Backward Uniqueness for Coupled Ultraparabolic Operators and an Application to Jerk-Driven Control Models

We prove backward uniqueness for a class of ultraparabolic operators with coupled linear drift. The main difficulty is that the Fourier transform in the degenerate variables turns the coupled drift into a transport operator in the dual frequency variables, so the classical Littlewood--Paley Carleman argument does not apply directly. We overcome this by introducing an invariant frequency variable and establishing a frequency-localized Carleman estimate adapted to the transport structure. The result gives a partial answer to the question of W. Wang and L. Zhang $\left[ \emph {Methods Appl. Anal.}, \ 20 \ (1) \ (2013) \ 79-88 \right]$ for constant coupled drift, with diffusion and lower-order coefficients depending on time and the diffusive variables. As an application, for a jerk-driven control model, we prove backward uniqueness for the equation describing the position, velocity, acceleration, or jerk error: under bounded lower-order coefficients, zero final error in $L^2$ implies zero error at all earlier times.