Growth of semigroups in discrete and continuous time

Gomilko, Alexander and Zwart, Hans and Besseling, Niels (2011) Growth of semigroups in discrete and continuous time. Studia Mathematica, 206 (3). pp. 273-292. ISSN 0039-3223

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Abstract:	We show that the growth rates of solutions of the abstract differential equations x˙(t)=Ax(t), x˙(t)=A −1 x(t) and the difference equation xd(n+1)=(A+I)(A−I)−1 xd(n) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup (e A−1t) t≥0 is O(√4t) and for ((A+I)(A−I)−1)n it is O(√4n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of ((A+I)(A−I) −1 )n is O(1), i.e., the operator is power bounded.
Item Type:	Article
Copyright:	© 2011 Instytut Matematyczny PAN
Faculty:	Electrical Engineering, Mathematics and Computer Science (EEMCS)
Research Group:	Mathematical Systems and Control Theory (MSCT)
Link to this item:	http://purl.utwente.nl/publications/78767
Official URL:	http://dx.doi.org/10.4064/sm206-3-3
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