Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay

Abstract

This paper considers a one-dimensional Euler-Bernoulli beam equation where two collocated actuators/sensors are presented at the internal point with pointwise feedback shear force and angle velocity at the arbitrary position ξ in the bounded domain (0,1). The boundary x = 0 is simply supported and at the other boundary x = 1 there is a shear hinge end. Both of the observation signals are subjected to a given time delay τ ( >0). Well-posedness of the open-loop system is shown to illustrate availability of the observer. An observer is then designed to estimate the state at the time interval when the observation is available, while a predictor is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controllers are introduced to guarantee the closed-loop system to be exponentially stable for the smooth initial values when ξ ∈ (0, 1) is a rational number satisfying ξ ≠ 2l∕(2m − 1) for any integers l, m. Simulation results demonstrate that the proposed feedback design effectively stabilizes the performance of the pointwise control system with time delay.

Output feedback stabilization of cascaded ODE‐Wave equations with time delay in observation

“…In control system, cascaded PDE-ODE and PDE-PDE systems describe fundamental laws of physics and mechanic, such as ODE-Schrödinger equation [1][2][3], ODE-Heat equation [4][5][6][7], ODE-Wave equation [6,8,9], coupled strings [10], coupled Timoshenko beam [11] etc. The stabilization of systems utilizing PDEs subject to time delay is drawn more attention [12][13][14][15][16][17]. In general, time delay may destroy the stability for a control system [18].…”

“…To our knowledge, the stabilization of the cascaded ODE-PDE control systems with time delay, especially that with delayed observation, has been rarely involved in even recent research. Even through some successful efforts have been made to the simple Euler-Bernoulli beam [12], unstable wave equation [16,17], and the nonlinear ODE systems [6], there are few work on the unstable PDE-ODE systems with time delay in observation and some unsolved problem in subject. In this paper, we consider the cascaded Wave-ODE equations with time delay in the observation.…”

“…This paper provides a generalization of PDE systems on the wave equation in [13,16,17], the beam equation [12] and the Schrödinger equation in [2,3] by applying the separation principle for the boundary condition. Our work in this paper contributes to addressing this mathematical challenge.…”

Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

“…For problems with delay in different contexts we cite [9,10,30,32] with references therein. In absence of delay (µ 2 (t) = 0), the problem (1) is exponentially stable provided that µ 1 (t) is constant, see, for instance [5,6,16,17,21] and reference therein.…”

Blow‐up of solution for beam equation with delay and dynamic boundary conditions

“…It is well known that time delay effects often appear in many chemical, physical, and economical phenomena because these phenomena depend not only on the present state but also on the past history of the system. Recently, beam equations with time delay have become an active area of research; see, for example, previous studies . Yang studied the Euler‐Bernoulli viscoelastic equation with a delay as follows: where and are arbitrary real numbers.…”