Nonlocal controllability of fractional measure evolution equation

Abstract

In this paper, we consider the following kind of fractional evolution equation driven by measure with nonlocal conditions: C D α 0+ x(t) = Ax(t) dt + (f (t, x(t)) + Bu(t)) dg(t), t ∈ (0, b], x(0) + p(x) = x 0. The regulated proposition of fractional equation is obtained for the first time. By noncompact measure method and fixed point theorems, we obtain some sufficient conditions to ensure the existence and nonlocal controllability of mild solutions. Finally, an illustrative example is given to show practical usefulness of the analytical results.

Controllability of fractional measure evolution systems with state-dependent delay and nonlocal condition

“…Moreover, let (t) = t 0 α (s)dg(s), where α (s) = (t − s) α−1 φ r * (s). From the Proposition 2.3 in [8], one can obtain that (t) : J → X is a regulated function. Thus,…”

“…However, MDEs are not as smooth or continuous as ordinary differential equations. As [8,14] said, they are only right continuous and bounded, which brings a lot of difficulties to further research. Furthermore, time delay will also cause obstacles to the estimation of measures of noncompactness.…”

Total Controllability of Non-Autonomous Measure Evolution Systems with Non-Instantaneous Impulses and State-Dependent Delay

“…A frequently used method for solving the controllability problem is to transform it into a fixed point problem for an appropriate operator in a function space. Existence and control problems for various types of measure differential systems have been studied by many authors in [19][20][21][22][23][24][25]. For example, Wan and Sun [24] investigated the approximate controllability for a class of abstract MDEs by applying ρ-set contractive fixed point theorem.…”

“…As the authors of [10,22] said, the main difficulty to deal with the MDEs is that they are not as smooth or continuous as ordinary differential equations. This fact means that further study for MEDs is more difficult, mainly because they are only right continuous and bounded.…”

Approximate Controllability of Neutral Measure Evolution Equations with Nonlocal Conditions

“…On the other hand, the complete controllability of several nonlinear dynamic systems, such as stochastic systems of fractional order and dynamic systems of impulsive differential equations, has been extensively studied. Recently, some authors had discussed existence, stability, and nonlocal controllability of the measure evolution equation [9,15,[19][20][21][22]. In the past few years, the existence and controllability of fractional abstract functional differential development systems with nonlocal conditions have been fully studied [2,[23][24][25][26][27][28][29][30][31].…”