Necessary conditions of optimality are obtained via the minimum principle, cf., e. g., . 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
The functions are second and the are first order state constraints. Thus the Hamiltonian becomes (cf. Bryson, Denham, Dreyfus )
where is a multiplier. The necessary conditions from the minimum principle yield for the state and adjoint variables among others (cf. )
The optimal control is determined by the minimum principle
where denotes the control space. If or then in because
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. , ).
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. , ).
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 .