Nonlocal controllability of fractional measure evolution equation

• Main Authors: Haibo Gu, Yu Sun
• Format: Article
• Language: English
• Published: SpringerOpen 2020-03-01
• Series: Journal of Inequalities and Applications
• Subjects: Fractional calculus; Evolution equation; Mild solution; Measure of noncompactness; Nonlocal controllability
• Online Access: http://link.springer.com/article/10.1186/s13660-020-02328-6

Abstract In this paper, we consider the following kind of fractional evolution equation driven by measure with nonlocal conditions: { D 0 + α C x ( t ) = A x ( t ) d t + ( f ( t , x ( t ) ) + B u ( t ) ) d g ( t ) , t ∈ ( 0 , b ] , x ( 0 ) + p ( x ) = x 0 . $$ \textstyle\begin{cases} {}^{C} D_{0+}^{\alpha }x(t)=Ax(t)\,dt+ (f(t,x(t))+Bu(t) )\,dg(t), \quad t\in (0, b],\\ x(0)+p(x)=x_{0}. \end{cases} $$ 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.