If A: D(A)C X→,X is a densely defined and closed linear operator, which generates a linear semigroup S(t) in Banach space X. The nonlocal controllability 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 functions f 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.%设A:D(A)CX→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.
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机译：If A: D(A)C X→,X is a densely defined and closed linear operator, which generates a linear semigroup S(t) in Banach space X. The nonlocal controllability 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 functions f 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.%设A:D(A)CX→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.
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