On optimal strain paths in linear viscoelasticity

For a viscoelastic material the work $W\left ( e \right )$ needed to produce a given strain ${e_0}$ in a given time $T$ depends on the strain path $e\left ( t \right ), 0 \le t \le T$, connecting the unstrained state with ${e_0}$ . We here ask the question: Of all strain paths of this type, is there one which is optimal, $^{1}$ that is, one which renders $W$ a minimum? In answer to this question we show that: (i) There is no smooth optimal strain path. (ii) There exists a unique optimal path in ${L_2}\left ( {0,T} \right )$; this path is smooth on the open interval $\left ( {0,T} \right )$, but suffers jump discontinuities $^{2}$ at the end points 0 and $T\left ( {i.e.,e\left ( {{0^ + }} \right ) \ne 0, \\ e\left ( {{T^ - }} \right ) \ne {e_0}} \right )$. (iii) For a Maxwell material the optimal path is linear on $\left ( {0,T} \right )$.