Lemma 1 Let p, v : [ a, b ] → C two bounded functions on the compact interval [ a, b ]. (i) If p is continuous and v is of bounded variation, then the Riemann-Stieltjes integral ∫ a b p ( t ) d v ( t ) exists and | ∫ a b p ( t ) d v ( t ) | ≤ max t ∈ [ a, b ] | p ( t ) | ⋁ a b ( v ), (3.1) .
It is clear that any bounded function on is in, but the converse is not true.
Assume that t σ f ( t, ⋅ ) is a bounded function on ( 0, ∞ ), and define the Hammerstein integral operator T : P → X by T u ( t ) = 1 Γ ∫ 0 t ( t − s ) α − 1 f ( s, u ( s ) ) d s + b Γ t α − 1 + a t α − 2. (8).
Let F be a Borel bounded function on [−a,a], SpL2T⊂[−a,a].
The robust controllers are synthesized via Lyapunov's direct method and utilize bounding functions on the magnitude of the uncertain terms.
The iteration process (3.1) is subject to a forcing term generated by a set of Lipschitzian mappings where is a sequence of means on, with the subset (defined in Definition 3.5 below) containing unity, where is the Banach space of all bounded functions on endowed with the supremum norm, such that where is the dual of.
Let (operatorname {C}_{mathrm{B}}[0,infty)) denote the space of all continuous and bounded functions on ([0,infty)), where the norm is defined by Vert hVert =sup_{[0,infty)}biglvert h(x bigrvert. For every (hin operatorname {C}_{mathrm{B}}[0,infty)), we have biglVert P_{n,alpha}^{beta,c}(h cdot bigrVert leq Vert h Vert. Lemma 3 can easily be proved using (2.1).
Let Λ be the set of all continuous and bounded functions on ([t_{0},infty)) with the sup norm.
Let Ω be the set of all continuous and bounded functions on ([t_{0}, infty)) and the norm be (|x t)|=sup_{t_{0}leq t< +infty}x t)).
