Consider the semilinear system defined by

x(i+1) = Ax(i) + f(x(i)), i≥ 0

x(0) = x0 ϵ ℜn

and the corresponding output signal y(i)=Cx(i), i ≥ 0, where A is a n x n matrix, C is a p x n matrix and f is a nonlinear function. An initial state x(0) is output admissible with respect to A, f, C and a constraint set Ω in ℜp if the output signal (y(i))i associated to our system satisfies the condition y(i) in Ω, for every integer i ≥ 0. The set of all possible such initial conditions is the maximal output admissible set Γ(Ω). In this paper we will define a new set that characterizes the maximal output set in various systems (controlled and uncontrolled systems). Therefore, we propose an algorithmic approach that permits to verify if such set is finitely determined or not. The case of discrete delayed systems is taken into consideration as well. To illustrate our work, we give various numerical simulations.

In this article, we extended the concept of controllability, traditionally used to control the final state of a system, to the exact control of its final speed. Inspired by Kalman’s theory, we have established some conditions to characterize the control that allows the system to reach a desired final speed exactly. When the assumptions ensuring speed-controllability are not met, we adopt a regulation strategy that involves determining the control law to make the system’s final speed approach as closely as possible to the predefined final speed, and this at a lower cost. The theoretical results obtained are illustrated through three examples.