Boundary Control of Coupled Non-Constant Parameter Systems of Time Fractional PDEs with Different-Type Boundary Conditions

doi:10.1007/s11424-023-0204-6

This paper addresses a boundary state feedback control problem for a coupled system of time fractional partial differential equations (PDEs) with non-constant (space-dependent) coefficients and different-type boundary conditions (BCs). The BCs could be heterogeneous-type or mixed-type. Specifically, this coupled system has different BCs at the uncontrolled side for heterogeneous-type and the same BCs at the uncontrolled side for mixed-type. The main contribution is to extend PDE backstepping to the boundary control problem of time fractional PDEs with space-dependent parameters and different-type BCs. With the backstepping transformation and the fractional Lyapunov method, the Mittag-Leffler stability of the closed-loop system is obtained. A numerical scheme is proposed to simulate the fractional case when kernel equations have not an explicit solution.

Key words: Boundary control, coupled systems, heterogeneous, mittag-leffler stability, time fractional PDEs with space-dependent coefficients

[1] Podlubny I, Fractional Differential Equations, Academic Press, San Diego, 1999.
[2] Chen J, Zhuang B, Chen Y Q, et al., Backstepping-based boundary feedback control for a fractional reaction diffusion system with mixed or Robin boundary conditions, IET Control Theory & Applications, 2017, 11(17):2964-2976.
[3] Laskin K, Fractional schrödinger equation, Physical Review E, 2002, 66(5):056108.
[4] Pagnini G, Nonlinear time-fractional differential equations in combustion science, Fractional Calculus and Applied Analysis, 2011, 14(1):80-93.
[5] Murio D A, Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Computers & Mathematics with Applications, 2007, 53(10):1492-1501.
[6] Sun H G, Chen W, Li C P, et al., Fractional differential models for anomalous diffusion, Physica A:Statistical Mechanics and Its Applications, 2010, 389(14):2719-2724.
[7] Laskin N, Fractional quantum mechanics, Physical Review E, 2000, 62(3):3135.
[8] Krstic M and Smyshlyaev A, Boundary Control of PDEs:A Course on Backstepping Designs, Siam, Philadelphia, 2008.
[9] Matignon D, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems Applications, 1996, 2:963-968.
[10] Ge F D, Chen Y Q, and Kou C H, Boundary feedback stabilisation for the time fractional-order anomalous diffusion system, IET Control Theory & Applications, 2016, 10(11):1250-1257.
[11] Zhou H C and Guo B Z, Boundary feedback stabilization for an unstable time fractional reaction diffusion equation, SIAM Journal on Control and Optimization, 2018, 56(1):75-101.
[12] Mai V T and Dinh C H, Robust finite-time stability and stabilization of a class of fractional-order switched nonlinear systems, Journal of Systems Science and Complexity, 2019, 32(6):1479-1497.
[13] Dinh C H and Mai V T, New Results on stability and stabilization of delayed Caputo fractional order systems with convex polytopic uncertainties, Journal of Systems Science and Complexity, 2020, 33(3):563-583.
[14] Gafiychuk V, Datsko B, Meleshko V, et al., Analysis of the solutions of coupled nonlinear fractional reaction-diffusion equations, Chaos, Solitons & Fractals, 2009, 41(3):1095-1104.
[15] Li Y, Chen Y Q, and Podlubny I, Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag-Leffler stability, Computers & Mathematics with Applications, 2010, 59(5):1810-1821.
[16] Li H L, Jiang Y L, Wang Z L, et al., Global mittag-leffler stability of coupled system of fractional-order differential equations on network, Applied Mathematics and Computation, 2015, 270:269-277.
[17] Ge F D, Meurer T, and Chen Y Q, Mittag-leffler convergent backstepping observers for coupled semilinear subdiffusion systems with spatially varying parameters, Systems & Control Letters, 2018, 122:86-92.
[18] Chen J, Tepljakov A, Petlenkov E, et al., Boundary state and output feedbacks for underactuated systems of coupled time-fractional pdes with different space-dependent diffusivity, International Journal of Systems Science, 2020, 51(15):2922-2942.
[19] Deutscher J and Kerschbaum S, Backstepping control of coupled linear parabolic PIDEs with spatially-varying coefficients, IEEE Transactions on Automatic Control, 2018, 63(12):4218-4233.
[20] Orlov Y, Pisano A, Pilloni A, et al., Output feedback stabilization of coupled reaction-diffusion processes with constant parameters, SIAM Journal on Control and Optimization, 2017, 55:4112-4155.
[21] Xu Q W and Xu Y F, Extremely low order time-fractional differential equation and application in combustion process, Communications in Nonlinear Science and Numerical Simulation, 2018, 64:135-148.
[22] Li J and Liu Y G, Stabilization of coupled PDE-ODE systems with spatially varying coefficient, Journal of Systems Science and Complexity, 2013, 26(2):151-174.
[23] Henry B I and Wearne S L, Fractional reaction-diffusion, Physica A:Statistical Mechanics and Its Applications, 2000, 276(3-4):448-455.
[24] Uchaikin V V and Sibatov R T, Fractional theory for transport in disordered semiconductors, Communications in Nonlinear Science and Numerical Simulation, 2008, 13(4):715-727.
[25] Chen J, Cui B T, and Chen Y Q, Backstepping-based boundary control design for a fractional reaction diffusion system with a space-dependent diffusion coefficient, ISA Transactions, 2018, 80:203-211.
[26] Zhou Y J, Chen J, and Cui B T, Observer design for boundary coupled fractional order distributed parameter systems, International Conference on Control, Mechatronics and Automation (ICCMA 2019), Delft, 2019:384-388.
[27] Kilbas A A A, Srivastava H M, and Trujillo J J, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, Amsterdam, 2006.
[28] Bazhlekova E, The abstract cauchy problem for the fractional evolution equation, Fractional Calculus and Applied Analysis, 1998, 1(3):255-270.
[29] Friedman A, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964.
[30] Curtain R F and Zwart H, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1995.
[31] Duarte-Mermoud M A, Aguila-Camacho N, Gallegos J A, et al., Using general quadratic lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Communications in Nonlinear Science & Numerical Simulation, 2015, 22(1-3):650-659.
[32] Miller K S and Samko S G, Completely monotonic functions, Integral Transforms & Special Functions, 2001, 12(4):389-402.
[33] Baccoli A, Pisano A, and Orlov Y, Boundary control of coupled reaction-diffusion processes with constant parameters, Automatica, 2015, 54:80-90.
[34] Li C, Wang J C, Lu J G, et al., Observer-based stabilisation of a class of fractional order non-linear systems for 0<α<2 case, IET Control Theory & Applications, 2014, 8(13):1238-1246.
[35] Saad K M and Gómez-Aguilar J F, Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense, Revista Mexicana de Fısica, 2018, 64(5):539-547.
[36] Rasheed A and Anwar M S, Simulations of variable concentration aspects in a fractional nonlinear viscoelastic fluid flow, Communications in Nonlinear Science and Numerical Simulation, 2018, 65:216-230.
[37] Luchko Y, Mainardi F, and Povstenko Y, Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation, Computers & Mathematics with Applications, 2013, 66(5):774-784.
[38] Li H F, Cao J X, and Li C P, High-order approximation to Caputo derivatives and Caputotype advection-diffusion equations (III), Journal of Computational & Applied Mathematics, 2016, 299(3):159-175.
[39] Ablowitz M J, Kruskal M D, and Ladik J, Solitary wave collisions, SIAM Journal on Applied Mathematics, 1979, 36(3):428-437.
[40] Wang K and Fridman E, Boundary control of delayed ODE-heat cascade under actuator saturation, Automatica, 2017, 83:252-261.

[1]	HU Chuanfeng, HU Hui, LIN Hongwei, YAN Jiacong. Isogeometric Analysis-Based Topological Optimization for Heterogeneous Parametric Porous Structures [J]. Journal of Systems Science and Complexity, 2023, 36(1): 29-52.
[2]	FENG Xiaodan, ZHANG Zhifei. Boundary Control of Coupled Wave Systems with Spatially-Varying Coefficients [J]. Journal of Systems Science and Complexity, 2022, 35(4): 1310-1329.
[3]	YANG Ruohan, ZHOU Deyun. Distributed Event-Triggered Tracking Control of Heterogeneous Discrete-Time Multi-Agent Systems with Unknown Parameters [J]. Journal of Systems Science and Complexity, 2022, 35(4): 1330-1347.
[4]	MO Lipo, YU Yongguang, REN Guojian, YUAN Xiaolin. Constrained Consensus of Continuous-Time Heterogeneous Multi-Agent Networks with Nonconvex Constraints and Delays [J]. Journal of Systems Science and Complexity, 2022, 35(1): 105-122.
[5]	KUANG Xiong · WANG Qian · CHEN Xia. The Effectiveness of Structural Monetary Policy and Macro-Prudential Policies — Based on the DSGE Model That Includes Bank Heterogeneous Credit [J]. Journal of Systems Science and Complexity, 2021, 34(6): 2267-2290.
[6]	ZHAO Zhixue,GUO Baozhu. Boundary Control Method to Identification of Elastic Modulus of String Equation from Neumann-Dirichlet Map [J]. Journal of Systems Science and Complexity, 2016, 29(5): 1212-1225.
[7]	WEN Ruili,CHAI Shugen. Regularity for Euler-Bernoulli Equations with Boundary Control and Collocated Observation [J]. Journal of Systems Science and Complexity, 2015, 28(4): 788-798.
[8]	FENG Xu , ZHANG Wei , ZHANG Yongjie , XIONG Xiong. INFORMATION IDENTIFICATION IN DIFFERENT NETWORKS WITH HETEROGENEOUS INFORMATION SOURCES [J]. Journal of Systems Science and Complexity, 2014, 27(1): 92-116.
[9]	Yifen MU , Lei GUO. TOWARDS A THEORY OF GAME-BASED NON-EQUILIBRIUM CONTROL SYSTEMS [J]. Journal of Systems Science and Complexity, 2012, 25(2): 209-226.
[10]	Hongquan LI, Shouyang WANG, Wei SHANG. HETEROGENEITY, NONLINEARITY AND ENDOGENOUS MARKET VOLATILITY [J]. Journal of Systems Science and Complexity, 2011, 24(6): 1130-1142.
[11]	Orazio ARENA;Walter LITTMAN. BOUNDARY CONTROL OF TWO PDE'S SEPARATED BY INTERFACE CONDITIONS [J]. Journal of Systems Science and Complexity, 2010, 23(3): 431-437.
[12]	W.L.Ohan;Guo Baozhu. APPROXIMATE OPTIMAL BIRTH CONTROL OF POTULATION SYSTEMS [J]. Journal of Systems Science and Complexity, 1990, 1(1): 46-052.
[13]	Ding Zhonghai;Feng Dexing. EXACT CONTROLLABILITY OF A CLASS OF DISTRIBUTED PARAMETER SYSTEMS [J]. Journal of Systems Science and Complexity, 1989, 2(3): 257-265.

Viewed

Full text

Abstract