Exact controllability of a structural acoustic system with variable coefficient and curved wall
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- 英文论文题名：Exact controllability of a structural acoustic system with variable coefficient and curved wall
- 论文作者：Fengyan Yang ; Pengfei Yao
- 英文论文作者：Fengyan Yang ; Pengfei Yao ; Key Laboratory of Systems and Control ; Academy of Mathematics and Systems Science ; Chinese Academy of Sciences
- 年：2017
- 作者机构：Key Laboratory of Systems and Control,Academy of Mathematics and Systems Science,Chinese Academy of Sciences;
- 英文论文关键词：exact controllability ; structural acoustic system ; variable coefficient ; curved wall ; Riemannian geometry
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：O42
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：143k
- 原文格式：D
- 会议级别：全国
摘要
This paper is concerned with the exact controllability of a structural acoustic system with two controls, which consists of an interior acoustic wave equation with variable coefficient and a coupled Kirchoff plate with a curved middle surface. We use the Riemannian geometrical approach and the multiplier technique to prove the corresponding observability inequality. And the exact controllability of this system is finally established under verifiable assumptions on the geometry of the interior domain and the interface.
This paper is concerned with the exact controllability of a structural acoustic system with two controls, which consists of an interior acoustic wave equation with variable coefficient and a coupled Kirchoff plate with a curved middle surface. We use the Riemannian geometrical approach and the multiplier technique to prove the corresponding observability inequality. And the exact controllability of this system is finally established under verifiable assumptions on the geometry of the interior domain and the interface.
This paper is concerned with the exact controllability of a structural acoustic system with two controls, which consists of an interior acoustic wave equation with variable coefficient and a coupled Kirchoff plate with a curved middle surface. We use the Riemannian geometrical approach and the multiplier technique to prove the corresponding observability inequality. And the exact controllability of this system is finally established under verifiable assumptions on the geometry of the interior domain and the interface.
引文
[1]B.Allibert and S.Micu,Controllability of analytic functions for a wave equation coupled with a beam,Rev.Mat.Iberoamericana,15(3):547-592,1999.
[2]G.Avalos and I.Lasiecka,Differential Riccati equation for the active control of a problem in structural acoustics,J.Optim.Theory Appl.,91(3):695-728,1996.
[3]G.Avalos and I.Lasiecka,Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls,Adv.Differential Equations,10(8):901-930,2005.
[4]G.Avalos,I.Lasiecka,R.Rebarber,Well-posedness of a structural acoustics control model with point observation of the pressure,J.Differential Equations,173(1):40-78,2001.
[5]G.Avalos,I.Lasiecka,R.Rebarber,Boundary controllability of a coupled wave/Kirchoff system,Systems Control Lett.,50(5):331-341,2003.
[6]H.T.Banks,W.Fang,R.J.Silcox,R.C.Smith,Approximation methods for control of acoustic/structure models with piezoceramic actuators,Contract Report 189578,NASA,1991.
[7]H.T.Banks and R.C.Smith,Feedback control of noise in a 2Dnonlinear structural acoustics model,Discrete Contin.Dynam.Systems,1(1):119-149,1995.
[8]J.Cagnol and C.Lebiedzik,Optimal control of a structural acoustic model with flexible curved walls,in Control and Boundary Analysis,Lecture Notes in Pure and Applied Mathematics(239),New York:Marcel Dekker,1995:37-52.
[9]M.Camurdan,Uniform stabilization of a coupled structural acoustic system by boundary dissipation,Abstr.Appl.Anal.,3(3-4):377-400,1998.
[10]M.Camurdan and R.Triggiani,Sharp regularity of a coupled system of a wave and Kirchoff equation with point control arising in noise reduction,Differential Integral Equations.,12(1):101-118,1999.
[11]X.M.Cao,Energy decay of solutions for a variablecoefficient viscoelastic wave equation with a weak nonlinear dissipation.J.Math.Phys.,57(2):1-16,2016.
[12]S.Dolecki and D.L.Russell,A general theory of observation and control,SIAM J.Control Optim.,15(2):185-220,1977.
[13]M.Grobbelaar-Van Dalsen,On a structural acoustic model with interface a Reissner-Mindlin plate or a Timishenko beam,J.Math.Anal.Appl.,320(1):121-144,2006.
[14]Y.X.Guo and P.F.Yao,Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback,J.Math.Anal.Appl.,317(1):50-70,2006.
[15]I.Lasiecka and R.Triggiani,Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control with no geometrical conditions,Appl.Math.Optim.,25(2):189-224,1992.
[16]I.Lasiecka and R.Triggiani,Recent advances in regularity of second-order hyperbolic mixed problems,and applications,in Dynamics Reported:Expositions in Dynamical Systems,Vol.3,New York:Springer-Verlag,1994:104-158.
[17]I.Lasiecka,R.Triggiani,X.Zhang,Nonconservative wave equations with unobserved Neumann B.C.:global uniqueness and observability on one shot,in Differential geometric methods in the control of partial differential equations,Contemp.Math.268,Providence,RI:Amer.Math.Soc.,2000:227-325.
[18]C.Lebiedzik,Uniform stability of a coupled structural acoustic system with thermoelastic effects,Dynamics of Continuous,Discrete,and Impulsive Systems,7(3):369-383,2000.
[19]B.Liu and W.Littman,On the spectral properties and stabilization of acoustic flow,SIAM J.Appl.Math.,59(1):7-34,1998.
[20]Y.X.Liu,B.Bin-Mohsin,H.Hajaiej,P.F.Yao,G.Chen,Exact controllability of structural acoustic interactions with variable coefficients,SIAM J.Control Optim.,54(4):2132-2153,2016.
[21]L.Q.Lu,S.J.Li,G.Chen,P.F.Yao,Control and stabilization for the wave equation with variable coefficients in domains with moving boundary,Systems Control Lett.,80:30-41,2015.
[22]S.Micu and E.Zuazua,Boundary controllability of a linear hybrid system arising in the control of noise,SIAM J.Control Optim.,35(5):1614-1637,1997.
[23]S.Miyatake,Mixed problem for hyperbolic equation of second order,J.Math.Kyoto Univ.,13(3):435-487,1973.
[24]Z.H.Ning,Q.X.Yan,Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback,J.Math.Anal.Appl.,367(1):167-173,2010.
[25]K.Seto,M.Z.Ren,F.Doi,Modeling and feedback structural acoustics control of a flexible plate,Journal of Vibration and Acoustics,123(1):18-23,2001.
[26]J.D.Sprofera,R.H.Cabell,G.P.Gibbs,R.L.Clark,Structural acoustic control of plates with variable boundary conditions:Design methodology,Journal of the Acoustical Society of America,122(1):271-279,2007.
[27]J.Q.Wu,F.Feng,S.G.Chai,Energy decay of a variablecoefficient wave equation with nonlinear time-dependent localized damping,Electron.J.Differential Equations,226:1-11,2015.
[28]F.Y.Yang,Exact controllability of the Euler-Bernoulli plate with variable coefficients and simply supported boundary condition,Electron.J.Differential Equations,257:1-19,2016.
[29]F.Y.Yang,B.Bin-Mohsin,G.Chen,P.F.Yao,Exactapproximate boundary controllability of the thermoelastic plate with a curved middle surface.J.Math.Anal.Appl.,451(2017):405-433.
[30]P.F.Yao,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37(5):1568-1599,1999.
[31]P.F.Yao,Boundary controllability for the quasilinear wave equation.Appl.Math.Optim.,61(2):191-233,2010.
[32]P.F.Yao,Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation,Chin.Ann.Math.Ser.B,31(1):59-70,2010.
[33]P.F.Yao,Modeling and Control in Vibrational and Structual Dynamics-A Differential Geometric Approach,in:Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series.Boca Raton,FL:CRC Press,2011.
[34]Z.F.Zhang,P.F.Yao,Global smooth solutions of the quasilinear wave equation with internal velocity feedback,SIAM J.Control Optim.,47(4):2044-2077,2008.
[35]Y.l.Wei,J.B.Qiu,H.R.Karimi,M.Wang.New results on H∞dynamic output feedback control for Markovian jump systems with time-varying delay and defective mode information,Optimal Control,Applications and Methods,35(6):656-675,2014.
[36]Y.l.Wei,J.B.Qiu,S.S.Fu.Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay,Nonlinear Analysis:Hybrid Systems,16:52-71,2015.
[2]G.Avalos and I.Lasiecka,Differential Riccati equation for the active control of a problem in structural acoustics,J.Optim.Theory Appl.,91(3):695-728,1996.
[3]G.Avalos and I.Lasiecka,Exact controllability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls,Adv.Differential Equations,10(8):901-930,2005.
[4]G.Avalos,I.Lasiecka,R.Rebarber,Well-posedness of a structural acoustics control model with point observation of the pressure,J.Differential Equations,173(1):40-78,2001.
[5]G.Avalos,I.Lasiecka,R.Rebarber,Boundary controllability of a coupled wave/Kirchoff system,Systems Control Lett.,50(5):331-341,2003.
[6]H.T.Banks,W.Fang,R.J.Silcox,R.C.Smith,Approximation methods for control of acoustic/structure models with piezoceramic actuators,Contract Report 189578,NASA,1991.
[7]H.T.Banks and R.C.Smith,Feedback control of noise in a 2Dnonlinear structural acoustics model,Discrete Contin.Dynam.Systems,1(1):119-149,1995.
[8]J.Cagnol and C.Lebiedzik,Optimal control of a structural acoustic model with flexible curved walls,in Control and Boundary Analysis,Lecture Notes in Pure and Applied Mathematics(239),New York:Marcel Dekker,1995:37-52.
[9]M.Camurdan,Uniform stabilization of a coupled structural acoustic system by boundary dissipation,Abstr.Appl.Anal.,3(3-4):377-400,1998.
[10]M.Camurdan and R.Triggiani,Sharp regularity of a coupled system of a wave and Kirchoff equation with point control arising in noise reduction,Differential Integral Equations.,12(1):101-118,1999.
[11]X.M.Cao,Energy decay of solutions for a variablecoefficient viscoelastic wave equation with a weak nonlinear dissipation.J.Math.Phys.,57(2):1-16,2016.
[12]S.Dolecki and D.L.Russell,A general theory of observation and control,SIAM J.Control Optim.,15(2):185-220,1977.
[13]M.Grobbelaar-Van Dalsen,On a structural acoustic model with interface a Reissner-Mindlin plate or a Timishenko beam,J.Math.Anal.Appl.,320(1):121-144,2006.
[14]Y.X.Guo and P.F.Yao,Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback,J.Math.Anal.Appl.,317(1):50-70,2006.
[15]I.Lasiecka and R.Triggiani,Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control with no geometrical conditions,Appl.Math.Optim.,25(2):189-224,1992.
[16]I.Lasiecka and R.Triggiani,Recent advances in regularity of second-order hyperbolic mixed problems,and applications,in Dynamics Reported:Expositions in Dynamical Systems,Vol.3,New York:Springer-Verlag,1994:104-158.
[17]I.Lasiecka,R.Triggiani,X.Zhang,Nonconservative wave equations with unobserved Neumann B.C.:global uniqueness and observability on one shot,in Differential geometric methods in the control of partial differential equations,Contemp.Math.268,Providence,RI:Amer.Math.Soc.,2000:227-325.
[18]C.Lebiedzik,Uniform stability of a coupled structural acoustic system with thermoelastic effects,Dynamics of Continuous,Discrete,and Impulsive Systems,7(3):369-383,2000.
[19]B.Liu and W.Littman,On the spectral properties and stabilization of acoustic flow,SIAM J.Appl.Math.,59(1):7-34,1998.
[20]Y.X.Liu,B.Bin-Mohsin,H.Hajaiej,P.F.Yao,G.Chen,Exact controllability of structural acoustic interactions with variable coefficients,SIAM J.Control Optim.,54(4):2132-2153,2016.
[21]L.Q.Lu,S.J.Li,G.Chen,P.F.Yao,Control and stabilization for the wave equation with variable coefficients in domains with moving boundary,Systems Control Lett.,80:30-41,2015.
[22]S.Micu and E.Zuazua,Boundary controllability of a linear hybrid system arising in the control of noise,SIAM J.Control Optim.,35(5):1614-1637,1997.
[23]S.Miyatake,Mixed problem for hyperbolic equation of second order,J.Math.Kyoto Univ.,13(3):435-487,1973.
[24]Z.H.Ning,Q.X.Yan,Stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback,J.Math.Anal.Appl.,367(1):167-173,2010.
[25]K.Seto,M.Z.Ren,F.Doi,Modeling and feedback structural acoustics control of a flexible plate,Journal of Vibration and Acoustics,123(1):18-23,2001.
[26]J.D.Sprofera,R.H.Cabell,G.P.Gibbs,R.L.Clark,Structural acoustic control of plates with variable boundary conditions:Design methodology,Journal of the Acoustical Society of America,122(1):271-279,2007.
[27]J.Q.Wu,F.Feng,S.G.Chai,Energy decay of a variablecoefficient wave equation with nonlinear time-dependent localized damping,Electron.J.Differential Equations,226:1-11,2015.
[28]F.Y.Yang,Exact controllability of the Euler-Bernoulli plate with variable coefficients and simply supported boundary condition,Electron.J.Differential Equations,257:1-19,2016.
[29]F.Y.Yang,B.Bin-Mohsin,G.Chen,P.F.Yao,Exactapproximate boundary controllability of the thermoelastic plate with a curved middle surface.J.Math.Anal.Appl.,451(2017):405-433.
[30]P.F.Yao,On the observability inequalities for exact controllability of wave equations with variable coefficients,SIAM J.Control Optim.,37(5):1568-1599,1999.
[31]P.F.Yao,Boundary controllability for the quasilinear wave equation.Appl.Math.Optim.,61(2):191-233,2010.
[32]P.F.Yao,Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation,Chin.Ann.Math.Ser.B,31(1):59-70,2010.
[33]P.F.Yao,Modeling and Control in Vibrational and Structual Dynamics-A Differential Geometric Approach,in:Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series.Boca Raton,FL:CRC Press,2011.
[34]Z.F.Zhang,P.F.Yao,Global smooth solutions of the quasilinear wave equation with internal velocity feedback,SIAM J.Control Optim.,47(4):2044-2077,2008.
[35]Y.l.Wei,J.B.Qiu,H.R.Karimi,M.Wang.New results on H∞dynamic output feedback control for Markovian jump systems with time-varying delay and defective mode information,Optimal Control,Applications and Methods,35(6):656-675,2014.
[36]Y.l.Wei,J.B.Qiu,S.S.Fu.Mode-dependent nonrational output feedback control for continuous-time semi-Markovian jump systems with time-varying delay,Nonlinear Analysis:Hybrid Systems,16:52-71,2015.