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A minimum energy problem

This well-known problem is reported in Sec.3.11, Example 2, of Bryson, Ho [5]. 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 [5] for details). For l;SPMgt;1/4 the inequality constraint does not become active, for tex2html_wrap_inline2113 the inequality constraint becomes active at a touch point at t=0.5, and for tex2html_wrap_inline2117 the inequality constraint becomes active a along a whole subarc of [0,1].


Figure: Optimal x(t) for an upper bound of tex2html_wrap_inline2123

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.

Oskar von Stryk
Tue Feb 1 13:50:42 CET 2000