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Necessary Conditions of Optimality

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].

next up previous
Next: Multiple Shooting Method Up: Numerical Methods Previous: Numerical Methods

Oskar von Stryk
Fri Apr 5 21:57:02 MET DST 1996