一个复合系统边界反馈的Riesz基性质

Abstract

Abstract:

An Euler-Bernoulli beam equation with one end fixed and a tip mass at another end is considered. It is shown that the closedloop system under boundary linear feedback control is a riesz spectral system. That is, there is a sequence of generalized eigenfunctions, which forms a riest basis for the state Hilberf space. The spectrum－determined growth condition is hence established. In the meanwhile, the asymptotic expansion of the eigenvolues is derived and the exponential seability of the system is concluded.

Key words: Distributed system, Riesz basis, Beam equation, Stability

CLC Number:

93C20，35B35.

Yao Cuizhen. On the Riest Basis Property of a Hybrid System with Boundary Feedback Control[J].Acta mathematica scientia,Series A, 2000, 20(4): 568-576.

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References

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２　ＣｈｅｎＧ，ＫｒａｎｔｚＳＧ，ＭａＤＷ，ＷａｙｎｅＣＥ．ＴｈｅＥｕｌｅｒＢｅｒｎｏｕｌｌｉｂｅａｍｅｑｕａｔｉｏｎｗｉｔｈｂｏｕｎｄａｒｙｅｎｅｒｇｙｄｉｓｓｉｐａｔｉｏｎ，ＯｐｅｒａｔｏｒＭｅｔｈｏｄｓｆｏｒＯｐｔｉｍａｌＣｏｎｔｒｏｌＰｒｏｂｌｅｍｓ（ＳＪＬｅｅｅｄ）．ＮｅｗＹｏｒｋ：ＭａｒｃｅｌＤｅｋｋｅｒ，１９８８，６７－９６
３　ＧｏｈｂｅｒｇＩＣ，ＫｒｅｉｎＭＧ．Ｉｎｔｒｏｄｕｃｔｉｏｎｔｏｔｈｅｔｈｅｏｒｙｏｆｌｉｎｅａｒｎｏｎｓｅｌｆａｄｊｏｉｎｔｏｐｅｒａｔｏｒｓ．ＴｒａｎｓａｃｔｉｏｎｏｆＭａｔｈｅｍａｔｉｃａｌＭｏｎｏｇｒａｐｈｓ，ＡＭＳＰｒｏｖｉｄｅｎｃｅ，１９６９．１８
４　ＧｕｏＢＺ．ＴｈｅＲｉｅｓｚｂａｓｉｓｐｒｏｐｅｒｔｙｏｆｄｉｓｃｒｅｔｅｏｐｅｒａｔｏｒｓａｎｄａｐｐｌｉｃａｔｉｏｎｔｏａＥｕｌｅｒＢｅｒｎｏｕｌｌｉｂｅａｍｅｑｕａｔｉｏｎｗｉｔｈｂｏｕｎｄａｒｙｌｉｎｅａｒｆｅｅｄｂａｃｋｃｏｎｔｒｏｌ．ＩＭＡＪＭａｔｈＣｏｎｔｒｏｌＩｎｆｏｒｍａｔｉｏｎ，ｔｏａｐｐｅａｒ
５　ＤｕｎｆｏｒｄＮ，ＳｃｈｗａｒｔｚＪＴ．ＬｉｎｅａｒＯｐｅｒａｔｏｒｓ．ＰａｒｔⅢ，ＷｉｌｅｙＩｎｔｅｒｓｃｉｅｎｃｅ，１９７１
６　ＬｕｏＺＨ，ＧｕｏＢＺ，ＭｏｒｇｕｌＯ．Ｓｔａｂｉｌｉｔｙａｎｄｓｔａｂｉｌｉｚａｔｉｏｎｏｆｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌｓｙｓｔｅｍｗｉｔｈａｐｐｌｉｃａｔｉｏｎｓ．ＳｐｒｉｎｇｅｒＶｅｒｌａｇ，１９９９

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On the Riest Basis Property of a Hybrid System with Boundary Feedback Control

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