Melnikov’s method for controlled stochastic oscillations of a rocking block with fractional derivative

Abstract

The problem of controlling a rocking block with fractional dissipation so as to prevent it from leaving a prescribed set of bounded oscillations has a number of interesting applications in many scientific and technological fields. This work develops an analytical framework to describe the influence of control on the behavior of a fractionally dissipative rocking block subject to random acceleration of the ground. Control equations are developed and control strategies leading to optimal suppression of instability are asymptotically described under the assumptions of weak dissipation along with a small energy noise. Under these assumptions, the behavior of the block is interpreted as weakly perturbed rocking oscillations close to a generic periodic solution, and the Melnikov approximations are considered as the necessary conditions of instability. The stochastic Melnikov criterion derived in this paper allows for an approximate calculation of an upper bound to the overturning probability as well as to the representation of an optimal control strategy. The Melnikov criterion for stochastic rocking oscillations in the presence of convenient control strategies are derived explicitly for the first time, using an extension of the Melnikov theory to a model with fractional derivative.