## Abstract and Applied Analysis

### Approximate Controllability of Semilinear Impulsive Evolution Equations

Hugo Leiva

#### Abstract

We prove the approximate controllability of the following semilinear impulsive evolution equation: $z\mathrm{\text{'}}=Az+Bu(t)+F(t,z,u),$ $z\in Z,$ $t\in (0,\tau ],$ $z(0)={z}_{0},$ $z({t}_{k}^{+})=z({t}_{k}^{-})+{I}_{k}({t}_{k},z({t}_{k}),u({t}_{k})),$ $k=1,2,3,\dots ,p,$ where $0<{t}_{1}<{t}_{2}<{t}_{3}<\cdots <{t}_{p}<\tau$, $Z$ and $U$ are Hilbert spaces, $u\in {L}^{2}(0,\tau ;U)$, $B:U\to Z$ is a bounded linear operator, ${I}_{k},F:[0,\tau ]\timesZ\timesU\to Z$ 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

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

#### References

• A. E. Bashirov and N. Ghahramanlou, “On partial approximate controllability of semilinear systems,” Cogent Engineering, vol. 1, no. 1, 2014.
• A. E. Bashirov and M. Jneid, “On partial complete controllability of semilinear systems,” Abstract and Applied Analysis, vol. 2013, Article ID 521052, 8 pages, 2013.
• A. E. Bashirov, N. Mahmudov, N. Şem\i, and H. Et\ikan, “Partial controllability concepts,” International Journal of Control, vol. 80, no. 1, pp. 1–7, 2007.
• H. Leiva and N. Merentes, “Approximate controllability of the impul sive semilinear heat equationčommentComment on ref. [11?]: Please update the information of this reference, if possible.,” to appear in Journal of Mathematics and Applications.
• D. N. Chalishajar, “Controllability of impulsive partial neutral functional differential equation with infinite delay,” International Journal of Mathematical Analysis, vol. 5, no. 5–8, pp. 369–380, 2011.
• B. Radhakrishnan and K. Balachandran, “Controllability results for semilinear impulsive integrodifferential evolution systems with nonlocal conditions,” Journal of Control Theory and Applications, vol. 10, no. 1, pp. 28–34, 2012.
• S. Selvi and M. Mallika Arjunan, “Controllability results for impulsive differential systems with finite delay,” Journal of Nonlinear Science and Its Applications, vol. 5, no. 3, pp. 206–219, 2012.
• L. Chen and G. Li, “Approximate controllability of impulsive differential equations with nonlocal conditions,” International Journal of Nonlinear Science, vol. 10, no. 4, pp. 438–446, 2010.
• H. Leiva and N. Merentes, “Controllability of second-order equations in ${L}^{2}(\Omega )$,” Mathematical Problems in Engineering, vol. 2010, Article ID 147195, 11 pages, 2010.
• H. Lárez, H. Leiva, and J. Uzcátegui, “Controllability of block diagonal systems and applications,” International Journal of Sys-tems, Control and Communications, vol. 3, no. 1, 2011.
• R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Sys-tems, vol. 8 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1978.
• R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimen-sional Linear Systems Theory, vol. 21 of Text in Applied Mathematics, Springer, New York, NY, USA, 1995.
• S. P. Chen and R. Triggiani, “Proof of extensions of two conjec-tures on structural damping for elastic systems,” Pacific Journal of Mathematics, vol. 136, no. 1, pp. 15–55, 1989. \endinput