半线性板方程的一类能控性性问题
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摘要
本文是对于半线性板方程的一类能控性问题的研究,半线性板方程是有深刻物理背景的系统,长期以来受到许多数学家及物理学家的关注。在这方面已取得了一些结果,其中包括这类系统的适定性结果及一些零散的能控性结果。作者是在前人的基础之上,研究了一类能控性问题,并取得了一些结果。
全文大体分三章:
第一章我们大体介绍了本文用到的结果及这类系统边界控制问题的相关已知结果,另外我们还介绍了我们在后面证明中用到的一个关键的定理。
第二章讨论了半线性板方程单变量边界精确能控性问题。首先我们把方程转化为抽象方程,并在抽象系统的框架下,结合[1]中定理,把原系统的能控性问题转化为其线性化系统的对偶系统的一个能观性问题。得到了我们想要的结果。
第三章讨论了半线性板方程Neumann边界精确能控性结果。大体方法类似于第一章的处理手法,但这一问题讨论的难度及讨论的必要性显然要比第一个问题大的多。因为这一边界条件显然来的更自然一些。
In this paper we consider a kind of controllability problems about semilinear plate equa-tion,which has profoundly physical background and has been paid much attention by many mathematicians and physical scientists for a long time. There have been some results in the search field including properties and some controllability about this system. Based on these results we study a kind of controllability problems and obtain some satisfying results.The paper is composed of the following three sections.In section l,we generally introduce the known correlative results about the problems of boundary controllability.And we refer to the key theorem we use in my paper.In section 2,we analyze exact controllability problem about semilinear plate equation with one-variable boundary. At first, we transform the eqations into the abstract equation,under the framework of abstract equations,combining the theorems in [l],we convert the controllability problems in original system.Finally using the method in [8],we study observability problems and obtain the desired results.In section 3 ,we analyze the exact controllability problem about semilinear plate equation with Neumann-boundary. The martingale is similar with one used in section 2. Obviously compared with the first problem the difficulty is much larger,and it is pretty necessary because this boundary condition is more spontaneous.
全文大体分三章:
第一章我们大体介绍了本文用到的结果及这类系统边界控制问题的相关已知结果,另外我们还介绍了我们在后面证明中用到的一个关键的定理。
第二章讨论了半线性板方程单变量边界精确能控性问题。首先我们把方程转化为抽象方程,并在抽象系统的框架下,结合[1]中定理,把原系统的能控性问题转化为其线性化系统的对偶系统的一个能观性问题。得到了我们想要的结果。
第三章讨论了半线性板方程Neumann边界精确能控性结果。大体方法类似于第一章的处理手法,但这一问题讨论的难度及讨论的必要性显然要比第一个问题大的多。因为这一边界条件显然来的更自然一些。
In this paper we consider a kind of controllability problems about semilinear plate equa-tion,which has profoundly physical background and has been paid much attention by many mathematicians and physical scientists for a long time. There have been some results in the search field including properties and some controllability about this system. Based on these results we study a kind of controllability problems and obtain some satisfying results.The paper is composed of the following three sections.In section l,we generally introduce the known correlative results about the problems of boundary controllability.And we refer to the key theorem we use in my paper.In section 2,we analyze exact controllability problem about semilinear plate equation with one-variable boundary. At first, we transform the eqations into the abstract equation,under the framework of abstract equations,combining the theorems in [l],we convert the controllability problems in original system.Finally using the method in [8],we study observability problems and obtain the desired results.In section 3 ,we analyze the exact controllability problem about semilinear plate equation with Neumann-boundary. The martingale is similar with one used in section 2. Obviously compared with the first problem the difficulty is much larger,and it is pretty necessary because this boundary condition is more spontaneous.
引文
[1] I.Laciecka and R.Triggiani, Exact Controllability of semilinear Absrtact systems with application to waves and plates boundary control problems,Appl.Math.Optim.1991,23:109-154.
[2] I.Laciecka and R.Triggiani, Regularity of hyperbolic equations under L~2 (O, T; L~2 (Γ)) boundary terms,Appl.Math.Optim.1983,10:275-286.
[3] I.Laciecka and R.Triggiani, Exact controliability and uniform stabilization of Euler-Bernouli equations with only one active control in Δω|Σ, Boll.Un.Mat.Ital.to appear.short announcement in proceeding of Uorau Conference,July 1988,International Serise of Numerical Mathematics.Vol.91,Birkhauser,Bash,391-400;and Rendiconti Accad. Nazionali Lincei,Roma.
[4] I.Laciecka and R.Triggiani, Exact controllability of the Euler-Bernouli equation with controls in the Dirichlet and neumann boundary conditions:A nonconservativc caze, SIAM.J.Control.and Optim.1989 Vol.27,No.2.330-373.
[5] J.L.Lions and E.Magenes,Nonhomogenous boundary value problems and applications,Vols Ⅰ and Ⅱ. Springer-Verlag, Berlin 1972.
[6] 周鸿兴,王连文,《线性算子半群理论及应用》山东科学出版社,1994.
[7] I.Laciecka and R.Triggiani, Regularity theory for a class of Euler-Bernouli equations a cosine operator approach,Boll.Un.Mat.Ital.B(7)3,1989, 199-228.
[8] 张旭,《半线性分布参数系统的精确能控性》高等教育出版社2004.
[9] Lebeau G, Conrtole de l'equation de schrodinger,J.Math.Pures Appl.1992,71:267-291.
[10] Li L,Zhang X,Exact controllability for semilinear wave equations, J.Math.Anal.Appl.2000,250 597.
[11] Liu W,Locally boundary controllability for the semilinear plate equation,Comm.PDE 1998, 23:201-221.
[12] Pazy A,Semigroup of linear operators and applications to PDE, New York:Springerverlay, 1983.
[13] Tataru D,Boundary controllability for conservative PDEs, Appl.Math.Optim.1995,31:257-295.
[14]Tataru D,Boundary observability and controllability for evolutions governed by higher order PDEs, J.Math.Anal.Appl.1995,193:632-658.
[15]ZhangX,Exact controllability of the semilinear plate equtions, Asymptotic.Analysis.2001,27:95-125.
[16]ZhangX,ZuaZua.Exact controllability of the semilinear wave equation,In:Blondel VD,Megretski A,ed.Sixty open problems in the mathematics of systems and control. Princeton,Princeton University Press.to appear.
[17]I.Laciecka and R.Triggiani,Exact controllability of Euler-Bernouli equations with boundary controls for displacement and moments , J.Math.Anal.Appl.145,1990.1-33.
[18]R.Triggiani,Conrtollablity and observability in Banach space with bounded operators, SIAM.J.Conrtol.Optim.13,1975,462-491.
[19]Weijiu Liu,Local boundary controllability for the semilinear plate eqution.Commun,in PDE,1998.23,201-221.
[20]J.L.Lions,Controllability exacte perturbations et stablisation de systemes disturbues.,Tomel, Controllability Exacte, Masson, paris,1988.
[21]J.L.Lions ,Exact Controllability,Stabilization and Perturbations, SIAM Rev.,30(1988),pp. 1-68.
[22]V.Komornik,Controlabiliteexacte en un temps minimal, C.R. Acad. Sci. Paris Ser. I Math.,304(1987).
[2] I.Laciecka and R.Triggiani, Regularity of hyperbolic equations under L~2 (O, T; L~2 (Γ)) boundary terms,Appl.Math.Optim.1983,10:275-286.
[3] I.Laciecka and R.Triggiani, Exact controliability and uniform stabilization of Euler-Bernouli equations with only one active control in Δω|Σ, Boll.Un.Mat.Ital.to appear.short announcement in proceeding of Uorau Conference,July 1988,International Serise of Numerical Mathematics.Vol.91,Birkhauser,Bash,391-400;and Rendiconti Accad. Nazionali Lincei,Roma.
[4] I.Laciecka and R.Triggiani, Exact controllability of the Euler-Bernouli equation with controls in the Dirichlet and neumann boundary conditions:A nonconservativc caze, SIAM.J.Control.and Optim.1989 Vol.27,No.2.330-373.
[5] J.L.Lions and E.Magenes,Nonhomogenous boundary value problems and applications,Vols Ⅰ and Ⅱ. Springer-Verlag, Berlin 1972.
[6] 周鸿兴,王连文,《线性算子半群理论及应用》山东科学出版社,1994.
[7] I.Laciecka and R.Triggiani, Regularity theory for a class of Euler-Bernouli equations a cosine operator approach,Boll.Un.Mat.Ital.B(7)3,1989, 199-228.
[8] 张旭,《半线性分布参数系统的精确能控性》高等教育出版社2004.
[9] Lebeau G, Conrtole de l'equation de schrodinger,J.Math.Pures Appl.1992,71:267-291.
[10] Li L,Zhang X,Exact controllability for semilinear wave equations, J.Math.Anal.Appl.2000,250 597.
[11] Liu W,Locally boundary controllability for the semilinear plate equation,Comm.PDE 1998, 23:201-221.
[12] Pazy A,Semigroup of linear operators and applications to PDE, New York:Springerverlay, 1983.
[13] Tataru D,Boundary controllability for conservative PDEs, Appl.Math.Optim.1995,31:257-295.
[14]Tataru D,Boundary observability and controllability for evolutions governed by higher order PDEs, J.Math.Anal.Appl.1995,193:632-658.
[15]ZhangX,Exact controllability of the semilinear plate equtions, Asymptotic.Analysis.2001,27:95-125.
[16]ZhangX,ZuaZua.Exact controllability of the semilinear wave equation,In:Blondel VD,Megretski A,ed.Sixty open problems in the mathematics of systems and control. Princeton,Princeton University Press.to appear.
[17]I.Laciecka and R.Triggiani,Exact controllability of Euler-Bernouli equations with boundary controls for displacement and moments , J.Math.Anal.Appl.145,1990.1-33.
[18]R.Triggiani,Conrtollablity and observability in Banach space with bounded operators, SIAM.J.Conrtol.Optim.13,1975,462-491.
[19]Weijiu Liu,Local boundary controllability for the semilinear plate eqution.Commun,in PDE,1998.23,201-221.
[20]J.L.Lions,Controllability exacte perturbations et stablisation de systemes disturbues.,Tomel, Controllability Exacte, Masson, paris,1988.
[21]J.L.Lions ,Exact Controllability,Stabilization and Perturbations, SIAM Rev.,30(1988),pp. 1-68.
[22]V.Komornik,Controlabiliteexacte en un temps minimal, C.R. Acad. Sci. Paris Ser. I Math.,304(1987).