Application of Pontryagin's Principle to Bimodal Optimization of Nano Rods

By using the Pontryagin's maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of characteristic length. Several concrete examples are treated in detail and the increase in buckling capacity is determined.