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.
_________________________________________________________________________
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.