Optimal Control for a Second Order Semilinear Sobolev Type Equation in the Sectorial Case

The paper investigates the optimal control of solutions to an initial-boundary value problem for a second-order semilinear Sobolev-type equation with a relatively sectorial operator. The equation models low-frequency electromagnetic waves in a dispersive transmission line with nonlinear capacitance. Using the Galerkin method, as well as compactness and monotonicity techniques, the existence and uniqueness of a solution on an arbitrary time interval are proved. The phase space of the problem is constructed. As an application of the abstract theory, a mathematical model of a nonlinear evolutionary dispersive line is considered; the Galerkin sums are constructed using the eigenfunctions of the Dirichlet problem for the Laplace operator. Furthermore, for this model, the optimal control problem with a balance-type cost functional is solved.

Keywords

Sobolev type equations; initial-boundary value problem; Galerkin method; *-weak convergence.

References

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