Fakhreddin Abedi

Dynamic robust stabilization of stochastic differential control systems

The stabilization of stochastic differential control systems by means of dynamic feedback laws is provided. We extend the well-known Artstein–Sontag theorem to derive the necessary and sufficient conditions for the dynamic robust stabilization of stochastic differential systems. An explicit formula for feedback law exhibiting dynamic robust stability in probability is established, and a numerical example is also given to illustrate our results.

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchingers control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.

On the Asymptotical and Practical Stability of Stochastic Control Systems

The asymptotical and practical stability in probability of stochastic control systems by means of feedback laws is provided. The main results of this work enable us to derive the sufficient conditions for the existence of control Lyapunov function that play a leading role in the existence of stabilizing feedback laws. Particularly, the sufficient conditions for practical stability in probability are established and numerical examples are also given to illustrate the usefulness of our results.

Lyapunov Characterization for the Stability of Stochastic Control Systems

Lyapunov-like characterization for the problem of input-to-state stability in the probability of nonautonomous stochastic control systems is established. We extend the well-known Artstein-Sontag theorem to derive the necessary and sufficient conditions for the input-to-state stabilization of stochastic control systems. Illustrating example is provided.

Quadrupole mass filter with fuzzy initial conditions

The employment of the fuzzy method to solve differential equations has been well studied. In this article, Mathieu differential equations of the quadrupole mass filter (QMF) have been solved by using the fuzzy method. This method has not been yet investigated in the QMF with fuzzy initial conditions. We survey the physical properties of the confined ion. The results of numerical simulations are presented and discussed.

A notion of stability in probability of stochastic nonlinear systems

The concepts of stability in probability of nontrivial solutions for stochastic nonlinear systems are analyzed in terms of a control Lyapunov function which is smooth except possibly at the origin. We show under certain hypothesis that the neighborhood of the origin is stable in probability. An illustrating example is provided.MSC:60H10, 93C10, 93D05, 93D15, 93D21, 93E15.

Exponential input-to-state stability of composite stochastic systems

Sufficient conditions for the exponential input-to-state stability in probability in r th mean and for the almost sure exponential input-to-state stability in probability of a composite stochastic system are established. Illustrating example is provided to validate our results.MSC:60H10, 93C10, 93D05, 93D15, 93D21, 93E15.

On Exponential Stability of Stochastic Control Systems

In this paper, we study the exponential input-to-state stability in probability of a wider class of composite stochastic control system. Our aim is to establish sufficient conditions for exponential input-to-state stability in probability of this composite system. We also give a numerical example illustrating our results.