## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2015, Special Issue (2015), Article ID 797439, 7 pages.

### Approximate Controllability of Semilinear Impulsive Evolution Equations

#### Abstract

We prove the approximate controllability of the following semilinear impulsive evolution equation: $z\mathrm{\text{'}}=Az+Bu(t)+F(t,z,u),\hspace{0.17em}$ $z\in Z,\hspace{0.17em}$ $t\in (0,\tau ],\hspace{0.17em}$ $z(0)={z}_{0},\hspace{0.17em}$ $z({t}_{k}^{+})=z({t}_{k}^{-})+{I}_{k}({t}_{k},z({t}_{k}),u({t}_{k})),\hspace{0.17em}$ $k=\mathrm{1,2},3,\dots ,p,$ where $$, $Z$ and $U$ are Hilbert spaces, $u\in {L}^{2}(0,\tau ;U)$, $B:U\to Z$ is a bounded linear operator, ${I}_{k},F:[0,\tau ]\times Z\times U\to Z$ are smooth functions, and $A:D(A)\subset Z\to Z$ is an unbounded linear operator in $Z$ which generates a strongly continuous semigroup $\{T(t){\}}_{t\ge 0}\subset Z$. We suppose that $F$ is bounded and the linear system is approximately controllable on $[0,\delta ]$ for all $\delta \in (0,\tau )$. Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on $[0,\tau ]$.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 797439, 7 pages.

**Dates**

First available in Project Euclid: 15 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1434398629

**Digital Object Identifier**

doi:10.1155/2015/797439

**Mathematical Reviews number (MathSciNet)**

MR3339674

**Zentralblatt MATH identifier**

1352.93024

#### Citation

Leiva, Hugo. Approximate Controllability of Semilinear Impulsive Evolution Equations. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 797439, 7 pages. doi:10.1155/2015/797439. https://projecteuclid.org/euclid.aaa/1434398629