For control-constrained semilinear parabolic tracking problems without Tikhonov regularization, the value function is Frechet differentiable in the initial value along the optimal trajectory, with gradient equal to the adjoint state.
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Differentiability of the value function in control-constrained parabolic problems
For control-constrained semilinear parabolic tracking problems without Tikhonov regularization, the value function is Frechet differentiable in the initial value along the optimal trajectory, with gradient equal to the adjoint state.