In a number of optimal control problems (e.g., see [1,2] and the bibliography therein), admissible controls w(·), that is, measurable functions essentially bounded on a given time interval are represented in the form w(·) = (v(·), u(·)) in order to take account of geometric constraints on the control and mixed constrains (which are important in applications) simultaneously. As was mentioned, e.g., in [3,4], in optimal control problems, it is of interest from the practical viewpoint to consider controls of the form (q, u(·)), where q belongs to a given set Q ⊂ R k and u : R → U, U ∈ comp (R m ), is a measurable ω-periodic function. In the present paper, we continue the research initiated in [5][6][7]. We give necessary extremum conditions for an almost periodic optimal control problem under constraints on the means, with admissible controls being pairs (v(·), u(·)), where v(·) belongs to a given subset S of the space B (R, R n ) of Bohr almost periodic functions and u(·) belongs to the set S(R, U) of Stepanov almost periodic functions. These conditions can be derived from the corresponding necessary conditions for the convexified problem in which the set S(R, U) is extended to include measure-valued almost periodic functions.
MAIN DEFINITIONS AND NOTATIONLet R n be the n-dimensional Euclidean space with norm | · |, let orb(ϕ) be the closure (in R n ) of the orbit of the function ϕ : R → R n , and let Hom (R n , R m ) be the space of linear operators L : R n → R m [Hom (R n )= Hom (R n , R n )] with norm |L|= max |x|≤1 |Lx|. Next, by B(R, Y) and S(R, Y) (Y ⊂ R n ) we denote the set of mappings f : R → Y that are Bohr (Stepanov) almost periodic with respect to the metric d l (d= d 1 ) [8]. Recall that for each almost periodic function f (in the Bohr sense as well as in the Stepanov sense), there exists a mean M {f (t)}= lim T →∞