We consider in this work a class of finite dimensional time-varying linear disturbed systems. The main objective of this work is to studied the optimal control which ensures the remediability of a disturbance of time-varying disturbed systems. The remediability concept consist to find a convenient control which bringing back the corresponding observation of disturbed system to the normal one at the final time. We give firstly some characterisations of compensation and in second party we find a control which annul the output of the system and we show also that the Hilbert Uniqueness Method can be used to solve the optimal control which ensure the remediability.Ageneral approachwas given to minimize the linear quadratic problem. Examples and numerical simulations are given.

Many nonlinear dynamical systems can present a challenge for the stability analysis in particular the estimation of the region of attraction of an equilibrium point. The usual method is based on Lyapunov techniques. For the validity of the analysis it should be supposed that the initial conditions lie in the domain of attraction. In this paper, we investigate such problem for a class of dynamical systems where the origin is not necessarily an equilibrium point. In this case, a small compact neighborhood of the origin can be estimated as an attractor for the system. We give a method to estimate the basin of attraction based on the construction of a suitable Lyapunov function. Furthermore, an application to Lorenz system is given to verify the effectiveness of the proposed method.

As nonlinear optimization techniques are computationally expensive, their usage in the real-time era is constrained. So this is the main challenge for researchers to develop a fast algorithm that is used in real-time computations. This work proposes a fast nonlinear model predictive control approach based on particle swarm optimization for nonlinear optimization with constraints. The suggested algorithm divide and conquer technique improves computing speed and disturbance rejection capability, demonstrating its suitability for real-time applications. The performance of this approach under constraints is validated using a highly nonlinear fast and dynamic real-time inverted pendulum system. The solution presented through work is computationally feasible for smaller sampling times and it gives promising results compared to the state of art PSO algorithm