High-dimensional Semi-linear Hyperbolic Equations, The Exact Controllability Problem

In this paper, we obtain the global exact controllability for a class of multidi-mensional semilinear hyperbolic equations with a superlinear nonlinearity. For thispurpose, we establish a crucial explicit observability estimate for the linear (multi-dimensional) hyperbolic equation with an unbounded potential. Such an estimateis obtained by a combination of a point-wise estimate and a global Carleman esti-mate for the hyperbolic differential operators, and an analysis on the regularity ofthe optimal solution to an auxiliary optimal control problem.