Banach空间中点控系统的范数最优控制

The Norm Optimal Control Problems for the Point Control Systems in Banach Spaces

本文讨论自反 Banach 空间 X 中点控制分布参数系统:(d/dt)x(t)=Ax(f)+u(t)f,0<t≤τ,x(0)=0,u(·)∈L^p(0,r)(1<P<∞),f∈D(A~*)′,x(τ;u)∈X的范数最优控制问题,算子 A 为 X 中强连续算子半群 T(t),t≥0的无穷小生成元,A~*是算子 A 的对偶算子,D(A~*)是 A~*的定义域,D(A~*)′是 D(A~*)的对偶子空间.利用 L^p(0,τ)空间的自反、光滑、严格凸性,用Banach 空间的对偶映射方法,证明了点控制系统的范数最优控制的存在唯一性,并给出了范数最优控制的形式表达式.

In this paper,we discuss the norm optimal control problems for the distributed parameter systems with the point-control in the reflexive Banach space X:X′(t)=Ax(t)+u(t)f,0<t≤τ,x(0)=0,u(·)∈L^P(0,τ)(1<p<∞), x(t;u)∈X.f∈D(A*)′.Here A is the infinitesimal generator of a strongly continuous bounded linear semigroups T(t).t≥0 on X,A* is the dual operator of A.D(A*)is the domain of A*,D(A*)′is the dual space of D(A*).Utilizir- ng space L^P(0,τ)′s reflexivity,smoothness and strict convexity,we have proved the existence and uniqueness of the norm optimal control for the point- control systems and have given its formal representation by means of the methods of the dual mapping of Banach space.The obtained solution will be more convenient for applying than 4's solution.