×

An approach to non-linear control problems using fixed-point methods, degree theory and pseudo-inverses. (English) Zbl 0563.93013

This paper is devoted to the problem of state estimation and controllability of the system (1) \(\dot z=Az+N(z)+Bu,\), \(z(0)=z_ 0\), \(t\in [t_ 0,T]\), (2) \(y=Cz\), where \(z\in Z\) (Z a Banach space), \(u\in U\); A, B, C are bounded linear operators and N(z) is a nonlinear, continuous operator. The solution of (1) is understood in a mild sense. For \(B=0\), if the linear part of the system (1)-(2) is continuously initially observable, the state of the system (1) can be constructed, given an observation y, from a ball in the space Y, on [0,T]. The proof is based on the Leray-Schauder theorem. If the linear part of the system (1) is exactly controllable on [0,T] to some subspace V (from the origin), then the state of the system (1) can be steered from the origin to any final state z(T) lying in a ball in V. The proof relies on Nussbaum’s fixed point theorem. The radius of both balls is estimated. As examples, two wave equations with nonlinear perturbations are considered.
Reviewer: Z.Wyderka

MSC:

93B05 Controllability
93B07 Observability
93C10 Nonlinear systems in control theory
47H10 Fixed-point theorems
55M25 Degree, winding number
55M20 Fixed points and coincidences in algebraic topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carmichael N., Appl Math Optim 9 pp 133– (1982) · Zbl 0502.93035
[2] Istratescu, V. 1981. ”Fixed Point Theory”. Netherlands: D Reidel. · Zbl 0465.47035
[3] Kantorovich, L.W and Akilov, G.P. 1964. ”Functional analysis in normal spaces”. New York: Macmillan. · Zbl 0127.06104
[4] Krasnosel’skiì M.A, Amer Math Soc Translations 10 pp 345– (1958)
[5] Leray l.J, Ann Sci Ecole Norm Sup 51 pp 45– (1934)
[6] Magnusson, K., Pritchard, A.J and Quinn, M.D. Proceedings of 1981 Banach Centre Semester on Control. University of Warwick.
[7] Martin, R.H. 1976. ”Nonlinear operators and differential equations in Banach spaces”. New York: John Wiley.
[8] Nashed M.Z, J Math Mech 18 pp 767– (1969)
[9] Nussbauro R.P, Bull Amer Math Soc 75 pp 490– (1969) · Zbl 0174.45402
[10] Petryshyn W.V, Trans Amer Math Soc 182 pp 323– (1973)
[11] Rail, L.B. 1969. ”Computational solution of nonlinear operator equations.”. New York: J Wiley.
[12] Russell D.L, J Math Anal Appl 18 pp 542– (1967) · Zbl 0158.10201
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.