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Stabilization of a class of distributional convolution equations. (English) Zbl 0566.93048
The paper deals with the stabilization of the system \[ (1)\quad \dot x(t)=A_ 0x(t)+A_ 1x(t-h)+\int^{h}_{0}A(s)x(t-s)ds+B_ 0u(t) \] by using the linear state feedback \(u(t)=-Kx(t)\). It is shown, in particular, that if \((A_ 0,B_ 0)\) is controllable then the system (1) is stabilizable if \(\| A_ 1\|\) and \(\| A(s)\|\), \(0\leq s\leq h\), are sufficiently small. An illustrative example is also given.
Reviewer: V.Marchenko

MSC:
93D15 Stabilization of systems by feedback
34K35 Control problems for functional-differential equations
93C05 Linear systems in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
46F10 Operations with distributions and generalized functions
42A85 Convolution, factorization for one variable harmonic analysis
34K20 Stability theory of functional-differential equations
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[1] ARTSTEIN Z., I.E.E.E. Trans. autom. Control 27 pp 869– (1982) · Zbl 0486.93011 · doi:10.1109/TAC.1982.1103023
[2] DERESE I., Int. J. Control 31 pp 219– (1980) · Zbl 0439.93043 · doi:10.1080/00207178008961039
[3] FELIACHI A., I.E.E.E. Trans. autom. Control 26 pp 586– (1981) · Zbl 0477.93048 · doi:10.1109/TAC.1981.1102653
[4] GRUBER M., I.E.E.E. Trans. autom. Control 14 pp 465– (1969) · doi:10.1109/TAC.1969.1099279
[5] HALE J., Functional Differential Equations (1971) · Zbl 0222.34003 · doi:10.1007/978-1-4615-9968-5
[6] IKEDA M., I.E.E.E. Trans. autom. Control 24 pp 369– (1979) · Zbl 0399.93037 · doi:10.1109/TAC.1979.1102025
[7] KAMEN E. W., I.E.E.E. Trans. autom. Control 27 pp 367– (1982) · Zbl 0517.93047 · doi:10.1109/TAC.1982.1102916
[8] KWON W. H., I.E.E.E. Trans. autom. Control 25 pp 266– (1980) · Zbl 0438.93055 · doi:10.1109/TAC.1980.1102288
[9] MORI T., Automatica 19 pp 571– (1983) · Zbl 0544.93055 · doi:10.1016/0005-1098(83)90013-4
[10] PATEL N. K., Int. J. Control 36 pp 303– (1982) · Zbl 0482.93046 · doi:10.1080/00207178208932894
[11] PATEL R. V., Multivariable System Theory and Design (1982) · Zbl 0475.93005
[12] THOWSEN A., Int. J. Systems Sci. 12 pp 1485– (1981) · Zbl 0504.93049 · doi:10.1080/00207728108963833
[13] WATANABE K., I.E.E.E. Trans. autom. Control 28 pp 506– (1983) · Zbl 0506.93027 · doi:10.1109/TAC.1983.1103258
[14] Wu W. T., Int. J. Control 38 pp 211– (1983) · Zbl 0511.93040 · doi:10.1080/00207178308933069
[15] ZEMANIAN A. H., Distribution Theory and Transform Analysis (1965) · Zbl 0127.07201
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