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On the Asymptotic Stability of Reactors with Arbitrary Feedback
dc.contributor.authorAkcasu, A. Ziyaen_US
dc.contributor.authorAkhtar, P.en_US
dc.date.accessioned2010-05-06T20:38:08Z
dc.date.available2010-05-06T20:38:08Z
dc.date.issued1970-01en_US
dc.identifier.citationAkcasu, A. Z.; Akhtar, P. (1970). "On the Asymptotic Stability of Reactors with Arbitrary Feedback." Journal of Mathematical Physics 11(1): 155-162. <http://hdl.handle.net/2027.42/69459>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/69459
dc.description.abstractA theorem for the boundedness and asymptotic stability of a point reactor with an arbitrary feedback is stated and proved. The criteria obtained are shown to be essentially the same as those given by Akcasu and Dolfes. The theorem is applied to a reactor with an arbitrary linear feedback and to a xenon‐controlled reactor with a flux reactivity coefficient whose feedback mechanism involves quadratic non‐linearity. It is also compared to a criterion obtained by Corduneanu in the case when delayed neutrons are ignored and the feedback mechanism is linear.en_US
dc.format.extent3102 bytes
dc.format.extent527899 bytes
dc.format.mimetypetext/plain
dc.format.mimetypeapplication/pdf
dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleOn the Asymptotic Stability of Reactors with Arbitrary Feedbacken_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumDepartment of Nuclear Engineering, The University of Michigan, Ann Arbor, Michiganen_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/69459/2/JMAPAQ-11-1-155-1.pdf
dc.identifier.doi10.1063/1.1665041en_US
dc.identifier.sourceJournal of Mathematical Physicsen_US
dc.identifier.citedreferenceThe usual point reactor kinetic equations2 can be reduced to the form of Eqs. (1) and (2) by choosing z(t)  =  [P(t)−P0]/p0,z(t)=[P(t)−P0]∕p0, zi(t)  =  [Ci(t)−Ci0]/Ci0,zi(t)=[Ci(t)−Ci0]∕Ci0, and by letting →βt/l,→βt∕l, ai  =  βi/β,ai=βi∕β, and hi  =  lλi/β.hi=lλi∕β. P0P0 and Ci0Ci0 denote equilibrium values of reactor power and delayed neutron precursor densities, respectively, and l,λi,βi,l,λi,βi, β have their conventional meanings in the field of reactor engineering. P0P0 and Ci0Ci0 are determined by K0+Kf(Po)  =  0K0+Kf(Po)=0 and Ci0  =  aip0/λi,Ci0=aip0∕λi, where K0K0 and Kf(P0)Kf(P0) are external and equilibrium feedback reactivities. The incremental feedback reactivity functional δKf[z,t]δKf[z,t] appearing in Eq. (1) is then defined as δKf[z,t] ≡ Kf[P,t]−Kf(p0),δKf[z,t]≡Kf[P,t]−Kf(p0), where Kf[P,t]Kf[P,t] is the total feedback reactivity functional. We will assume that the algebraic equation relating K0K0 and Kf(P0)Kf(P0) has a unique solution. We also note that (1+z)(1+z) and (1+zi)(1+zi) are nonnegative, since P0P0 and Ci0Ci0 can never be negative.en_US
dc.identifier.citedreferenceA. F. Henry, Nucl. Sci. Eng. 3, 52 (1958).en_US
dc.identifier.citedreferencez(t)  =  Zi(t) ≡ −1z(t)=Zi(t)≡−1 for all t also represents an equilibrium state. Physically, this corresponds to a reactor in which there are no neutrons and delayed neutron precursors. In the course of a derivation we shall exclude this equilibrium state from discussion because it will be shown that, when conditions for boundedness as stated in the theorem in the next section are satisfied, z(t)z(t) and zi(t)zi(t) can never approach −1 [cf. Eqs. (14a) and (14b)] once the reactor is perturbed.en_US
dc.identifier.citedreferenceN. N. Krasovskii, Stability of Motion (Stanford University Press, Stanford, California, 1963).en_US
dc.identifier.citedreferenceA. Z. Akcasu and A. Dalfes, Nucl. Sci. Eng. 8, 89 (1960).en_US
dc.identifier.citedreferenceJ. Chernick, G. Lellouche, and W. Wollman, Nucl. Sci. Eng. 10, 120 (1960).en_US
dc.identifier.citedreferenceA. Z. Akcasu and P. Akhtar, J. Nucl. Energy 21, 341 (1967).en_US
dc.identifier.citedreferenceIt should be noted that there is no restriction on the bound of the test functions {y(t)}{y(t)} for t ≥ 0,t≥0, and they may diverge as t→∞.t→∞. Clearly {y(t)}{y(t)} contains all possible solutions of Eqs. (1) and (2) as a subset.en_US
dc.identifier.citedreferenceIt may be noted that the test functions {ω(t)}{ω(t)} form a subset of the functions {v(t)},{v(t)}, which in turn are a subset of test functions {y(t)}.{y(t)}.en_US
dc.identifier.citedreferenceI. Barbalat, Rev. Math. Pures Appl. 4, 267 (1959).en_US
dc.identifier.citedreferenceG. S. Lellouche, J. Nucl. Energy 21, 519 (1967).en_US
dc.identifier.citedreferenceThe condition (26) is necessary and sufficient13 for the linear functional δKf[z,t]δKf[z,t] in Eq. (25) to be bound for all t and for all bounded functions z(t)z(t) in (−∞,+∞).(−∞,+∞).en_US
dc.identifier.citedreferenceA. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw‐Hill Book Co., New York, 1965).en_US
dc.identifier.citedreferenceD. V. Widder, The Laplace Transform (Princeton University Press, Princeton, N.J., 1946).en_US
dc.identifier.citedreferencePasquantonia and Kappel have recently shown16 that the condition in Ref. 5 is sufficient for asymptotic stability using Hale’s theorem.17en_US
dc.identifier.citedreferenceF. Di Pasquantonia and F. Kappel, Energia Nucl. (Milan) 15, 761 (1968).en_US
dc.identifier.citedreferenceJ. K. Hale, J. Differential Equations 1, 452 (1965).en_US
dc.identifier.citedreferenceM. C. Corduneanu, C. R. Acad. Sci. (Paris) 256, 3564 (1963).en_US
dc.identifier.citedreferenceThe condition (ii) of Carduneanu’s theorem requires ρ>0,ρ>0, which by virtue of (51e) implies j(0)>0.j(0)>0. The latter is the condition for the existence of a finite equilibrium power level. The condition (53) alone can not guarantee asymptotic stability.en_US
dc.identifier.citedreferenceA. Z. Akcasu and L. M. Shotkin, Nucl. Sci. Eng. 28, 72 (1967).en_US
dc.owningcollnamePhysics, Department of


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