= Displacement stability of a symmetric semidiscrete wave equation
Let $K_h$ be a symmetric nonnegative <matrix> and consider $U''=(\alpha I-K_h)U$ with fixed real $\alpha$. In any <inner product> for which $K_h$ is self-adjoint, <diagonalization of a matrix> gives
$$
\|U(t)\|\leq e^{\sqrt{\max(\alpha,0)}t}
\bigl(\|U(0)\|+t\|U'(0)\|\bigr).
$$
Indeed each modal coefficient is either a cosine and sine divided by its frequency, a linear function at zero frequency, or a hyperbolic cosine and sine divided by its growth rate. The bounds $|\cos(\omega t)|\leq1$, $|\sin(\omega t)/\omega|\leq t$, and $\sinh(\beta t)/\beta\leq t e^{\beta t}$ prove the estimate. It is uniform in the spectrum of $K_h$ and therefore gives finite-time displacement <stability>. Uniform velocity estimates need additional control of the initial spatial energy. All-time displacement bounds require a strictly positive lower bound for $K_h-\alpha I$, a different condition.
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