Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 331 / 3 / iii / d

For $-1/2<\lambda <0$, logarithmic differentiation givesThe unique maximizing time is thereforeWriting $\alpha =-\lambda >0$ and $q=\sqrt{1-4{\alpha }^2}$, the maximum isConsequently, as $\lambda \to 0^{-}$,At $\lambda =0$ exactly there is no finite maximizing time: the Jordan-block shear produces unbounded quadratic growth. The formulas describe how the optimal transient moves to later times and becomes larger as the damping tends to zero.