Pontryagin Maximum Principle

It is considered that the behavior of an object at any instant of time is characterized by n real numbers x ¹,..., x ⁿ. The vector space X of the vector variable x = (x ¹,..., x ⁿ) is the phase space of the object under consideration. The motion of the object consists of the fact that the variables x ¹,..., x ⁿ change with time. It is assumed that the object’s motion can be controlled, i.e., that the object is equipped with certain controllers on whose position the motion of the object depends. The positions of the controllers are characterized by points u = (u ¹,..., u ^r) of a certain control region U, which may be any set in some r-dimensional Euclidean space. In applications, the case where U is a closed region in the space is especially important.

In the statement of the problem it is assumed that the object’s law of motion can be written in the form of a system of differential equations

(1)

or in vector from

(2)

where the functions f _i are defined for x ∈ X and u ∈ U. They are...