# Clamped beam

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

$Latex formula$
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 $Latex formula$, the constraint is not active and the solution is $Latex formula$.
If $Latex formula$, there is a touch point at $Latex formula$.
If $Latex formula$, 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 $Latex formula$, i.e., when a boundary arc occurs.

Numerical simulations: problem clamped_beam
Discretization: Gauss II with 100 steps.

Figure 2: Clamped beam $Latex formula$. 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.