# Stability and Asymptotic Behavior of the Causal Operator Dynamical Systems Using Nonlinear Variation of Parameters

### Abstract

The operator T from a domain D into the space of measurable functions is called a nonanticipating operator if the past informations is independent from the future outputs. We will use the solution to the operator di§erential equation y0(t) = A(t)y(t)+f(t; y(t); T(y)(t)) to analyze the solution of this operator di§erential equation which is generated by a perturbation (t) = g(t; yt ; T2(yt)). When this perturbation is from a measurable space then the existence and uniqueness of the solution to the operator di§erential equation will be studied. Finally, we use the nonlinear variation of parameters for nonanticipating operator di§erential equations to study the stability and asymptotic behavior of the equilibrium solution.

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### References

R. R. Ahangar, Nonanticipating Dynamical Model and Optimal Control, Applied Mathematics Letter, vol. 2, No. 1, pp.15-18, 1989.

R. R. Ahangar and E. Salehi : "Automatic Controls for Nonlinear Dynamical Systems with Lipschitzian Trajectories" Journal of Mathematical Analysis and Application (JMAA), 268, pp. 400-405 (2002).

R. R. Ahangar, "Optimal Control Solution to Operator Differential Equations Using Dynamic Programming", Proceedings of the 2005 International Conference on Scientific Computing, Las Vegas, Nevada, June 20-23, 2005, pp. 16-22.

R. R. Ahangar and E. Salehi : "Optimal Automatic Controls Solution to

Nonlinear Operator Dynamical Systems", Proceeding of The International

Conference on Scientific Computing, (CSC-06), June 26-29, 2006, Las Vegas, Nevada.

R. R. Ahangar, "Optimal Control Solution to Nonlinear Causal Operator Systems with Target State", FCS (Foundations of Computer Science), WORLD COMP, 2008, pp. 218-223.

R.R. Ahangar, "Variation of Parameters for Causal Operator Differential Equations", Journal of Applied Mathematics, ID:7403787,

http://www.scirp.org/journal/am, toapear Dec. 2017.

V.M. Bogdan, "Existence and Uniqueness of Solution for a class of Nonlinear Operator Differential Equations Arising in Automatic Spaceship Navigation," NASA technical Administration, Springfield, VA, 1981.

V.M. Bogdan, "Existence and Uniqueness of Solution to Nonlinear Operator Differential Equations Generalizing Dynamical Systems of Automatic Spaceship Navigation, "Nonlinear Phenomena in Mathematical Sciences, Academic Press, p. 123-136, 1982.

J. Hale Jack, "Theory of Functional Differential Equations", Spring Verlag, 1977.

J. Hale, J. Arrieta, A. N. Carvalho, "A Damped Hyperbolic Equation with Critical Exponent", Commun. in Partial Differential Equations, 17(5&6), 841-866 (1992).

V. M. Alekseev, "An estimate for the perturbations of the solution of ordinary differential equations, Vestn. Mosk. Univ. ser. 1, Math. Mek., No. 2 (1961), 28-36. MR: 23 # A2596.

V. Lakshmikantham V. and S. Leela, "Nonlinear Differential Equations in Abstract Spaces", Pergamon Press, 1981.

V. Lakshmikantham and G.E. Ladas, "Differential Equations in Abstract Space", Academic Press, 1972.

V. Lashmikantham and G. S. Ladde, "Random Differential Equations", Academic Press, 1980.

Lord M. E., "Stability and Asymptotic Equivalence of Perturbations of Nonlinear Systems of Differential Equations" Department of Mathematics, The University of Txas at Arlington, Technical Report No. 139, August 1980.

Lord M.E., Mitchell A. R., "A New Approach to the Method of Nonlinear Variation of Parameters", Appl. Math. Comp., 4, 95-105, Elsevier NorthHoland, Inc., 1978.

F. Brauer, "Perturbations of Nonlinear Systems of Differential Equations", J. Math. Anal. Appl., 14, 198-206, 1966.

F. Brauer, "Perturbation of Nonlinear Systems of Differential Equations II.", J. Math. Anal. Appl., 17, 418-434, (1967).

R. D. Driver, "Ordinary and Delay Differential Equations", Applied Mathematical Sciences 20, Springer-Verlag, 1977.

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