Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

Abstract In a separable Hilbert space X , we study the controlled evolution equation $$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0, \end{aligned}$$ u′(t)+Au(t)+p(t)Bu(t)=0, where $$A\ge -\sigma I$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">A≥-σI ( $$\sigma \ge 0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">σ≥0 ) is self-adjoint linear operator, B bounded operator on and $$p\in L^2_{loc}(0,+\infty )$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p∈Lloc2(0,+∞) bilinear control. We give sufficient conditions in order for above nonlinear control system to be locally controllable j th eigensolution any $$j\ge 1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">j≥1 . also derive semi-global controllability results large time discuss applications parabolic equations low dimension. Our method constructive all constants involved main can explicitly computed.