【作者】 王锦荣；

【导师】 韦维； 项筱玲；

【作者基本信息】 贵州大学 ， 基础数学， 2006， 硕士

【摘要】 本文主要讨论了Banach空间中受控系统是一类半线性积微分方程的时间最优控制存在性以及受控系统分别是半线性和非线性方程的时间最优控制的Meyer逼近。 首先，作者研究了如下的受控系统其中A是Banach空间X中C₀-半群{T（t），t≥0}的无穷小生成元，积分算子如下 （Sx）（t）=integral from n=0 to 1 k（t，τ）g（τ，x（τ））dτ。在得到了方程（1）的温和解的存在唯一性之后，作者着重对目标是固定点和活动靶两种情形的时间最优控制的存在性问题分别进行了详细的讨论，并得到了两个新的存在性结果。 其次，在Banach空间中，作者分别讨论了下面的半线性方程（2）以及非线性方程（3）的时间最优控制的Meyer逼近。 借助紧半群的一致算子拓扑收敛性和变换思想，作者构造了一串Meyer问题序列去逼近半线性发展方程（2）的时间最优控制闯题，从而揭示了时间最优控制问题与Meyer问题之间本质联系。同时，给出了时间最优控制的存在性证明的新方法。这里，算子B满足较弱的条件如B本征有界。 运用C₀-半群强算子拓扑收敛性和变换思想，作者重新构造了一串Meyer问题序列去逼近半线性发展方程（3）的时间最优控制问题，再次揭示了时间最优控制问题与Meyer问题之间的本质联系。作为结果，也得到了（3）的时间最优控制的存在性。这里，虽然半群的条件将低了，但是算子B要求强连续。关键词：C₀-半群；紧半群；存在性；温和解；时间最优控制；变换；Meyer逼近更多还原

【Abstract】 In this thesis, we pay great attention to discuss the existence of time optimal control of system governed by a class of semi-linear integro-differential equations and Meyer Approximation of time optimal control of system governed by semi-linear equations and nonlinear equations respectively on infinite dimensional Banach spaces.First, the following control system governed by semi-linear integro-differential equations (7) is consideredwhere A is the infinitesimal generator of a Co-semigroup, I = [0,1] and l is fixed and integro operator is given byAfter getting the extisence of mild soultion of equations (7), we pay great attention to study the existence of time optimal controls about the target which are corresponding to a fixed point and a set varying with time respectively. As conclusion, we obtain two new existence results.Second, another remarkable problem that Meyer approximation of time optimal control of system governed by semi-linear equationsand nonlinear equationsare also investigated respectively on infinite dimensional Banach spaces.Using the uniformly operator topology of compact semigroup theory and transformation idea, a sequence of solution of Meyer problems are constructed to approach to a class of time optimal control problems of system governed by semi-linear equations (8) on infinite dimensional Banach spaces. A deep relation between Meyer problem and time optimal control is also presented. Consequently, the existence of time optimal control is obtained. Here, operator B satisfies relative weak conditions such as B satisfies essence bounded.By virtue of the strongly operator topology of Co-semigroup theory and transformation idea, a sequence of solution of Meyer problems are constructed to approach to a class of time optimal control problems of system governed by nonlinear equations (9) on infinite dimensional Banach spaces. A deep relation between Meyer problem and time optimal control is also presented. Consequently, the existence of time optimal control is obtained. Though Co-semigroup is not as strong as compact semigroup, B satisfies relative strong condition such as B satisfies essence bounded and it is a strong continuous operator.更多还原