The periodic KdV with control on space-time measurable sets

TL;DR

This paper proves local exact controllability of the KdV equation on a torus using a new method for observability inequalities on space-time measurable sets, applicable to various dispersive equations.

Contribution

Introduces a novel strategy for establishing observability inequalities on space-time measurable sets, enabling controllability results for dispersive equations on torus.

Findings

Proves local exact controllability of KdV on torus.

Develops a new approach for observability inequalities.

Applicable to a broad class of dispersive equations.

Abstract

In this paper, we establish the local exact controllability of the KdV equation on torus around equilibrium states, where both the spatial control region and the temporal control region are sets of positive measure. The proof is based on a novel strategy for proving observability inequalities on space-time measurable sets. This approach is applicable to a broad class of dispersive equations on torus.