Infinite dimensional systems’ sliding motions. (English) Zbl 1293.93161
Summary: We show how, using differential inclusions and viability theory it is possible to define sliding modes for (feedback) controlled semilinear differential equations in Banach spaces. In order to compare this definition with the equivalent control method proposed by V. Utkin and Yu. Orlov for infinite dimensional systems, we introduce the notion of extended equivalent control. This allows the interpretation of the sliding motion by a classical semigroup approach. Then we are able to prove that, if the sliding manifold satisfies suitable regularity hypotheses, the projected evolution found by means of the extended equivalent control and our sliding mode do coincide. We then apply these results to the problem of stabilization of a heat equation and of a delay differential equation.
References:
| [1] | Balakrishnan, A. V., Applied functional analysis (1976), Springer-Verlag · Zbl 0333.93051 |
| [2] | Carja, O.; Vrabie, I. I., Some new viability results for semilinear differential inclusions, No. Dea Nonlinear Diff Equations Appl, 4, 401-424 (1997) · Zbl 0876.34069 |
| [3] | Carja, O.; Vrabie, I. I., Viability for semilinear differential inclusions via weak sequential tangency condition, J Math Anal Appl, 262, 1, 24-38 (2001) · Zbl 1011.34051 |
| [4] | Curtain, R. F.; Zwart, H. J., An introduction to infinite- dimensional linear systems theory, Texts in applied mathematics (1995), Springer-Verlag, vol. 21. · Zbl 0839.93001 |
| [5] | Filippov, A., Differential equations with discontinuous right-hand sides (1988), Kluwer Academic Publishers · Zbl 0664.34001 |
| [6] | Levaggi, L., Sliding modes in Banach spaces, Diff Integral Equations, 15, 2, 167-189 (2002) · Zbl 1031.93050 |
| [7] | Orlov Yu, V., Discontinuous unit feedback control of uncertain infinite-dimensional systems, IEEE Trans Autom Control, 45, 5, 834-843 (2000) · Zbl 0973.93018 |
| [8] | Orlov, Y.; Dochain, D., Discontinuous feedback stabiliza tion of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor, IEEE Trans Automat Control, 47, 8, 1293-1304 (2002) · Zbl 1364.93660 |
| [9] | Orlov Yu, V.; Utkin, V. I., Sliding mode control in indefinite-dimensional systems, Automatica, 23, 6, 753-757 (1987) · Zbl 0661.93036 |
| [10] | Orlov Yu, V.; Utkin, V. I., Unit sliding mode control in infinite-dimensional systems, Appl Math Comput Sci, 8, 1, 7-20 (1998) · Zbl 0907.93015 |
| [11] | Pazy, A., Semigroups of linear operators and applications to partial differential equations (1983), Springer-Verlag · Zbl 0516.47023 |
| [12] | Utkin, V. I., Sliding modes in control optimization, communications and control engineering series (1992), Springer-Verlag · Zbl 0748.93044 |
| [13] | Vrabie, I. I., Compactness methods for nonlinear evolutions, Pitman monographs and surveys in pure and applied mathematics (1987), John Wiley & Sons, Inc, vol. 32 · Zbl 0721.47050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
