【作者】 朱敏；

【导师】 田立新；

【作者基本信息】 江苏大学 ， 应用数学， 2005， 硕士

【摘要】 本文在变分不等式最优控制理论和分布参数系统的最优控制理论的基础上,研究了几类孤立波方程的一种很典型的最优控制问题,形式如下: minJ(y,u)满足e(y,u)=0这儿J:Y×U→R,e:Y×U→Z~*;都是充分光滑的函数,Y,U和Z都是Hilbert空间,Z~*是Z的对偶,集合Y,U分别表示状态和控制空间。 首先在前人研究的基础上,本文研究了Burgers方程最优控制的一些性质,包括给出J和e的F导数,证明了正则点条件,一阶必要最优条件和二阶充分最优条件。然后进一步的研究了在Dirichlet边界条件下,KdV-Burgers方程的最优控制问题,根据变分不等式最优控制理论和分布参数系统的最优控制理论,选择了合适的性能指标J(y,u),证明了解的范数与原方程的控制项和初始值有关;并且给出了方程的最优控制,还证明了最优解的存在性。同时文章还讨论了在Neumann边界条件下,非线性强度Burgers方程的最优控制问题。用Galerkin方法证明了非线性强度Burgers方程在Neumann边界条件下一个很短的时间区域内解的存在性。通过选择合适的性能指标J(y,u),进一步给出了最优控制,并证明了其最优解的存在性。更多还原

【Abstract】 According to the optimal control theory about variational inequality and distributed parameter system,we have studied a typical optimal control problem about some solitary wave equations, the form is as follows:min J(y,u) ande(y,u) = 0here J: Y×U → R,e: Y×U → Z~*;is abundant smooth function, Y,U and Z are all Hilbert space, Z~* is the dual of Z,set Y,U is state space and control space.On the basis of presented results ,the paper study some optimal control quantity about Burgers equation include giving the F-derivatives of J and e, proving the regular point condition, the First-order necessary optimality conditions and the Second-order sufficient optimality condition. Further, investigating optimal control problem of KdV-Burgers equation, according to the optimal control theory about variational inequality and distributed parameter system, choosing suitable performance index J(y, u),then it is proofed that the solution is related with control item and original value. The optimal control of KdV-Burgers equation under the Dirichlet boundary condition is given and the existence of optimal solution is proofed. At the same time, The paper study the optimal control for nonlinear strength Burgers equation under Neumann boundary condition. According to Galerkin method, it is proofed that the solution exist in a short interval time. Choosing suitable performance index J(y,u) , the optimal control of nonlinear strength Burgers equation under the Neumann boundary condition is given and the existence of optimal solution is proofed.更多还原