An algebraic method for boundary control systems of parabolic type

抛物型边界控制系统的代数方法

• 批准号: 10640207
• 负责人: NAMBU Takao
• 金额: $ 1.54万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640207/
• 关键词: boundary control; algebraic transform; linear parabolic systems; null controllability; HィイD1∞ィエD1-control; moving sphere method; population dynamics; dynamic compensator; boundary control systems; feedback control; algebraic method; spectrum as

1.Stabilization of linear parabolic systems by means of boundary feedback is studied. The boundary condition is partially of the first kind and partially of the third kind. An entirely new and simple algebraic transform (isomorphism in LィイD12ィエD1-spaces) is introduced for the complicated boundary condition. By introducing a finite-dimensional dynamic compensator of general type, stabilization is achieved for more complicated control systems. An algebraic structure of the so called Silvester equation with unbounded linear operators as coefficients is also studied.2.Approximate null controllability problem is studied for a class of linear second order parabolic systems in Hilbert spaces by means of the so called HUM method. A numerical approximation of solutions to the sine-Gordon equation is also studied by means of FEM.3.An integro-partial differential equation is established as a mathematical model for studing logistic growth of human population with migration. It is proven that the Cauchy problem admits the unique global solution in time. The asymptotic behavior of the solutions is also investigated. The property of solutions is used for explaining about the recent change of labor dynamics in European Continent.4.A class of nonlinear elliptic partial differential equations is studied, where the appearing coefficients and the spatial domain has a kind of symmetry. Existence and uniqueness of solutions are proven via ODE approach. A generalization of the so called moving sphere method is also obtained.5.Stabilization problem is studied for linear parabolic control systems by means of HィイD1∞ィエD1-control method. Robustness of the stabilizing feedback control scheme is proven in some topology. Another stabilization problem is also studied, where the sensors and actuators are periodic functions in time. By generalizing the result of Brunovsky (JDE, 1969), the stabilization is achieved.

1.研究了边界反馈稳定线性抛物线系统。边界条件部分为第一类，部分为第三类。针对复杂的边界条件，引入了一种全新且简单的代数变换（LィイD12ィエD1-空间中的同构）。通过引入通用类型的有限维动态补偿器，可以实现更复杂的控制系统的稳定性。还研究了以无界线性算子为系数的Silvester方程的代数结构。2.利用HUM方法研究了Hilbert空间中一类线性二阶抛物型系统的近似零可控性问题。利用有限元法研究了正弦-戈登方程解的数值逼近。3.建立了积分偏微分方程作为研究人口迁移过程中Logistic增长的数学模型。证明柯西问题及时承认全局唯一解。还研究了解的渐近行为。利用解的性质来解释欧洲大陆劳动力动态的近期变化。4.研究了一类非线性椭圆偏微分方程，其出现系数与空间域具有一定的对称性。通过ODE方法证明解的存在性和唯一性。并得到了动球法的推广。5.利用HィイD1∞ィエD1控制方法研究了线性抛物线控制系统的稳定性问题。稳定反馈控制方案的鲁棒性在某些拓扑中得到了证明。还研究了另一个稳定性问题，其中传感器和执行器是时间的周期函数。通过推广 Brunovsky (JDE, 1969) 的结果，实现了稳定性。

项目成果

S. Nakagiri: "Identifiability problems in distributed parameter systems"Nonlinear Functional Analysis and Applications. vol. 3. 135-150 (1998)

S. Nakagiri：“分布式参数系统中的可识别性问题”非线性泛函分析和应用。

T.Nambu: "A note on algebraic aspects of boundary feedback control systems of parabolic type"Proc. Japan Acad. Ser. A Math Sci.. vol.75, no.7. 137-140 (1999)

T.Nambu：“关于抛物型边界反馈控制系统的代数方面的说明”Proc。

J.Vanualailai, J.Ha, and S.Nakagiri: "A solution to the two-dimensional findpath problem"Dynamics and Stability. vol.13. 373-401 (1998)

J.Vanualailai、J.Ha 和 S.Nakagiri：“二维查找路径问题的解决方案”动力学和稳定性。

H.Sano and Y.Sakawa: "H control of diffusion systems by using a finite-dimensional controller"SIAM J.Control and Optimization. vol.37. 409-428 (1999)

H.Sano 和 Y.Sakawa：“使用有限维控制器对扩散系统进行 H 控制”SIAM J.控制与优化。

M.Tabata, N.Eshima, and I.Takagi: "The nonlinear integro-partial differential equation describing the logistic growth of human population with migration"Applied Mathematics and Applications. vol.98. 169-183 (1999)

M.Tabata、N.Eshima 和 I.Takagi：“描述人口随迁移的逻辑增长的非线性积分偏微分方程”应用数学与应用。