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Abstract: "In this paper we study a Hamilton-Jacobi equation related to the boundary control of a parabolic equation with Neumann boundary conditions. The state space of this problem is a Hilbert space and the equation is defined classically only on a dense subset of the state space. Moreover the Hamiltonian appearing in the equation contains fractional powers of an unbounded operator. These facts render the problem difficult. In this paper we give a revised definition of a viscosity solution to accommodate the unboundedness of the Hamiltonian. We then obtain existence and uniqueness results for viscosity solutions. In particular we show that under suitable assumptions the value function of the boundary control problem is the unique viscosity solution of the related Hamilton-Jacobi equation."