Finite time stabilisation of discrete systems under multiplicative or additive controls

We study the problem of finite-time stabilisation for a class of discrete-time dynamical systems subject to either multiplicative (bilinear) or additive (linear) control actions. Our approach relies on a structural decomposition of the dynamics into two complementary parts: one component whose finite-time stability is intrinsic and independent of the control input, and a second component that requires the explicit design of a stabilising feedback law. For this general framework, we provide necessary and sufficient conditions, ensuring finite-time stability and discuss their implications for both theory and practice. The results are illustrated through representative examples, including finite- and infinite-dimensional settings, and are further supported by numerical simulations that confirm the effectiveness of the proposed methodology.