Necessary Conditions for a Class of Optimal Control Problems on Time Scales

and Applied Analysis 3 Remark 2.2. For each t0 ∈ T \ {maxT}, the single-point set {t0} is Δ-measurable, and its Δmeasure is given by μΔ {t0} σ t0 − t0 μ t0 . 2.2 Obviously, E1 ⊂ A does not have any right-scattered points. For a set E ⊂ T, define the Lebesgue Δ-integral of f over E by ∫ Ef t Δt and let f ∈ LT E,R see 8 . Lemma 2.3 see 8 . Let f : a, b T → R. f̃ : a, b → R is the extension of f to real interval a, b , defined by f̃ t : ⎧ ⎨ ⎩ f t if t ∈ a, b T , f ti if t ∈ ti, σ ti , for some i ∈ I, 2.3 where {ti}i∈I , I N is the index of the set of all right-scattered points of a, b T. Then, f ∈ L1 T a, b T ,R if and only if f̃ ∈ L1 a, b ,R . In this case, ∫