Lyapunov equations for time-varying linear systems. (English) Zbl 0678.93051

The object of this paper is to give characterizations of the exponential stability of linear time-varying deterministic and stochastic systems in Hilbert spaces, using the Lyapunov differential equations. Generalizations of well-known results of Datko about exponential stability of evolutionary processes are obtained.
Reviewer: M.Megan


93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93E15 Stochastic stability in control theory
35B35 Stability in context of PDEs
34D20 Stability of solutions to ordinary differential equations
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[1] Arnold, L., Stochastic differential equations: theory and applications, (1974), John Wiley & Sons New York
[2] Barbu, V.; Da Prato, G., Hamilton—jacobi equations in Hilbert spaces, (1983), Pitman London · Zbl 0508.34001
[3] Da Prato, G., Some results on linear stochastic evolution equations in Hilbert spaces by the semi-group method, Stoch. anal. appl., 1, 57-88, (1983) · Zbl 0511.60055
[4] Da Prato, G.; Ichikawa, A., Stability and quadratic control for linear stochastic equations with unbonded coefficients, Boll. un. mat. ital. B, 4, 6, 987-1001, (1985) · Zbl 0598.93062
[5] Da Prato, G.; Ichikawa, A., Bounded solutions on the real line to non-autonomous Riccati equations, Atti acc. naz. lincei, 79, 5, 108-112, (1985) · Zbl 0651.34063
[6] G. Da Prato and A. Ichikawa, Optimal control for linear time varying systems (submitted for publication). · Zbl 0692.49006
[7] Datko, R., Extending a theorem of A.M. Liapunov to Hilbert space, J. math. anal. appl., 32, 610-616, (1970) · Zbl 0211.16802
[8] Datko, R., Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. math. anal., 3, 428-445, (1972) · Zbl 0241.34071
[9] Haussmann, U.G., Asymptotic stability of the linear ito equation in infinite dimensions, J. math. anal. appl., 65, 219-235, (1978) · Zbl 0385.93051
[10] Ichikawa, A., Optimal control of a linear stochastic evolution equation with state and control dependent noise, (), 383-401
[11] Ichikawa, A., Dynamic programming approach to stochastic evolution equations, SIAM J. control optim., 17, 153-174, (1979) · Zbl 0434.93069
[12] Ichikawa, A., Equivalence of Lp stability and exponential stability for a class of nonlinear semigroups, Nonlinear anal. T.M.A., 8, 805-815, (1984) · Zbl 0547.47041
[13] Ichikawa, A., Bounded solutions and periodic solutions of a linear stochastic evolution equation, () · Zbl 0422.93102
[14] Pazy, A., On the applicability of Liapunov theorem in Hilbert space, SIAM J. math. anal., 3, 291-294, (1972) · Zbl 0242.47028
[15] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer New York · Zbl 0516.47023
[16] Tanabe, H., Equations of evolution, (1979), Pitman London
[17] Zabczyk, J., Remarks on the control of discrete-time distributed parameter systems, SIAM J. control optim., 12, 721-735, (1974) · Zbl 0254.93027
[18] Zabczyk, J., On stability of infinite dimensional stochastic systems, (), 273-281
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