摘要
设A:D(A)X→X是Banach空间X上的线性稠定的闭算子,它是X上的强连续有界线性算子半群S(t)的无穷小生成元.对于Banach空间X中的含非局部初值条件u(0)=u0+g(u)的半线性Cauchy问题:u′(t)=Au(t)+Bx(t)+f(t,u(t)),在A生成的线性算子半群S(t)是非紧,映射f和g满足一定的紧性条件,控制算子B是有界线性算子时,证明了该问题是非局部可控的.并分别在半群是紧或强连续的条件下,证明了在控制算子B和W不是有界情形时上面的非局部Cauchy问题是非局部可控的.同时给出了在偏微分方程中的可控性问题的一个应用.
If A: D(A) X→X is a densely defined and closed linear operator, which generates a linear semigroup S (t) in Banach space X. The nonlocal control/ability for the following nonlocal semilinear problems: u' (t) = Au (t) + Bx( t) + f( t, u(t) ), 0≤t ≤ T with nonlocal initial condition u(0) = u0 + g(u) is discussed in Banach space X. The results show that if semigroup S(t) is strongly continuous, the functionsf and g are compact and the control B is bounded, then it is nonlocally controllable. The nonlocal controllability for the above nonlocal problem is also studied when B and W are unbounded and the semigroup S(t) is compact or strongly continuous. For illustration, a partial differential equation is worked out.
基金
the National Natural Science Foundation of China(No.10674024)
关键词
非局部问题
非局部可控
适度解
全连续
nonlocal problem
nonlocal controllability
mild solution
completely continuous