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.
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Copyright:© 2011 Instytut Matematyczny PAN
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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