On exact controllability for a class of 2-D Grushin hyperbolic equation

In this paper, we establish exact controllability for a class of two-dimensional Grushin hyperbolic equations, which are degenerate hyperbolic equations. Currently, research on degenerate equations primarily focuses on parabolic equations, with limited research on degenerate hyperbolic equations, mainly confined to one-dimensional cases. The primary method utilized in this paper for the hyperbolic Gurshin equation is through weighted Carleman estimates. Due to the degeneracy, classical Carleman estimates and regular solution spaces are not applicable to equations with degenerate coefficients. Therefore, we first introduce a weighted solution space. To prove controllability, we must first establish the existence of solutions, which is accomplished by adopting the standard Galerkin method. Once the existence of solutions is confirmed, we proceed to prove controllability by performing weighted Carleman estimates. Before calculations, we configure the weight function by adding a cut-off function to a solution of a similar but not dual equation. This enables us to truncate the degenerate part during calculations, turning it into zero, and eliminating the need to consider issues such as the regularity of the degenerate boundary and the meaningfulness of integrals on the degenerate boundary. Since we are integrating only in the non-degenerate region, we can proceed with Carleman estimates for regular non-degenerate hyperbolic equations. After obtaining the Carleman estimates, we use standard methods to prove the observability inequality, leading to the exact controllability.