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Boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction-diffusion systems. (English) Zbl 1415.93201

Summary: In this paper, the problem of fixed-time stabilization of constant-parameter reaction-diffusion partial differential equations by means of continuous boundary time-varying feedbacks is considered. Moreover, the time of convergence can be prescribed in the design. The design of time-varying feedbacks is carried out based on the backstepping approach. Using a suitable target system with a time varying-coefficient, one can state that the resulting kernel of the backstepping transformation is time-varying and rendering the control feedback to be time-varying as well. Explicit representations of the kernel solution in terms of generalized Laguerre polynomials and modified Bessel functions are derived. The fixed-time stability property is then proved. A simulation example is presented to illustrate the main results.

MSC:

93D15 Stabilization of systems by feedback
93C20 Control/observation systems governed by partial differential equations
35K57 Reaction-diffusion equations
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[1] Auriol, J.; Di Meglio, F., Minimum time control of heterodirectional linear coupled hyperbolic pdes, Automatica, 71, 300-307 (2016) · Zbl 1343.93049
[2] Baccoli, A.; Pisano, A.; Orlov, Y., Boundary control of coupled reaction-diffusion processes with constant parameters, Automatica, 54, 80-90 (2015) · Zbl 1318.93072
[3] Bhat, S. P.; Bernstein, D. S., Finite time stability of continuous autonomous systems, SIAM Journal on Control and Optimization, 38, 3, 751-766 (2000) · Zbl 0945.34039
[4] Boskovic, D. M.; Krstic, M.; Liu, W., Boundary control of an unstable heat equation via measurement of domain-averaged temperature, IEEE Transactions on Automatic Control, 46, 12, 2022-2028 (2001) · Zbl 1006.93039
[5] Colton, D., The solution of initial-boundary value problems for parabolic equations by the method of integral operators, Journal of Differential Equations, 26, 2, 181-190 (1977) · Zbl 0407.35043
[6] Coron, J.-M.; Hu, L.; Olive, G., Finite-time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation, Automatica, 84, 95-100 (2017) · Zbl 1376.93090
[7] Coron, J.-M.; Nguyen, H.-M., 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 Analysis, 225, 3, 993-1023 (2017) · Zbl 1417.93067
[8] Deutscher, J.; Kerschbaum, S., Backstepping control of coupled linear parabolic pides with spatially-varying coefficients, IEEE Transactions on Automatic Control, PP, 99 (2018), 1-1 · Zbl 1423.93155
[9] Espitia, N., Polyakov, A., Efimov, D., & Perruquetti, W. (2018). On continuous boundary time-varying feedbacks for fixed-time stabilization of coupled reaction-diffusion systems. In 57th IEEE conference on decision and control; Espitia, N., Polyakov, A., Efimov, D., & Perruquetti, W. (2018). On continuous boundary time-varying feedbacks for fixed-time stabilization of coupled reaction-diffusion systems. In 57th IEEE conference on decision and control
[10] Galaktionov, V. A.; Vasquez, J. L., Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equation, Archive for Rational Mechanics and Analysis, 129, 3, 225-244 (1995) · Zbl 0827.35055
[11] Haimo, V. T., Finite time controllers, SIAM Journal on Control and Optimization, 24, 4, 760-770 (1986) · Zbl 0603.93005
[12] Lopez-Ramirez, F.; Polyakov, A.; Efimov, D.; Perruquetti, W., Finite-time and fixed-time observer design: implicit lyapunov function approach, Automatica, 87, 52-60 (2018) · Zbl 1378.93095
[13] Meurer, T.; Krstic, M., Finite-time multi-agent deployment: a nonlinear pde motion planning approach, Automatica, 47, 11, 2534-2542 (2011) · Zbl 1228.93013
[14] Meurer, T.; Kugi, A., Tracking control for boundary controlled parabolic pdes with varying parameters: combining backstepping and differential flatness, Automatica, 45, 5, 1182-1194 (2009) · Zbl 1162.93016
[15] Michalska, M.; Szynal, J., A new bound for the laguerre polynomials, 5th Int. symp. on orthogonal polynomials, special functions and their applications. 5th Int. symp. on orthogonal polynomials, special functions and their applications, Journal of Computational and Applied Mathematics, 133, 1, 489-493 (2001) · Zbl 0993.33005
[16] Orlov, Y., Theory of optimal systems with generalized controls (1988), Nauka: Nauka Moscow, (in Russian)
[17] Orlov, Y., Schwartz’ distributions in nonlinear setting: applications to differential equations, filtering and optimal control, Mathematical Problems in Engineering, 8, 4-5, 367-387 (2002) · Zbl 1066.93023
[18] Orlov, Y., Discontinuous systems: Lyapunov analysis and robust synthesis under uncertainty conditions (2009), Springer-Verlag · Zbl 1180.37004
[19] Orlov, Y.; Pisano, A.; Pilloni, A.; Usai, E., Output feedback stabilization of coupled reaction-diffusion processes with constant parameters, SIAM Journal on Control and Optimization, 55, 6, 4112-4155 (2017) · Zbl 1386.93224
[20] Pazy, A., Semigroups of linear operators and applications to partial differential equations (1983), Springer · Zbl 0516.47023
[21] Perrollaz, V.; Rosier, L., Finite-time stabilization of \(2 \times 2\) hyperbolic systems on tree-shaped networks, SIAM Journal on Control and Optimization, 52, 1, 143-163 (2014) · Zbl 1295.35308
[22] Pisano, A.; Orlov, Y., On the iss properties of a class of parabolic dps’ with discontinuous control using sampled-in-space sensing and actuation, Automatica, 81, 447-454 (2017) · Zbl 1372.93071
[23] Pisano, A.; Orlov, Y.; Usai, E., Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques, SIAM Journal on Control and Optimization, 49, 2, 363-382 (2011) · Zbl 1217.93136
[24] Polianin, A. D., Handbook of linear partial differential equations for engineers and scientists (2002), Chapman & Hall/CRC · Zbl 1027.35001
[25] Polyakov, A., Coron, J.-M., & Rosier, L. (2017). On boundary finite-time feedback control for heat equation. In 20th IFAC world congress; Polyakov, A., Coron, J.-M., & Rosier, L. (2017). On boundary finite-time feedback control for heat equation. In 20th IFAC world congress
[26] Polyakov, A.; Coron, J.-M.; Rosier, L., On homogeneous finite-time control for evolution equation in hilbert space, IEEE Transactions on Automatic Control, 63, 9, 3143-3150 (2018) · Zbl 1423.93160
[27] Polyakov, A.; Efimov, D.; Perruquetti, W., Finite-time and fixed-time stabilization: implicit lyapunov function approach, Automatica, 51, 1, 332-340 (2015) · Zbl 1309.93135
[28] Smyshlyaev, A.; Krstic, M., Closed-form boundary state feedbacks for a class of 1-d partial integro-differential equations, IEEE Transactions on Automatic Control, 49, 12, 2185-2202 (2004) · Zbl 1365.93193
[29] Smyshlyaev, A.; Krstic, M., On control design for pdes with space-dependent diffusivity or time-dependent reactivity, Automatica, 41, 9, 1601-1608 (2005) · Zbl 1086.93023
[30] Song, Y.-D.; Wang, Y.-J.; Holloway, J.-C.; Krstic, M., Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time, Automatica, 83, 243-251 (2017) · Zbl 1373.93136
[31] Song, Y.-D.; Wang, Y.-J.; Krstic, M., Time-varying feedback for stabilization in prescribed finite time, Utkin 80-sliding mode control and observation. Utkin 80-sliding mode control and observation, International Journal of Robust and Nonlinear Control (2018)
[32] Szegö, G., Orthogonal polynomials (1975), American Mathematical Society Providence: American Mathematical Society Providence Rhode Island · JFM 61.0386.03
[33] Vazquez, R.; Krstic, M., Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients, IEEE Transactions on Automatic Control, 62, 4, 2026-2033 (2017) · Zbl 1366.93533
[34] Vazquez, R.; Trélat, E.; Coron, J.-M., Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow, Discrete & Continuous Dynamical Systems - B, 10, 4, 925-956 (2008) · Zbl 1147.93011
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