The classical example of second-order state constraint is the one of the Euler-Bernoulli beam, see Bryson et al. [1]

The exact solution, for various values of *a*, is displayed in figure 1.

Figure 1: Shape of a beam: the three cases and the locus of junction points

The qualitative behavior is as follows:

If , the constraint is not active and the solution is .

If , there is a touch point at .

If , there is a boundary arc without strict complementarity: the measure has its support at end points. The locus of switching points is piecewise affine.

Our numerical results are in accordance with the theory: we display in figure 2 the displacement and control when , i.e., when a boundary arc occurs.

*Numerical simulations: problem clamped_beam*

*Discretization: Gauss II with 100 steps.*

Figure 2: Clamped beam . Boundary and control.

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**References**

[1] A.E Bryson, W.F. Denham, and S.E Dreyfus. Optimal programming problems with inequality constraints I: necessary conditions for extremal solutions. AIAA Journal, 1:2544-2550, 1963.