Closed range weak or mild solution operators in the resonant case (Q2733852)

The author studies the operators \(W\) and \(M\) associated with the solutions to the first-order abstract evolution equation NEWLINE\[NEWLINEy'= Ay+ f,\quad ay(0)= y(T),\quad a=\pm 1,NEWLINE\]NEWLINE where \(A: D(A)\subset H\to H\) is a linear densely defined operator with closed range \(R(A)\) on a Hilbert space \(H\), \(T> 0\) and \(f\) is an element of \(L^2(\langle 0,T\rangle, H)\). Particularly, conditions are established implying that \(R(W)\) and \(R(M)\) are closed in \(H\). Applications to nonlinear nonconvex optimal control problems are discussed.