one (see e.g. [16, 26]), our main result provides a sufficient condition for quadratic growth of the cost for admissible controls whose associated states are uniformly near to the state of the nominal one. As a consequence of our results, for qualified problems with a strictly convex and quadratic Hamiltonian, we prove that both notions of quadratic growth coincide.