The paper is devoted to the study of boundary finite-time control for a reaction-diffusion (RD) system with switching time-delayed input. The RD system with switching time-delay input is converted to a switching system of RD equation cascaded with a transport equation with non-delay boundary input. Next, a novel switching controller is designed for the cascaded RD-transport system based on the backstepping technique, and this causes the closed-loop system to be convergence in a finite-time. Simulation results are provided to exhibit the effectiveness of the proposed method.
AmbrosinoRCalabreseFCosentinoC, et al. (2009)Sufficient conditions for finite-time stability of impulsive dynamical systems. IEEE Transactions on Automatic Control54(4): 861–865.
2.
BhatSBernsteinD (2000)Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization38(3): 751–766.
BurnsJAHerdmanTLZietsmanL (2013)Control of PDE systems with delays. IFAC Proceedings Volumes46(26): 79–84.
5.
CoronJMNguyenHM (2017)Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach. Archive for Rational Mechanics and Analysis225(3): 993–1023.
6.
CoronJMHuLOliveG (2017)Finite-time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation. Automatica84: 95–100.
7.
DiagneMBekiaris-LiberisNKrsticM (2017)Compensation of input delay that depends on delayed input. Automatica85: 362–373.
8.
EspitiaNPolyakovAEfimovD, et al. (2019a)Boundary time-varying feedbacks for fixed-time stabilization of constant parameter reaction-diffusion systems. Automatica103: 398–407.
9.
EspitiaNPolyakovAFridmanE (2019b)On local finite-time stabilization of the viscous burgers equation via boundary switched linear feedback. In: 58th IEEE Conference on Decision and Control, Nice, France, 11–13 December 2019 pp.: 4029–4034.
10.
GhaderiNKeyanpourM (2020a)Backstepping design for a class of coupled parabolic PDEs with spatially varying coefficient. Asian Journal of Control22(5): 2017–2028.
11.
GhaderiNKeyanpourM (2020b)Anti-collocated observer-based output feedback control of wave equation with cubic velocity nonlinear boundary and Dirichlet control. International Journal of Control, pp: 1–11. DOI: 10.1080/00207179.2020.1757761.
12.
GhaderiNKeyanpourM (2020c)Output feedback stabilization for 1-D unstable wave equations with boundary control matched disturbance and van der Pol nonlinear boundary. IFAC Journal of Systems and Control. IFAC Journal of System and Control12: 100085.
13.
GhaderiNKeyanpourMMojallaliH (2020)Observer-based finite-time output feedback control of heat equation with Neumann boundary condition. Journal of the Franklin Institute357(14): 9154–9173.
14.
GuinLNAcharyaS (2017)Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting. Nonlinear Dynamics88(2): 1501–1533.
15.
GuoBZYangKY (2009)Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica45(6): 1468–1475.
16.
HanXFridmanESpurgeonSK (2010)Sliding-mode control of uncertain systems in the presence of unmatched disturbances with applications. International Journal of Control83(12): 2413–2426.
17.
HouMZitekPPattonRJ (2002)An observer design for linear time-delay systems. IEEE Transaction Automatic Control47(1): 121–125.
18.
LiberzonD (2003)Switching in Systems and Control. Boston: Birkhauser.
19.
KrsticM (2009)Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Cambridge: Birkhaüser.
20.
KrsticM (2010)Compensation of infinite-dimensional actuator and sensor dynamics. IEEE Control System Society30(1): 22–41.
21.
KrsticMSmyshlyaevA (2008)Boundary Control of PDEs: A Course on Backstepping Designs. Philadelphia: SIAM.
22.
NicaiseSPignottiC (2008)Stabilization of the wave equation with boundary or internal distributed delay, Differential and Integral Equations21(9–10): 935–958.
23.
NicaiseSValeinJFridmanE (2009)Stability of the heat and of the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems2(3): 559–581.
24.
PerrollazVRosierL (2014)Finite-time stabilization of hyperbolic systems on tree-shaped networks. SIAM Journal of Control and Optimization52(1): 143–163.
25.
PolyakovACoronJMRosierL (2017)On boundary finite-time feedback control for heat equation. In: 20th IFAC World Congress, Toulouse, France, 9–14 July 2017, pp.: 671–676.
26.
PolyakovAEfimovDPerruquettiW (2015)Finite-time and fixed-time stabilization: Implicit Lyapunov function approach. Automatica51(1): 332–340.
27.
RajchakitG (2012)Robust stability and stabilization of nonlinear uncertain stochastic switched discrete-time systems with interval time-varying delays. Applied Mathematics and Information Sciences6: 555–565.
28.
RajchakitG (2014)Switching design for the robust stability of nonlinear uncertain stochastic switched discrete-time systems with interval time-varying delay. Journal of Computational Analysis and Applications16(1): 10–19.
29.
RajchakitGRojsiraphisalTRajchakitM (2012)Robust stability and stabilization of uncertain switched discrete-time systems. Advances in Difference Equations2012(1): 134.
30.
SteevesDKrsticMVazquezR (2019)Prescribed-time -stabilization of reaction-diffusion equations by means of output feedback. In: 18th European Control Conference (ECC), Naples, Italy, June 25–28 2019 pp.: 1932–1937.
31.
TemmeNM (1966)Special Functions: An Introduction to the Classical Functions of Mathematical Physics. New York: Wiley.
32.
WangJLWuHN (2013)Passivity of delayed reaction-diffusion networks with application to a food web model. Applied Mathematics and Computation219(24): 11311–11326.
33.
WangJMSunCY (2018)Delay-dependent exponential stabilization for linear distributed parameter systems with time-varying delay. Journal of Dynamic Systems Measurement and Control-Transactions of the ASME140(5): 051003.
34.
WangYEWuBWWuC (2017)Stability and -gain analysis of switched input delay systems with unstable modes under asynchronous switching. Journal of the Franklin Institute354(11): 4481–4497.
35.
WuKNSunHXShiP, et al. (2018a)Finite-time boundary stabilization of reaction-diffusion systems. International Journal of Robust and Nonlinear Control28(5): 1641–1652.
36.
WuKNSunHXShiP, et al. (2018b)Finite-time boundary control for delay reaction-diffusion systems. Applied Mathematics and Computation329: 52–63.
37.
WuYCaoJAlofiA, et al. (2015)Finite-time boundedness and stabilization of uncertain switched neural networks with time-varying delay. Neural Networks69: 135–143.
38.
ZhaoDXWangJM (2012)Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks. Journal of Dynamical and Control System18(2): 269–295.
