Optimal Control Theory of Enler-BernouIIi Equation

恩勒-伯努利方程的最优控制理论

• 批准号: 12640154
• 负责人: TSUJIOKA Kunio
• 金额: $ 1.15万
• 依托单位: Saitama University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640154/
• 关键词: controllability; Euler-Bernoulli equation; singular boundary condition; wave equation; optimal control theory; viscositv solution; Hamilton-Jacobi equation; variational equation; 発展方程式; 有限伝播速度; 梁の方程式; 数理ファイナンス; 近似可制御性; オイラー・ビーム; Kuhn-Tucker

(1) In the first year of the term of the project, the head investigator researched controllability of a system coupled by two Euler- Bernoulli beams with control at the coupled point. First he studied asymptotic behavior of eigenvalues of a fourth order differential operator related to the Euler-BernouIIi equation. Then controllability problem is reduced a moment problem in a Hilbert space. The method to solve controllability by reducing to a moment problem is called moment problem method. Our ultimate object of the project is to study controllability of a system coupled several Euler-Bernoulli beams. However corresponding moment problem method is too complicated and does not work to our problem. So this problem is still open. In the second and third year of the term of the project, he turned to controllability of evolution equations with singular boundary condition. A boundary condition in an evolution equation is called singular if its order in spatial derivative is as same as that of the equation. We investigated controllability of wave equation with singular boundary condition using moment problem method. A. similar result will be obtained for Euler-Bernoulli equation with singular boundary condition.(2) Investigator Koike contributed greatly to optimal control theory related to viscosity solution. Various problems are proposed and solved by him.

(1)在第一年的项目，首席研究员研究了系统的可控性耦合的两个欧拉-伯努利梁与控制在耦合点。首先，他研究了渐近行为的特征值的四阶微分算子有关的欧拉-伯努利方程。然后将能控性问题归结为Hilbert空间中的矩问题.将能控性问题归结为矩量问题的方法称为矩量问题法。本课题的最终目的是研究多个Euler-Bernoulli梁耦合系统的可控性。然而对应矩问题的求解方法过于复杂，对我们的问题不起作用。所以这个问题仍然是开放的。在第二和第三年的任期的项目，他转向可控性的发展方程奇异边界条件。发展方程中的边界条件，如果其空间导数的阶数与方程的阶数相同，则称为奇异边界条件。利用矩量问题方法研究了具有奇异边界条件的波动方程的能控性。A.对于具有奇异边界条件Euler-Bernoulli方程，也将得到类似的结果。(2)研究者小池对与粘度溶液相关的最优控制理论做出了巨大贡献。他提出并解决了许多问题。

项目成果

S. Koike, H. Morimoto: "On variational inequalities for leavable boundary-velocity control"Applied Mathematics and Optimization. To appear.

S. Koike、H. Morimoto：“关于可左边界速度控制的变分不等式”应用数学和优化。

S.Koike,M.Bardi,P.Soravia: "Pursuit-evasion games with state constraints dynamic programming and discrete-time approximations"Discrete and Continuous Dynamical Systems. 6・2. 361-380 (2000)

S.Koike、M.Bardi、P.Soravia：“具有状态约束动态规划和离散时间近似的追击游戏”离散和连续动力系统。 361-380（2000）。

K. Tsujioka, J. Uchiumi: "Approximate controllability of a system coupled by Euler-Bernoulli beams"Advances in Mathematical Science and Applications. 12(2). 645-663 (2002)

K. Tsujioka、J. Uchiumi：“欧拉-伯努利梁耦合系统的近似可控性”数学科学与应用进展。

K.Tsujioka, Y.-S..Shao: "On Proper-Efficiency for nonsmooth multiobjective optimal control problems"Bulitine of Informatics. 30. No 2.302. 139-155 (2000)

K.Tsujioka，Y.-S..Shao：“论非光滑多目标最优控制问题的适当效率”信息学Bulitine。

Y.Shao, K.Tsujioka: "On proper-efficiency for nonsmooth multiobjective optimal control problems"Bulletin of Information and Cybernetics. 32・2. 139-155 (2000)

Y.Shao，K.Tsujioka：“非平滑多目标最优控制问题的适当效率”，信息与控制论通报 32・2（2000）。