Delay Differential Equations in Sequence Spaces

Yıl 2019, Cilt: 2 Sayı: 3, 182 - 191, 30.09.2019

Luis Gerardo Mármol Carmen Judith Vanegas

https://doi.org/10.33434/cams.528305

Öz

The standard  delay equations are newly studied in the context of classical separable Banach Sequence Spaces.  As a classical solution is shown to exist, the associated semigroup and its infinitesimal generator are found, and some important properties of those operators are proven, including some spectral properties. As an application, it is shown how can these results be used to characterize the constrained null-controllability.

Anahtar Kelimeler

Delay Differential Equations, Sequence Spaces, Exact Controllability

Kaynakça

• [1] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I - Sequence Spaces, Springer Verlag, Berlin-Heidelberg- New York, 1977.
• [2] R. F. Curtain, H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics 21, Sringer Verlag, New York-Berlin, 1995.
• [3] I. Singer, Bases in Banach Spaces I, Springer Verlag, Berlin-Heidelberg- New York, 1970.
• [4] R. F. Brammer, Controllability in linear autonomous systems with positive controllers, SIAM J. Control, 10(2) (1972) 329-353.
• [5] D. Barcenas, J. Diestel, Constrained controllability in non reflexive Banach spaces, Quaest. Math.,18 (1995), 185-198.
• [6] J. Diestel, Grothendieck spaces in vector measures. Contained in Vector and Operator Valued Measures and Applications, Proc. Sympos. Alta Utah, 1972. Academic Press, NY, USA, (1973), 97-108.
• [7] D. Barcenas, L. G. M´armol, On the adjoint of a strongly continuous semigroup, Abstr. Appl. Anal., 2008 (2008), Article ID 651294.
• [8] P. Habala, P. Hajek, V. Zizler, Introduction to Banach Spaces I. Mat Fiz press vy davatelstvi Matematicko-Fyzik ˆ alni Fakulty Univerzity Karlovy, 1996.
• [9] R. Manzanilla, L. G. Marmol, C. J. Vanegas, On the controllability of a differential equation with delayed and advanced arguments, Abstr. Appl. Anal., 2010 (2010), 1-16.
• [10] V. Iakovleva, R. Manzanilla, L. G. Marmol, C. J. Vanegas, Solutions and Constrained null-controllability for a differentialdifference equation, Math. Slovaca, 66(1) (2016), 169-184.
• [11] V. Iakovleva, C. J. Vanegas, Spectral analysis of the semigroup associated to a mixed functional differential equation, Int. J. Pure Appl. Math. 72(4) (2011), 491-499.
• [12] V. Iakovleva, C. J. Vanegas, Smooth Solution of an Initial Value Problem for a Mixed-Type Differential Difference Equation. Contained in Current trends in Analysis and its Applications. Vol. XVI, Proceedings of the 9th ISAAC Congress, Krakow 2013. Editors Mityushev, Vladimir, Ruzhansky, Michael. Birkh¨auser Basel, (2015), 649-654.