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Keywords:
delay-differential system; algebraic methods; general convolution equations; noncommensurate delays; delay-differential systems; behavioral approach
delay-differential system; algebraic methods; general convolution equations; noncommensurate delays; delay-differential systems; behavioral approach
Summary:
This paper presents a survey on the recent contributions to linear time- invariant delay-differential systems in the behavioral approach. In this survey both systems with commensurate and with noncommensurate delays will be considered. The emphasis lies on the investigation of the relationship between various systems descriptions. While this can be understood in a completely algebraic setting for systems with commensurate delays, this is not the case for systems with noncommensurate delays. In the study of this class of systems functional analytic methods need to be introduced and general convolutional equations have to be incorporated. Whenever it is possible, the results will be linked to the relevant control theoretic notions.
References:
[1] Becker T., Weispfennig V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, New York 1993 MR 1213453
[2] Berenstein C. A., Dostal M. A.: The Ritt theorem in several variables. Ark. Mat. 12 (1974), 267–280 DOI 10.1007/BF02384763 | MR 0377111 | Zbl 0293.33001
[3] Berenstein C. A., Struppa D. C.: Complex analysis and convolution equations. Several complex variables. Encyclopedia Math. Sci. 54 (1993), 1–108
[4] Berenstein C. A., Yger A.: Ideals generated by exponential-polynomials. Adv. in Math. 60 (1986), 1–80 DOI 10.1016/0001-8708(86)90002-2 | MR 0839482 | Zbl 0586.32019
[5] Brezis H.: Analyse fonctionnelle: theorie et applications. Masson, Paris 1983 MR 0697382 | Zbl 1147.46300
[6] Cohen A. M., Cuypers, H., (eds.) H. Sterk: Some Tapas of Computer Algebra. Springer, Berlin 1999 MR 1679917 | Zbl 0924.13021
[7] Cohn P. M.: Free Rings and Their Relations. Academic Press, London 1985. Second edition MR 0800091 | Zbl 0659.16001
[8] Diab A.: Sur les zéros communs des polynômes exponentiels. C. R. Acad. Sci. Paris Sér. A 281 (1975), 757–758 MR 0390186 | Zbl 0323.30003
[9] Ehrenpreis L.: Solutions of some problems of division. Part III. Division in the spaces ${\mathcal D}^{\prime },\,{\mathcal H},\,{\mathcal Q}_A,\,{\mathcal O}$. Amer. J. Math. 78 (1956), 685–715 MR 0083690 | Zbl 0072.32801
[10] Folland G. B.: Fourier Analysis and its Applications. Wadsworth & Brooks, Pacific Grove 1992 MR 1145236 | Zbl 1222.42001
[11] Gluesing–Luerssen H.: A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability. Math. Control Signal Systems 13 (2000), 22–40 DOI 10.1007/PL00009859 | MR 1742138 | Zbl 0954.93007
[12] Gluesing–Luerssen H.: A behavioral approach to delay differential equations. SIAM J. Control Optim. 35 (1997), 480–499 DOI 10.1137/S0363012995281869 | MR 1436634
[13] Gluesing–Luerssen H.: Linear delay-differential systems with commensurate delays: An algebraic approach. Habilitationsschrift at the University of Oldenburg 2000. Accepted for publication as Lecture Notes in Mathematics, Springer MR 1874340 | Zbl 0989.34001
[14] Habets L. C. G. J. M.: System equivalence for AR-systems over rings – With an application to delay-differential systems. Math. Control Signal Systems 12 (1999), 219–244 DOI 10.1007/PL00009851 | MR 1707899 | Zbl 0951.93015
[15] Habets L. C. G. J. M., Eijndhoven S. J. L.: Behavioral controllability of time-delay systems with incommensurate delays. In: Proc. IFAC Workshop on Linear Time Delay Systems (A. M. Perdon, ed.), Ancona 2000, pp. 195–201
[16] Helmer O.: The elementary divisor theorem for certain rings without chain condition. Bull. Amer. Math. Soc. 49 (1943), 225–236 DOI 10.1090/S0002-9904-1943-07886-X | MR 0007744 | Zbl 0060.07606
[17] Jacobson N.: Basic Algebra I. Second edition. W. H. Freeman, New York 1985 MR 0780184 | Zbl 0557.16001
[18] Kamen E. W.: On an algebraic theory of systems defined by convolution operators. Math. Systems Theory 9 (1975), 57–74 DOI 10.1007/BF01698126 | MR 0395953 | Zbl 0318.93003
[19] Kamen E. W., Khargonekar P. P., Tannenbaum A.: Proper stable Bezout factorizations and feedback control of linear time-delay systems. Internat. J. Control 43 (1986), 837–857 DOI 10.1080/00207178608933506 | MR 0828360 | Zbl 0599.93047
[20] Kaplansky I.: Elementary divisors and modules. Trans. Amer. Math. Soc. 66 (1949), 464–491 DOI 10.1090/S0002-9947-1949-0031470-3 | MR 0031470 | Zbl 0036.01903
[21] Kelley J. L., Namioka I.: Topological Vector Spaces. Van Nostrand, 1963 MR 0166578
[22] Lang S.: Algebra. Second edition. Addison–Wesley, Reading, N.J. 1984 MR 0783636 | Zbl 1063.00002
[23] Lezama O., Vasquez O.: On the simultaneous basis property in Prüfer domains. Acta Math. Hungar. 80 (1998), 169–176 DOI 10.1023/A:1006537212456 | MR 1624566
[24] Malgrange B.: Existence et approximations des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier 6 (1955/1956), 271–355 DOI 10.5802/aif.65 | MR 0086990
[25] Meisters G. H.: Periodic distributions and non-Liouville numbers. J. Funct. Anal. 26 (1977), 68–88 DOI 10.1016/0022-1236(77)90016-7 | MR 0448068 | Zbl 0359.46027
[26] Mounier H.: Algebraic interpretations of the spectral controllability of a linear delay system. Forum Math. 10 (1998), 39–58 DOI 10.1515/form.10.1.39 | MR 1490137 | Zbl 0891.93014
[27] Niven I.: Irrational Numbers. Wiley, New York 1956 MR 0080123 | Zbl 0146.27703
[28] Oberst U.: Multidimensional constant linear systems. Acta Appl. Math. 20 (1990), 1–175 DOI 10.1007/BF00046908 | MR 1078671 | Zbl 0715.93014
[29] Olbrot A. W., Pandolfi L.: Null controllability of a class of functional differential systems. Internat. J. Control 47 (1988), 193–208 DOI 10.1080/00207178808906006 | MR 0929735 | Zbl 0662.93008
[30] Parreau F., Weit Y.: Schwartz’s theorem on mean periodic vector-valued functions. Bull. Soc. Math. France 117 (1989), 3, 319–325 MR 1020109 | Zbl 0704.46011
[31] Polderman J. W., Willems J. C.: Introduction to Mathematical Systems Theory. A behavioral approach. Springer, Boston 1998 MR 1480665 | Zbl 0940.93002
[32] Rocha P., Wood J.: Trajectory control and interconnection of 1D and $n$D systems. SIAM J. Control Optim. 40 (2001), 107–134 DOI 10.1137/S0363012999362797 | MR 1855308
[33] Schwartz L.: Théorie génerale des fonctions moyennes-périodiques. Ann. of Math. (2) 48 (1947), 857–929 DOI 10.2307/1969386
[34] Treves F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York 1967 MR 0225131 | Zbl 1111.46001
[35] Eijndhoven S. J. L. van, Habets L. C. G. J. M.: Equivalence of Convolution Systems in a Behavioral Framework. Report RANA 99-25. Eindhoven University of Technology 1999
[36] Poorten A. J. van der, Tijdeman R.: On common zeros of exponential polynomials. Enseign. Math. (2) 21 (1975), 57–67 MR 0379387
[37] Vettori P.: Delay Differential Systems in the Behavioral Approach. Ph. D. Thesis, Università di Padova 1999
[38] Vettori P., Zampieri S.: Controllability of systems described by convolutional or delay-differential equations. SIAM J. Control Optim. 39 (2000), 728–756 DOI 10.1137/S0363012999359718 | MR 1786327 | Zbl 0976.34070
[39] Vettori P., Zampieri S.: Some results on systems described by convolutional equations. IEEE Trans. Automat Control. AC–46 (2001), 793–797 DOI 10.1109/9.920803 | MR 1833038 | Zbl 1009.93013
[40] Willems J. C.: On interconnection, control, and feedback. IEEE Trans. Automat. Control AC-42 (1997), 326–339 DOI 10.1109/9.557576 | MR 1435822
[41] Zemanian A. H.: Distribution Theory and Transform Analysis. McGraw–Hill, New York 1965 MR 0177293 | Zbl 0643.46028
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