Clamped beam

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

$\begin{array}{ll} Min \frac{1}{2} \int_0^1 u(t)^2 dt \\ \ddot{x}(t) = u(t); \;\; x(t) \leq a \\ x(0)= x(1)=0; \quad \dot x(0) = - \dot x(1) = 1. \end{array}$
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 $a \geq 1/4$ , the constraint is not active and the solution is $x(t)=t(1-t)$ .
If $a\in [1/6,1/4]$ , there is a touch point at $t = 1/2$ .
If $a < 1/6$ , 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 $a=0.1$ , i.e., when a boundary arc occurs.

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

Figure 2: Clamped beam $a=0.1$ . 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.

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