Second order analysis for the optimal control of parabolic equations under control and final state constraints

Francisco José Silva (1)

1 XLIM-DMI - DMI

We consider the optimal control of a semilinear parabolic equation with pointwise bounds constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson's constraint qualification [37], 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 ∞ topologies.

Mots clés: Optimal control; parabolic equations; box constraints for the control; finitely many con-straints for the state; Robinson constraint qualification; second order optimality conditions; quadratic Hamiltonian.

Domaines: Optimisation et contrôle [math.OC]

Francisco José Silva. Second order analysis for the optimal control of parabolic equations under control and final state constraints. 2014. ⟨hal-01096660⟩