Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space

Mathematical Biosciences and Engineering

Volumn 4, Issue 3, 2007, Pages 403-430

  Banks, H T ^a   Dediu, S ^a,b   Nguyen, H K ^a,c  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b Statistical and Applied Mathematical Sciences Institute USA   (United States)

^c CNA CORPORATION   (United States)

Author keywords

Approximation; Differential equations with distributional delays; Nonlinear dynamical systems with uncertain parameters; Sensitivity with respect to probability measures

Indexed keywords

EID: 34547457646     PISSN: 15471063     EISSN: 15510018     Source Type: Journal    
DOI: 10.3934/mbe.2007.4.403     Document Type: Article

References (64)

1. H. M. ADELMAN AND R.T. HAFTKA, Sensitivity analysis of discrete structural systems, A.I.A.A. Journal, 24 (1986), 823-832.
2. H.T. BANKS, Approximation of nonlinear functional differential equation control systems, J. Optimization Theory and Applications, 24 (1979), 383-408.
3. H.T. BANKS, J.E. BANKS, L.K. DICK AND J.D. STARK, Estimation of dynamic rate parameters in insect populations undergoing sublethal exposure to pesticides, CRSC-TR05-22, NCSU, May, 2005; Bulletin of Mathematical Biology, to appear.
4. H.T. BANKS AND K. L. BIHARI, Modeling and estimating uncertainty in parameter estimation, CRSC-TR99-40, NCSU, August, 1999; Inverse Problems, 17 (2001), 1-17.
5. H.T. BANKS, V.A. BOKIL, S. HU, A.K. DHAR, R.A. BULLIS, C.L. BROWDY AND F.C.T. ALLNUTT, Shrimp biomass and viral infection for production of biological countermeasures. CRSC-TR05-45, NCSU, December, 2005; Math. Biosci. and Engr., 3 (2006), 635-660.
6. H.T. BANKS AND D. M. BORTZ, A parameter sensitivity methodology in the context of HIV delay equation models, CRSC-TR02-24, NCSU, August, 2002; J. Mathematical Biology, 50 (2005), 607-625.
7. H.T. BANKS, D.M. BORTZ AND S.E. HOLTE, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics, Mathematical Biosciences, 183 (2003), 63-91.
8. H.T. BANKS, D.M. BORTZ, G.A. PINTER AND L.K. POTTER, Modeling and imaging techniques with potential for application in bioterrorism, chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, (H.T. Banks and C. Castillo-Chavez, eds.), Frontiers in Applied Mathematics FR28, SIAM, Philadelphia, 2003, pp. 129-154.
9. H.T. BANKS, M. BUKSAS AND T. LIN, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts, Frontiers in Applied Mathematics FR21, SIAM, Philadelphia, 2000.
10. H.T. BANKS AND J. A. BURNS, Hereditary control problems: Numerical methods based on averaging approximations , SIAM J. Control Optimization, 16 (1978), 169-208.
11. H.T. BANKS AND N.L. GIBSON, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters, CRSC-TR04-01, NCSU, January, 2004; Applied Math. Letters, 18 (2005), 423-430.
12. H.T. BANKS AND N.L. GIBSON, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters (with N.L. Gibson), CRSC-TR05-29, NCSU, August, 2005; Quarterly of Applied Mathematics, 64 (2006), 749-795.
13. H.T. BANKS AND F. KAPPEL, Spline approximations for functional differential equations, J. Differential Equations, 34 (1979), 496-522.
14. H.T. BANKS AND K. KUNISCH, Estimation Techniques for Distributed Parameter Systems, Boston: Birkhauser, Boston, 1989.
15. H.T. BANKS, A.J. KURDILA AND G. WEBB, Identification of hysteretic control influence operators representing smart actuators, part I: Formulation, Mathematical Problems in Engineering, 3 (1997), 287-328.
16. H.T. BANKS, A.J. KURDILA AND G. WEBB, Identification of hysteretic control influence operators, part II: Convergent approximations, J. Intelligent Material Systems and Structures, 8 (1997), 536-550.
17. H.T. BANKS, N.G. MEDHIN AND G.A. PINTER, Multiscale considerations in modeling of nonlinear elastomers, CRSC-TR03-42, NCSU, October, 2003; J. Comp. Meth. Engr. Sci. and Mech., 8 (2007), 53-62.
18. H.T. BANKS, N.G. MEDHIN AND G.A. PINTER, Nonlinear reptation in molecular based hysteresis models for polymers, CRSC-TR03-45, NCSU, December, 2003; Quart. Appl. Math., 62 (2004), 767-779.
19. H.T. BANKS AND H. K. NGUYEN, Sensitivity of dynamical systems to Banach space parameters, CRSC-TR05-13, NCSU, February, 2005; Journal of Mathematical Analysis and Applications, 323 (2006), 146-161.
20. H.T. BANKS AND G.A. PINTER, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, CRSC-TR04-03, NCSU, January, 2004; SIAM J. Multiscale Modeling and Simulation, 3 (2005), 395-412.
21. H.T. BANKS, R.C. SMITH AND Y. WANG, Smart Material Structures: Modeling, Estimation and Control, Masson/John Wiley, Paris/Chichester, 1996.
22. J.J. BATZEL, F. KAPPEL, D. SCHNEDITZ AND H.T. TRAN, Cardiovascular and Respiratory Systems: Modeling, Estimation and Control, Frontiers in Applied Mathematics, SIAM, Philadelphia, 2006.
23. P. BILLINGSLEY, Convergence of Probability Measures, Wiley, NY, 1968.
24. S.M. BLOWER AND H. DOWLATABADI, Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model as an example, International Statistics Review, 62 (1994), 229-243.
25. J. BORGGAARD AND J.A. BURNS, A PDE sensitivity equation method for optimal aerodynamic design, Journal of Computational Physics, 136 (1997), 367-384.
26. J. BORGGAARD AND A. VERMA, On efficient solutions to the continuous sensitivity equation using automatic differentiation, SIAM Journal on Scientific Computing, 22 (2001), 39-62.
27. D.M. BORTZ AND P.W. NELSON, Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics, Bulletin of Mathematical Biology, 66 (2004), 1009-1026.
28. C. J. F. BÖTTCHER AND P. BORDEWIJK, Theory of Electric Polarization, Vol. II, Elsevier, New York, 1978.
29. M. BROKATE AND J. SPREKELS, Hysteresis and Phase Transitions, Springer-Verlag, New York, NY, 1996.
30. K.P. BURNHAM AND D.R. ANDERSON, Model Selection and Multimodal Inference: A Practical Information-Theoretic Approach, Springer-Ver lag, New York, NY, 2002.
31. J.A. BURNS, T. LIN AND L. STANLEY, A Petrov-Galerkin finite element method for interface problems arising in sensitivity computations, Computers and Mathematics with Applications, 49 (2005), 1889-1903.
32. J. A. BURNS, D. RUBIO AND M. I. TROPAREVSKY, Sensitivity computations for elliptic equations with interfaces, in Proceedings ICNPAA-2006 Conference on Mathematical Problems in Engineering and Aerospace Sciences, to appear.
33. J. A. BURNS AND L. STANLEY, A note on the use of transformations in sensitivity computations for elliptic systems, Mathematical and Computer Modelling, 33 (2001), 101-114.
34. E. M. CLIFF, M. HEINKENSCHLOSS AND A.R. SHENOY, Adjoint-based methods in aerodynamic design-optimization, in Computational Methods in Optimal Design and Control, J.T. Borggaard, J. Burns, E. Cliff, and S. Schreck, eds, Birkhauser-Boston, 1998, 91-112.
35. J.B. CRUZ, ED., System Sensitivity Analysis, Dowden, Hutchinson & Ross, Inc., Stroudsberg, PA, 1973.
36. M. DAVIDIAN AND D.M. GILITAN, Nonlinear Models for Repeated Measurement Data, Chapman & Hall, London, 1995.
37. M. ESLAMI, Theory of Sensitivity in Dynamic Systems: An Introduction, Springer-Verlag, Berlin, 1994.
38. K.R. FOSTER AND H.P. SCHWAN, Dielectric properties of tissues and biological materials: A critical review, Critical Rev. in Biomed. Engr. 17 (1989), 25-104.
39. P.M. FRANK, Introduction to System Sensitivity Theory, Academic Press, Inc., New York, NY, 1978.
40. K.W. HASSON, D.V LIGHTNER, L.L. MOHNEY, R.M. REDMAN, B.T. POULOS AND B.M. WHITE, Taura Syndrome Virus (TSV) lesion development and the disease cycle in the Pacific white shrimp Penaeus vannamei, Diseases of Aquatic Organisms, 36 (1999), 81-93.
41. E. J. HAUG, K.K. CHOI AND V. KOMKOV, Design Sensitivity Analysis of Structural Systems, Academic Press, New York, NY, 1986.
42. M. KLEIBER, H. ANTUNEZ, T.D. HIEN AND P. KOWALCZYK, Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations, John Wiley &: Sons, New York, NY, 1997.
43. MARK KOT, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.
44. M. KRASNOSELSKII AND A. POKROVSKII, Systems with Hysteresis, Nauka/Springer-Verlag, 1983/1989.
45. E.L. LEHMANN, Elements of Large-Sample Theory, Springer-Verlag, New York, NY, 1999.
46. I. MAYERGOYZ, Mathematical Models of Hysteresis, Springer-Verlag, New York, NY, 1991.
47. E.J. MCSHANE, Generalized curves, Duke Math J., 6 (1940), 513-536.
48. E.J. MCSHANE, Relaxed controls and variational problems, SIAM J. Control, 5 (1967), 438-484.
49. J.A. J. METZ AND E.O. DIEKMANN, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Vol. 68, Springer, Heidelberg, 1986.
50. The NCSU Entomology Homepage, on line at http://ipm.ncsu.edu/ AG271/forages/pea_aphid.html.
51. A. SALTELLI, K. CHAN AND E.M. SCOTT, EDS., Sensitivity Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, New York, NY, 2000.
52. W.W. SCHMAEDEKE, Optimal control theory for nonlinear vector differential equations containing measures, SIAM J. Control, 3 (1965), 231-280.
53. G.A.F. SEBER AND C.J. WILD, Nonlinear Regression, John Wiley & Sons, Inc., New York, NY, 1989.
54. L.B. SHEINER, B. ROSENBERG AND K. MELMON, Modeling of individual pharmacokinetics for computer-aided drug dosage, Comp. Biomed. Res., 5 (1972), 441-459.
55. L. STANLEY AND D. STEWART, Design Sensitivity Analysis: Computational Issues of Sensitivity Equation Methods, Frontiers in Applied Mathematics, FR25, SIAM, Philidelphia, 2002.
56. J. D. STARK AND T. M. RANGUS, Lethal and sublethal effects of the neem insecticide formulation, 'Margosan-O', on the pea aphid, Pestic. Sci., 41 (1994), 155-160.
57. E.R VON SCHWEIDLER, Studien über anomalien im verhalten der dielektrika, Ann. Physik, 24 (1907), 711-770.
58. K.W. WAGNER, Zur theorie der unvollkommenen dielektrika, Ann. Physik, 40 (1913), 817-855.
59. J. WARGA, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128.
60. J. WARGA, Functions of relaxed controls, SIAM J. Control, 5 (1967), 628-641.
61. J. WARGA, Optimal Control of Differential and Functional Equations, Acadedmic Press, New York, NY, 1972.
62. A. WIERZBICKI, Models and Sensitivity of Control Systems, Studies in Automation and Control 5, Elsevier Science Publ. Co., Inc., New York, NY, 1984.
63. L.C. YOUNG, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. et Lettres, Varsovie, Cl. III, 30 (1937), 212-234.
64. L.C. YOUNG, Necessary conditions in the calculus of variations, Acta Math., 69 (1938), 239-258.