Approximate controllability of semilinear measure driven systems

Abstract

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. By using the semigroup theory and Schauder fixed point theorem, sufficient conditions for approximate controllability of measure driven equations are established. The obtained results are a generalization and continuation of the recent results on this issue. Finally, an example is provided to illustrate the application of the obtained results.

Approximate Controllability of Neutral Measure Evolution Equations with Nonlocal Conditions

“…Definition 2 (See [7]). Let X be a Banach space with a norm ‖ • ‖ and [a, b] be a closed interval of the real line.…”

“…Lemma 1 (See [7]). Consider the functions f: J ⟶ X and g: J ⟶ R such that g is regulated and b a fdg exists.…”

“…Assume that A ⊂ G([a, b]; X) is equiregulated, and for every t ∈ [a, b], the set x(t): t ∈ A { } is relatively compact in X. en, the set A is relatively compact in G([a, b]; X). Lemma 3 (See [7]). Let X be a Banach space and E⊆X be a bounded, closed, and convex set.…”

“…Let X be a Banach space and E⊆X be a bounded, closed, and convex set. If the operator N: E ⟶ E is completely continuous, then N has a fixed point on E. Lemma 4 (See [7]). Let x n ∞ n�1 be a sequence of functions from [a, b] to X.…”

“…As a result, it cannot simulate some complex phenomena, such as Zeno's behavior. However, the dynamic system with discontinuous trajectory is modeled by a measure differential equation or measure-driven equation [4][5][6][7][8][9][10]. Measure differential equations (MDEs) were studied in the early days [11][12][13][14][15][16][17][18].…”