Abstract
We consider the optimal control of a semilinear parabolic equation with pointwise bound constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson’s constraint qualification, we provide a second order necessary condition over a set of strictly critical directions. The main feature of this result is that the qualification condition needed for the second order analysis is the same as for classical finite-dimensional problems and does not imply the uniqueness of the Lagrange multiplier. We establish also a second order sufficient optimality condition which implies, for problems with a quadratic Hamiltonian, the equivalence between solutions satisfying the quadratic growth property in the L 1 and $L^{\infty }$ topologies.
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