Limit 2‐Cycles for a Discrete‐Time Bang‐Bang Control Model

Abstract

A discrete-time periodic model with bang-bang feedback control is investigated. It is shown that each solution tends to one of four different types of limit 2-cycles. Furthermore, the accompanying initial regions for each type of solutions can be determined. When a threshold parameter is introduced in the bang-bang function, our results form a complete bifurcation analysis of our control model. Hence, our model can be used in the design of a control system where the state variable fluctuates between two state values with decaying perturbation.

Nonnegative Periodic Solutions of a Three-Term Recurrence Relation Depending on Two Real Parameters

“…Some of these behaviors can be explained (see [1][2][3]) but some not (at least to the best of our knowledge). Similar models of piecewise constant dynamic systems which exhibit similar behaviors with parameters can be found in many recent investigations; see for examples [4][5][6][7][8][9][10] and the references therein.…”

“…The proof is complete. Let = { } ∈Z be a solution of (5). Then is uniquely determined by two consecutive terms and +1 .…”

“…Since ∈ R in (5) and (6), we can further restrict ourselves in the sequel to the following difference equation:…”