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

Abstract

SUMMARYThe 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.

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

“…Necessary optimality conditions have been obtained by Jacobson, Lele and Speyer [42] including the maximization condition of the Hamiltonian. Similar conditions can be found for example in the articles of Lastman [45]), Maurer [49,51], Norris [53], etc. This field of research has been the subject of a comprehensive survey by Hartl, Sethi and Vickson [38] in 1995.…”

JuMP: A Modeling Language for Mathematical Optimization

“…9.1. clnlbeam. The first model, clnlbeam, is a nonlinear beam control problem obtained from Hans Mittelmann's AMPL-NLP benchmark set 6 ; see also [60]. It can be scaled by increasing the discretization of the one-dimensional domain through the parameter n. We test with n ∈ {5000, 50000, 500000}.…”

Continuation and path following

“…Mittelmann and collaborators have made extensive applications of path following and bifurcation methods in the context of minimal surfaces, free boundary problems, obstacle problems and variational inequalities, see, e.g., Hornung and Mittelmann (1991), Maurer and Mittelmann (1991), Mittelmann (1989-1992) and Mittelmann (1990).…”