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].
