This paper addresses the null controllability problem for bilinear boundary control systems. Due to the inherent nonlinearity introduced by the bilinear coupling, exact null controllability in the full state space is generally unattainable. We therefore introduce a weaker concept, termed \(j\) -null controllability, which aims at steering the solution to zero along a prescribed eigenmode of the unperturbed generator. Under Riesz-spectral assumptions, we establish sufficient conditions for \(j\) -null controllability using spectral decompositions and the moment method, and we provide explicit estimates of the control cost. We also discuss the necessity of the main non-vanishing coupling condition and clarify the role of the summability assumption in the moment-method construction. The theoretical results are applied to several examples including bilinear boundary-controlled heat equations, parabolic beam systems, and Euler–Bernoulli beams with structural damping, proving their \(1\) -null controllability in any positive time. In particular, we include a non-self-adjoint heat equation with convection, for which the biorthogonal family is computed explicitly and the Riesz-basis constant appearing in the control-cost estimate is shown to be nontrivial.
