# American Institute of Mathematical Sciences

2013, 2013(special): 477-487. doi: 10.3934/proc.2013.2013.477

## Existence of sliding motions for nonlinear evolution equations in Banach spaces

 1 Free University of Bolzano/Bozen, Piazza Università 1, 39100 Bolzano, Italy

Received  September 2012 Revised  September 2013 Published  November 2013

In this paper the issue of existence of sliding motions for a class of control systems of parabolic type is considered. The operator satisfies standard hemicontinuity, monotonicity and coercivity assumptions; the control law is finite-dimensional and enters linearly in the equation. By using a Faedo-Galerkin approach, a family of finite-dimensional ODEs is constructed and an approximating sequence of sliding motions is obtained using classical variable structure control techniques. Previous results on the convergence of the approximations are extended, by taking into consideration more general growth assumptions on the feedbacks. A detailed description of the approach for semilinear partial differential equations with Neumann boundary control is discussed.
Citation: Laura Levaggi. Existence of sliding motions for nonlinear evolution equations in Banach spaces. Conference Publications, 2013, 2013 (special) : 477-487. doi: 10.3934/proc.2013.2013.477
##### References:
 [1] G. Bartolini and T. Zolezzi, Control of nonlinear variable structure systems,, J. Math. Anal. Appl., 118 (1986), 42. [2] S. Drakunov and Ü. Özgüner, Generalized sliding modes for manifold control of distributed parameter systems,, in, 193 (1994), 109. [3] S. V. Drakunov and V. I. Utkin, Sliding mode control in dynamic systems,, Internat. J. Control, 55 (1992), 1029. [4] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Mathematics and its Applications (Soviet Series), (1988). [5] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, (1985). [6] L. Levaggi, Infinite dimensional systems' sliding motions,}, Eur. J. Control, 8 (2002), 508. [7] L. Levaggi, Sliding modes in Banach spaces,, Differ. Integral Equ., 15 (2002), 167. [8] L. Levaggi, High-gain feedback and sliding modes in infinite dimensional systems,, Control Cybernet., 33 (2004), 33. [9] L. Levaggi, Variable structure control for parabolic evolution equations,, in, (2005), 1234. [10] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires,", (French) Dunod; Gauthier-Villars, (1969). [11] J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations,", Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, (1971). [12] Y. Orlov, Discontinuous unit feedback control of uncertain infinite-dimensional systems,, IEEE Trans. Automat. Control, 45 (2000), 834. [13] Y. Orlov and D. Dochain, Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor,, IEEE Trans. Automat. Control, 47 (2002), 1293. [14] Y. Orlov, Y. Lou and Panagiotis D. Christofides, Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control,, Internat. J. Control, 77 (2004), 1115. [15] Y. Orlov, A. Pisano and E. Usai, Continuous state-feedback tracking of an uncertain heat diffusion process,, Systems Control Lett., 59 (2010), 754. [16] Y. Orlov and V. Utkin, Use of sliding modes in distributed system control problems,, Automat. Remote Control, 43 (1982), 1127. [17] Y. Orlov and V. Utkin, Sliding mode control in indefinite-dimensional systems,, Automatica J. IFAC, 23 (1987), 753. [18] Y. Orlov and V. Utkin, Unit sliding mode control in infinite-dimensional systems,, Adaptive learning and control using sliding modes. Appl. Math. Comput. Sci., 8 (1998), 7. [19] A. Pisano, Y. Orlov and E. Usai, Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques,, SIAM J. Control Optim., 49 (2011), 363. [20] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Mathematical Surveys and Monographs, (1997). [21] V. Utkin, "Sliding Modes in Control and Optimization,", Communications and Control Engineering Series, (1992). [22] E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/A,", Linear monotone operators. Translated from the German by the author and Leo F. Boron. Springer-Verlag, (1990). [23] E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B,", Nonlinear monotone operators. Translated from the German by the author and Leo F. Boron. Springer-Verlag, (1990). [24] T. Zolezzi, Variable structure control of semilinear evolution equations,, in, (1989), 997.

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##### References:
 [1] G. Bartolini and T. Zolezzi, Control of nonlinear variable structure systems,, J. Math. Anal. Appl., 118 (1986), 42. [2] S. Drakunov and Ü. Özgüner, Generalized sliding modes for manifold control of distributed parameter systems,, in, 193 (1994), 109. [3] S. V. Drakunov and V. I. Utkin, Sliding mode control in dynamic systems,, Internat. J. Control, 55 (1992), 1029. [4] A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Mathematics and its Applications (Soviet Series), (1988). [5] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, (1985). [6] L. Levaggi, Infinite dimensional systems' sliding motions,}, Eur. J. Control, 8 (2002), 508. [7] L. Levaggi, Sliding modes in Banach spaces,, Differ. Integral Equ., 15 (2002), 167. [8] L. Levaggi, High-gain feedback and sliding modes in infinite dimensional systems,, Control Cybernet., 33 (2004), 33. [9] L. Levaggi, Variable structure control for parabolic evolution equations,, in, (2005), 1234. [10] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires,", (French) Dunod; Gauthier-Villars, (1969). [11] J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations,", Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, (1971). [12] Y. Orlov, Discontinuous unit feedback control of uncertain infinite-dimensional systems,, IEEE Trans. Automat. Control, 45 (2000), 834. [13] Y. Orlov and D. Dochain, Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor,, IEEE Trans. Automat. Control, 47 (2002), 1293. [14] Y. Orlov, Y. Lou and Panagiotis D. Christofides, Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control,, Internat. J. Control, 77 (2004), 1115. [15] Y. Orlov, A. Pisano and E. Usai, Continuous state-feedback tracking of an uncertain heat diffusion process,, Systems Control Lett., 59 (2010), 754. [16] Y. Orlov and V. Utkin, Use of sliding modes in distributed system control problems,, Automat. Remote Control, 43 (1982), 1127. [17] Y. Orlov and V. Utkin, Sliding mode control in indefinite-dimensional systems,, Automatica J. IFAC, 23 (1987), 753. [18] Y. Orlov and V. Utkin, Unit sliding mode control in infinite-dimensional systems,, Adaptive learning and control using sliding modes. Appl. Math. Comput. Sci., 8 (1998), 7. [19] A. Pisano, Y. Orlov and E. Usai, Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques,, SIAM J. Control Optim., 49 (2011), 363. [20] R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Mathematical Surveys and Monographs, (1997). [21] V. Utkin, "Sliding Modes in Control and Optimization,", Communications and Control Engineering Series, (1992). [22] E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/A,", Linear monotone operators. Translated from the German by the author and Leo F. Boron. Springer-Verlag, (1990). [23] E. Zeidler, "Nonlinear Functional Analysis and its Applications. II/B,", Nonlinear monotone operators. Translated from the German by the author and Leo F. Boron. Springer-Verlag, (1990). [24] T. Zolezzi, Variable structure control of semilinear evolution equations,, in, (1989), 997.
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