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We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of  time‐dependent domains , with  timelike moving boundaries , (b) they apply to linear wave equations in  any spatial dimension  and with  general time‐dependent lower order coefficients , and (c) they allow for  smaller time‐dependent regions of observations  than allowed from existing Carleman estimate methods. As a standard application, we establish exact controllability for general linear waves, again in the setting of time‐dependent domains and regions of control.

On Carleman and observability estimates for wave equations on time‐dependent domains