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

Abstract

AbstractIn a separable Hilbert spaceX, 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$$A\ge -\sigma I$$A≥-σI($$\sigma \ge 0$$σ≥0) is a self-adjoint linear operator,Bis a bounded linear operator onX, and$$p\in L^2_{loc}(0,+\infty )$$p∈Lloc2(0,+∞)is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to thejth eigensolution for any$$j\ge 1$$j≥1. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.