Total Controllability of the Second Order Semi-Linear Differential Equation with Infinite Delay and Non-Instantaneous Impulses
Abstract
:1. Introduction
2. Preliminaries and Assumptions
- (i)
- for all
- (ii)
- (iii)
- is continuous in t on for each fixed point
- (A1)
- A is the infinitesimal generator of a strongly continuous cosine family . In addition, there exists a constant such that , for all .
- (A2)
- , is a continuous function and there exist positive constant such that
- (A3)
- B is continuous operator from U to X and the linear operator is defined by It has a bounded invertible operator which takes values in and there exist positive constants such that .
- (A4)
- There exist positive constants such that and where
- (A5)
- and there are positive constants , such that
3. Total Controllability
4. Integrodifferential Equation
- (A6)
- The real-valued function is piecewise continuous on and there exists a positive constant such that
- (A7)
- , is a continuous function and there exist positive constant such that
5. Application
6. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Chalishajar, D.N.; Kumar, A. Total Controllability of the Second Order Semi-Linear Differential Equation with Infinite Delay and Non-Instantaneous Impulses. Math. Comput. Appl. 2018, 23, 32. https://doi.org/10.3390/mca23030032
Chalishajar DN, Kumar A. Total Controllability of the Second Order Semi-Linear Differential Equation with Infinite Delay and Non-Instantaneous Impulses. Mathematical and Computational Applications. 2018; 23(3):32. https://doi.org/10.3390/mca23030032
Chicago/Turabian StyleChalishajar, Dimplekumar N., and Avadhesh Kumar. 2018. "Total Controllability of the Second Order Semi-Linear Differential Equation with Infinite Delay and Non-Instantaneous Impulses" Mathematical and Computational Applications 23, no. 3: 32. https://doi.org/10.3390/mca23030032