This well-known problem is reported in Sec.3.11, Example 2, of Bryson, Ho . The task is to minimize
subject to the differential equations and boundary conditions
and subject to the state variable inequality constraint
First the objective has to be transformed into Mayer type by introducing a third state variable
Then the objective to be minimized is
Now the dynamic model has the dimensions
NY = DIMY = 3, NV = DIMV = 0, NU = DIMU = 1, NP = NPAR = 0, NRB = NRB = 5, NZB = NZB = 1.
A formula of the solution can be given explicitly (see  for details). For l;SPMgt;1/4 the inequality constraint does not become active, for the inequality constraint becomes active at a touch point at t=0.5, and for the inequality constraint becomes active a along a whole subarc of [0,1].
Figure: Optimal x(t) for an upper bound of
The unconstrained problem (NZB = 0) can be investigated first. Next, the state constrained problem can be investigated by setting NZB = 1 and l properly (for example, l=1/6 or 1/9). The behaviour of the solution may now be studied for a sequence of increased or decreased values of l.