Necessary conditions of optimality are obtained via the minimum principle, cf., e. g., [3]. With the adjoint variables

the Hamiltonian function of the unconstrained problem is

To have a uniform treatment of active upper or lower state constraints we introduce the new state constraints

With the abbreviations for the total time derivatives of **S**

we find

The functions are second and the are first order state constraints. Thus the Hamiltonian becomes (cf. Bryson, Denham, Dreyfus [2])

where is a multiplier. The necessary conditions from the minimum principle yield for the state and adjoint variables among others (cf. [20])

The optimal control is determined by the minimum principle

where denotes the control space. If or then in because

and and
**j=1,2,3**, and the right hand sides of the first order
differential equations (13), (14)
do not depend on .

Furthermore,
it can be easily shown (for all three objectives) that
and ,
**i=1,2,** or **3**,
cannot occur at the same time
(cf. [20],
[22]).

In the time optimal motion ()
the controls appear linearly in **H**. Thus **H** is not regular.
If no state constraint is active the **i**-th optimal control
of bang-bang type
is determined by the sign of the switching function

The case of a * singular* control, i. e.,
in a whole subinterval, did not occur
in the point-to-point trajectories considered here,
but might be possible, too.

The minimum time is determined by

As , ,
it follows that along the time optimal trajectory.

If the minimum power consumption ()
or the minimum energy criterion ()
are chosen the unbounded optimal control
is determined by

if and .

If it can be shown
that if (**i=1,2,** or **3**) then
there exists such that
, for ,
and in the same way at
(cf. [20],
[22]).

If a state constraint is active, e. g., or
then is determined from
or , resp., for any objective.

All in all, the necessary conditions can be
stated as a well-defined multi-point boundary value problem
if the optimal * switching structure*
of state and control constraints is known.
For more details we refer the interested reader
to [20].

Fri Apr 5 21:57:02 MET DST 1996