Exact uniform control time for a centered viscous Burgers shock

We determine the minimal time for boundary null control of the Burgers equation linearized at the stationary shock $-\tanh(x/(2\eps))$ on $(-L,L)$, with a control at the left endpoint. For arbitrary $L^2$ initial data, the worst-case $L^2$ control cost remains bounded as the viscosity tends to zero at every time $T>4\sqrt2L$ and diverges at every time $0<T<4\sqrt2L$. Thus the exact uniform control time is $4\sqrt2L$. The upper bound separates the near-zero shock mode from the fast spectral modes and estimates their coefficient norms through the exact spectral determinant. The lower bound uses an inverse exponential Gram matrix: its alternating coefficients combine with the eigenfunctions' endpoint signs to prevent cancellation in a common boundary layer.