# Riesz Basis Generation: Comparison Method

Chapter
Part of the Communications and Control Engineering book series (CCE)

## Abstract

This chapter provides a panoramic view on the comparison method. It discusses systematically how the comparison method is used to derive the Riesz basis generation for systems described by partial differential equations. A basic assumption for comparison method to be working is that the feedback can be considered as a perturbation of the system itself, that is, the order of the feedback is lower than the order of original system, which is clear from the spectrum or transfer function point of view. It starts with a constant beam equation with collocated boundary feedback control, and then the beam equation with variable coefficients. A beam equation with span pointwise control is presented to show its Riesz basis and exponential stability. A one-dimensional thermoelastic system is fully discussed. The Riesz basis property has been developed for these systems, which implies particularly that the dynamics of the system is completely determined by vibration frequencies. Mathematically, all the operators are of compact resolvent. In the last section, however, an example of the Boltzmann integral model is presented where the resolvent is not compact and the continuous spectrum exists. Two different types of Boltzmann integrals for the dynamics of vibrating systems are discussed and the Riesz basis property has been developed.

## References

1. Bilalov BT (2003) Bases of exponentials, cosines, and sines formed by eigenfunctions of differential operators. Differ Equ 39:652–657
2. Chen G, Krantz SG, Russell DL, Wayne CE, West HH, Coleman MP (1989) Analysis, designs, and behavior of dissipative joints for coupled beams. SIAM J Appl Math 49:1665–1693
3. Chen X, Chentouf B, Wang JM (2014) Nondissipative torque and shear force controls of a rotating flexible structure. SIAM J Control Optim 52:3287–3311
4. Chentouf B, Wang JM (2008) A Riesz basis methodology for proportional and integral output regulation of a one-dimensional diffusive-wave equation. SIAM J Control Optim 47:2275–2302
5. Chentouf B, Wang JM (2009) Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system with $$L^\infty$$-coefficients. J Differ Equ 246:1119–1138Google Scholar
6. Conrad F (1990) Stabilization of beams by pointwise feedback control. SIAM J Control Optim 28:423–437
7. Conrad F, Mörgül O (1998) On the stabilization of a flexible beam with a tip mass. SIAM J Control Optim 36:1962–1986
8. Conrad F, Saouri FZ (2002) Stabilization of a beam: study of the optimal decay rate of elastic energy. ESAIM Control Optim Calc Var 7:567–595
9. Guilliemin EA (1957) Synthesis of passive networks. Wiley, New YorkGoogle Scholar
10. Guo BZ (1999) On the exponential stability of $$C_0$$-semigroups on Banach spaces with compact perturbations. Semigroup Forum 59:190–196Google Scholar
11. Guo BZ (2001) Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J Control Optim 39:1736–1747
12. Guo BZ (2002a) Further results for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions. ANZIAM J 43:449–462Google Scholar
13. Guo BZ (2002b) On the boundary control of a hybrid system with variable coefficients. J Optim Theory Appl 114:373–395Google Scholar
14. Guo BZ (2002c) Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. SIAM J Control Optim 40:1905–1923Google Scholar
15. Guo BZ, Wang JM (2006) Riesz basis generation of an abstract second-order partial differential equation system with general non-separated boundary conditions. Numer Funct Anal Optim 27:291–328
16. Guo BZ, Xu GQ (2004) Riesz bases and exact controllability of $$C_0$$-groups with one-dimensional input operators. Syst Control Lett 52:221–232Google Scholar
17. Guo BZ, Yu R (2001) The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control. IMA J Math Control Inform 18:241–251
18. Guo BZ, Yung SP (1997) The asymptotic behavior of the eigenfrequency of a one-dimensional linear thermoelastic system. J Math Anal Appl 213:406–421
19. Guo BZ, Zhang GD (2012) On spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping. ESAIM Control Optim Calc Var 18:889–913
20. Hardy GH, Wright EM (1979) Introduction to the theory of numbers, 5th edn. The Clarendon Press, New YorkGoogle Scholar
21. Ho LF (1984) Uniform basis properties of exponential solutions of functional differential equations of retarded type. Proc R Soc Edinb Sect A 96:79–94
22. Ho LF (1986) Spectral assignability of systems with scalar control and application to a degenerate hyperbolic system. SIAM J Control Optim 24:1212–1231
23. Ho LF (1993) Controllability and stabilizability of coupled strings with control applied at the coupled points. SIAM J Control Optim 31:1416–1437
24. Ho LF, Russell DL (1983) Admissible input elements for systems in Hilbert space and a Carleson measure criterion. SIAM J Control Optim 21:614–640
25. Özkan OA, Hansen SW (2011) Exact controllability of a Rayleigh beam with a single boundary control. Math Control Signals Syst 23:199–222
26. Özkan OA, Hansen SW (2014) Exact boundary controllability results for a multilayer Rao-Nakra sandwich beam. SIAM J Control Optim 52:1314–1337
27. Li SJ, Yu J, Liang Z, Zhu G (1999) Stabilization of high eigenfrequencies of a beam equation with generalized viscous damping. SIAM J Control Optim 37:1767–1779
28. Naimark MA (1967) Linear differential operators, vol I. Ungar, New YorkGoogle Scholar
29. Pandolfi L (2009) Riesz systems and controllability of heat equations with memory. Integr Equ Oper Theory 64:429–453
30. Pandolfi L (2010) Riesz systems and moment method in the study of viscoelasticity in one space dimension. Discret Contin Dyn Syst Ser B 14:1487–1510
31. Rao BP (1995) Uniform stabilization of a hybrid system of elasticity. SIAM J Control Optim 33:440–454
32. Rao BP (1997) Optimal energy decay rate in a damped Rayleigh beam. Contemporary mathematics, vol 209. American Mathematical Society, RI, Providence, pp 221–229Google Scholar
33. Rebarber R (1989a) Spectral assignability for distributed parameter systems with unbounded scalar control. SIAM J Control Optim 27:148–169Google Scholar
34. Rebarber R (1989b) Spectral determination for a cantilever beam. IEEE Trans Autom Control 34:502–510Google Scholar
35. Rebarber R (1995) Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J Control Optim 33:1–28
36. Russell DL (1978) Controllability and stabilizability theory for linear PDE’s: recent progress and open questions. SIAM Rev 20:639–739
37. Shkalikov AA (1986) Boundary problems for ordinary differential equations with parameter in the boundary conditions. J Sov Math 33:1311–1342
38. Shubov MA (1996) Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string. Integr Equ Oper Theory 25:289–328
39. Shubov MA (1997) Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string. Trans Am Math Soc 349:4481–4499
40. Shubov MA (1999) Spectral operators generated by Timoshenko beam model. Syst Control Lett 38:249–258
41. Shubov MA (2006a) Generation of Gevrey class semigroup by non-selfadjoint Euler-Bernoulli beam model. Math Methods Appl Sci 29:2181–2199Google Scholar
42. Shubov MA (2006b) Riesz basis property of mode shapes for aircraft wing model (subsonic case). Proc R Soc Lond Ser A Math Phys Eng Sci 462:607–646Google Scholar
43. Shubov MA (2017) Spectral analysis of a non-selfadjoint operator generated by an energy harvesting model and application to an exact controllability problem. Asymptot Anal 102:119–156
44. Tretter C (2000a) Linear operator pencils $$A-\lambda B$$ with discrete spectrum. Integr Equ Oper Theory 37:357–373Google Scholar
45. Tretter C (2000b) Spectral problems for systems of differential equations $$y^{\prime } +A_0 y=\lambda A_1 y$$ with $$\lambda$$-polynomial boundary conditions. Math Nachr 214:129–172Google Scholar
46. Tretter C (2001) Boundary eigenvalue problems for differential equations $$N\eta =\lambda P \eta$$ and $$\lambda$$-polynomial boundary conditions. J Differ Equ 170:408–471Google Scholar
47. Tucsnak M, Weiss G (2009) Observation and control for operator semigroups. Birkhäuser, Basel
48. Wang JM, Guo BZ (2006) On the stability of swelling porous elastic soils with fluid saturation by one internal damping. IMA J Appl Math 71:565–582
49. Wang JM, Guo BZ (2008) Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network. Math Methods Appl Sci 31:289–314
50. Wang JM, Guo BZ, Chentouf B (2006) Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach. ESAIM Control Optim Calc Var 12:12–34
51. Wang JM, Krstic M (2015) Stability of an interconnected system of Euler-Bernoulli beam and heat equation with boundary coupling. ESAIM Control Optim Calc Var 21:1029–1052
52. Wang JM, Su L, Li HX (2015) Stabilization of an unstable reaction-diffusion PDE cascaded with a heat equation. Syst Control Lett 76:8–18
53. Wang JM, Xu GQ, Yung SP (2004) Exponential stability of variable coefficients Rayleigh beams under boundary feedback controls: a Riesz basis approach. Syst Control Lett 51:33–50
54. Wang JM, Xu GQ, Yung SP (2005a) Exponential stabilization of laminated beams with structural damping and boundary feedback controls. SIAM J Control Optim 44:1575–1597Google Scholar
55. Wang JM, Xu GQ, Yung SP (2005b) Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beam with indefinite damping. IMA J Appl Math 70:459–477Google Scholar
56. Weiss G (1989) Admissible observation operators for linear semigroups. Isr J Math 65:17–43
57. Weiss G, Xu CZ (2005) Spectral properties of infinite-dimensional closed-loop systems. Math Control Signals Syst 17:153–172
58. Xu CZ, Sallet G (1996) On spectrum and Riesz basis assignment of infinite-dimensional linear systems by bounded linear feedback. SIAM J Control Optim 34:521–541
59. Xu CZ, Weiss G (2011) Eigenvalues and eigenvectors of semigroup generators obtained from diagonal generators by feedback. Commun Inf Syst 11:71–104
60. Yao CZ, Guo BZ (2003) Pointwise measure, control and stabilization of elastic beams. Control Theory Appl 20(3):351–360. (in Chinese)Google Scholar