Pointwise stabilisation of a string with time delay in the observation

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.…”

“…These delays often create stability problem [18]. Various research has been done examining the problem of stability and stabilization of time-delay systems has been dealt with a number of different ways [15,17]. There are some results that are independent of the size of the time delays and the stability is satisfied for any value of the time delays.…”

“…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.…”

“…In this subsection, we predict the state {(X(s), u(x, s), u s (x, s)), t − < s ≤ t} for t > , which can be obtained by solving (16) in time interval [t − , t] with estimated initial condition {(X(t − ),û(x, t − ),û s (x, t − )), t > } obtained from (17) at s = t − : for t > , and t − < s < t, and…”