The non‐linear beam via optimal control with bounded state variables

The non‐linear beam with bounded deflection is considered as an optimal control problem with bounded state variables. The theory of necessary optimality conditions leads to boundary value problems with jump conditions which are solved by multiple‐shooting techniques. A branching analysis is performed which exhibits the different solution structures. In particular, the second bifurcation point is determined numerically. The bifurcation diagram reveals a hysteresis‐like behaviour and explains the jumping to a different state at this bifurcation point.