Inverse problems with integral conditions for the generalized Korteweg-de Vries equation

TL;DR

This paper establishes well-posedness results for inverse problems involving the generalized Korteweg-de Vries equation with integral conditions, considering controls on the right-hand side and boundary data without growth restrictions.

Contribution

It provides the first well-posedness results for these inverse problems without restrictions on nonlinearity growth, using regular solutions and smallness assumptions.

Findings

Proved well-posedness for inverse problems with integral conditions.

Handled controls on the right-hand side and boundary data.

Solutions are regular in time under smallness assumptions.

Abstract

Results on well-posedness of three inverse problems with integral conditions on a bounded interval for the generalized Korteweg-de Vries equation without any restrictions on the growth rate of nonlinearity are established. Either the right-hand side of equation or the boundary data, or both are chosen as controls. The considered solutions are regular with respect to the time variable. Assumptions on smallness of the input data or smallness of a time interval are required.