Optimal Control Theory of Enler-BernouIIi Equation

(1)在项目的第一年，首席研究员研究了两根欧拉-伯努利梁耦合系统的可控性，并在耦合点进行控制。首先，他研究了与Euler-BernouIIi方程相关的四阶微分算子的特征值的渐近行为。然后将可控性问题化为希尔伯特空间中的矩问题。将可控性问题简化为矩问题的方法称为矩问题法。我们项目的最终目标是研究耦合几个欧拉-伯努利梁的系统的可控性。但是对应矩问题方法过于复杂，对我们的问题并不适用。所以这个问题还没有解决。在项目的第二年和第三年，他转向了奇异边界条件下进化方程的可控性。如果演化方程的边界条件在空间导数上的阶数与演化方程的阶数相同，则称其为奇异边界条件。用矩问题方法研究了具有奇异边界条件的波动方程的可控性。A.对于具有奇异边界条件的Euler-Bernoulli方程，可以得到类似的结果。(2)小池研究员对粘度解相关的最优控制理论做出了重要贡献。各种各样的问题都是他提出并解决的。

(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.