Exact and approximate controllability of abstract semilinear control systems (Q1860737)

The authors study exact and approximate controllability of an abstract semilinear control system of the form \[ \dot x(t)=Ax(t)+u(t)+f(t,x(t)), \quad 0\leq t\leq T, \qquad x(0)=x_{0}, \] where \(A: D(A)\subseteq V\to V\) is a closed linear operator with dense domain \(D(A)\) generating a \(C_{0}\)-semigroup, \(f: [0,T]\times V\to V\) a nonlinear operator which satisfies Carathéodory's conditions, \(u\in L^{2}([0,T],\hat{V})\) and \(V, \hat{V}\) Hilbert spaces.