Abstract and Applied Analysis

Approximate Controllability of Semilinear Impulsive Evolution Equations

Hugo Leiva

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Abstract

We prove the approximate controllability of the following semilinear impulsive evolution equation: z ' = A z + B u ( t ) + F ( t , z , u ) ,  z Z ,  t ( 0 , τ ] ,  z ( 0 ) = z 0 ,  z ( t k + ) = z ( t k - ) + I k ( t k , z ( t k ) , u ( t k ) ) ,  k = 1,2 , 3 , , p , where 0 < t 1 < t 2 < t 3 < < t p < τ , Z and U are Hilbert spaces, u L 2 ( 0 , τ ; U ) , B : U Z is a bounded linear operator, I k , F : [ 0 , τ ] × Z × U Z are smooth functions, and A : D ( A ) Z Z is an unbounded linear operator in Z which generates a strongly continuous semigroup { T ( t ) } t 0 Z . We suppose that F is bounded and the linear system is approximately controllable on [ 0 , δ ] for all δ ( 0 , τ ) . Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on [ 0 , τ ] .

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


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