Inverse final observation problems for Maxwell's equations in the quasi-stationary magnetic approximation and stable sequential Lagrange principles for their solving

A. V. Kalinin   ; M. I. Sumin   ; A. A. Tyukhtina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 2, pp. 187-209

Résumé

An initial-boundary value problem for Maxwell's equations in the quasi-stationary magnetic approximation is investigated. Special gauge conditions are presented that make it possible to state the problem of independently determining the vector magnetic potential. The well-posedness of the problem is proved under general conditions on the coefficients. For quasi-stationary Maxwell equations, final observation problems formulated in terms of the vector magnetic potential are considered. They are treated as convex programming problems in a Hilbert space with an operator equality constraint. Stable sequential Lagrange principles are stated in the form of theorems on the existence of a minimizing approximate solution of the optimization problems under consideration. The possibility of applying algorithms of dual regularization and iterative dual regularization with a stopping rule is justified in the case of a finite observation error.

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