Optimal Control Of Two Nonlinear Partial Differential Equations

The optimal distributed control problems for the Novikov equation with strong viscosity and the optimal boundary and distributed control problems for the Fitzhugh-Nagumo equation are concerned in this paper.For the optimal distribution control problem of the Novikov equation with strong viscosity and with periodic boundary condition.We apply Galerkin method to obtain the existence of a weak solution to the corresponding initial boundary value problem.Using the variational theory,we derive existence of an optimal solution.The deduced surjectivity of the linearized operator help us to obtain the first-order necessary optimal condition.Finally,utilizing the augmented Lagrangian functional method,two second-order sufficient optimality conditions are established.For the optimal boundary and distributed control problem of the the Fitzhugh-Nagumo equation with nonlinear boundary condition,we first prove the well-posedness of the corresponding initial value problem by Schauder's fixed-point theorem.Then applying the variational theory,we derive existence of an optimal boundary and distributed control solution.Finally,the deduced differentiability of the control-to-state mapping allows us to obtain the first-order necessary optimal condition.