# Local Exact Controllability to the Trajectories of the Korteweg-de Vries-Burgers Equation on a Bounded Domain with Mixed Boundary Conditions

## TL;DR

This paper establishes local exact controllability to trajectories for the nonlinear Korteweg-de Vries-Burgers equation on a bounded domain with mixed boundary conditions, using Carleman estimates and duality methods.

## Contribution

It introduces a novel Carleman estimate for the linearized system and applies an inversion theorem to achieve controllability for the nonlinear equation.

## Key findings

- Proved controllability for the linearized KdVB system.
- Extended results to the nonlinear KdVB equation.
- Developed new Carleman estimates for mixed boundary conditions.

## Abstract

This paper studies the internal control of the Korteweg-de Vries-Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear.