Time‐Varying Stabilizers for Stochastic Systems with no Unforced Dynamics

Abstract

This paper is concerned with the stabilizability of nonlinear stochastic systems with no unforced dynamics. Sufficient conditions allowing to design explicitly time–varying feedback laws which render such systems asymptotically stable in probability are given. The techniques used in this work involve the stochastic Lyapunov analysis combined with the stochastic version of the La Salle invariance principle. The interest of our results is that the systems considered in the present paper cannot in general be stabilized via time‐invariant feedback laws.

Stabilization of nonlinear stochastic systems without unforced dynamics via time--varying feedback

“…A geometric interpretation on why the topological obstruction can be weakened by using time-varying feedback has been given by Sepulchre, Campion and Wertz [18] and Sontag [19]. Note that the results obtained by Pomet in [16] has been extended to the stochastic context by Florchinger in [11].…”

“…But, if LV (t, x t ) = 0 for every t ≥ 0, inequality (11) implies that ΛV (t, x t ) = 0 for every t ≥ 0, that is Λ i V (t, x t ) = 0 for every t ≥ 0 and i ∈ {0, . .…”

Model‐Reference Adaptive Moment Control of Uncertain Nonlinear Stochastic Systems

“…Note that, V raug (t) is a vector of the expected and covariance of v r (t), which is the stochastic input of reference model (10). Now in order to control the expected and covariance of uncertain nonlinear stochastic system (1), the nonlinear control input u X; b θ; v r , which was determined based on adaptive model-reference system in Fig.1, is imposed on the associated nonlinear system (4). The control input u X; b θ; v r is determined at each time by two control loops, b θ is determined by adaptive control loop and v r is determined by moment control loop simultaneously.…”

“…The algorithm accounts for random wind dynamics and convective weather areas with changing size. studied the problems of stabilization of uncertain singular Markovian jump systems with dependent noise and nonlinear stochastic systems with no unforced dynamics respectively. presented exponential stability and delayed‐state‐feedback stabilization criteria for a class of nonlinear uncertain stochastic time‐delay systems.…”

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

“…Many real systems are nonlinear and are also prone to stochastic phenomena. These features have naturally attracted attention in the control literature with studies examining stabilization problems for stochastic nonlinear systems, and some interesting results obtained during the past few decades, see [1][2][3][4][5][6][7] and references therein. These results were obtained using classical stochastic theory as found, for instance, in [8] and [9].…”