Hansen, Scott W. Bounds on functions biorthogonal to sets of complex exponentials. Control of damped elastic systems. (English) Zbl 0742.93004 J. Math. Anal. Appl. 158, No. 2, 487-508 (1991). The paper is devoted to the problem of obtaining the explicit bounds on the norms of biorthogonal functions to sets of complex exponentials \(\exp(-\lambda_ kt)\), where \(\{\lambda_ k\}\) belong to a sector \(\{\lambda\in C: | \arg \lambda|\leq \theta\}\). The bounds obtained are uniform within a class of sequences which have some growth and separation properties. The main theorem leads to some results concerning the solvability of moment problems arising in control theory. As an application an exact null controllability result for a special system (a structurally damped Euler-Bernoulli plate) is proposed. Reviewer: O.I.Nikonov (Sverdlovsk) Cited in 23 Documents MSC: 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 93C05 Linear systems in control theory Keywords:solvability of moment problems; null controllability; structurally damped Euler-Bernoulli plate PDFBibTeX XMLCite \textit{S. W. Hansen}, J. Math. Anal. Appl. 158, No. 2, 487--508 (1991; Zbl 0742.93004) Full Text: DOI References: [1] Chen, G.; Russell, D. L., A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. (Jan. 1982) [2] Fattorini, H. O.; Russell, D. L., Exact controllability theorems for linear parabolic problems in one space dimension, Arch. Rational Mech. Anal., 4, 272-292 (1971) · Zbl 0231.93003 [3] Fattorini, H. O.; Russell, D. L., Uniform bounds on biorthogonal functions to real exponentials with an application to control of parabolic equations, Quart. Appl. Math., 32, 45-69 (1974) · Zbl 0281.35009 [4] Hansen, S., Frequency Proportional Damping Models for the Euler-Bernoulli Beam Equation, (Ph.D. thesis (Dec. 1988), Univ. Wisconsin: Univ. Wisconsin Madison, WI) [5] Hille, E., (Analytic Function Theory, Vol. II (1962), Ginn & Co.,: Ginn & Co., New York) [6] Ho, L. F.; Russell, D. L., Admissible input elements for systems on Hilbert space and a Carleson measure criterion, Siam J. Control Optim., 21, 614-639 (1983) · Zbl 0512.93044 [7] Kaczmarz, S.; Steinhaus, H., Theorie der Orthogonalreihen, Monografje Matematyczne, Tom VI (1935), Warszawa · JFM 61.1119.05 [8] Krabs, W.; Leugering, G.; Seidman, T. I., On boundary controllability of a vibrating plate, Appl. Math. Optim., 13, 205-229 (1985) · Zbl 0596.49025 [9] Luxemburg, W. A.J; Korevaar, J., Entire functions and Muntz-Szasz type approximation, Trans. Amer. Math. Soc., 156 (1971) · Zbl 0224.30049 [10] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023 [11] Russell, D. L., Mathematical models for the elastic beam and their control-theoretic implications, (Brezis; Crandall; Kappel, Pitman Res. Notes Math. Ser., Vol. 152 (1986), Longman Sci. Tech.,: Longman Sci. Tech., Harlow), 177-216 [12] Russell, D. L., Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20, 700-709 (1978) [13] Redheffer, R. M., Completeness of sets of complex exponentials, Adv. in Math., 24, 1-62 (1977) · Zbl 0358.42007 [14] Schwartz, L., Étude des sommes d’exponentielles (1959), Hermann: Hermann Paris · Zbl 0092.06302 [15] Weiss, G., Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989) · Zbl 0685.93043 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.