Approximate controllability of second-order stochastic distributed implicit functional differential systems with infinite delay. (English) Zbl 1176.93013

Summary: In this paper, sufficient conditions for the approximate controllability of the following stochastic semilinear abstract functional differential equations with infinite delay are established \[ \begin{aligned} d\bigl[x^{\prime}(t)-g(t,x_{t})\bigr]&=\bigl[Ax(t)+f(t,x_{t})+Bu(t)\bigr]dt+G(t,x_{t})dW(t),\quad \text{a.e on}\;t\in J:=[0,b],\\x_{0}&=\varphi\in {\mathfrak{B}},\\x^{\prime}(0)&=\psi \in H,\end{aligned} \] where the state \(x(t)\in H,x_{t}\) belongs to phase space \({\mathfrak{B}}\) and the control \(u(t)\in L^{\mathcal F}_2 (J,U)\), in which \(H,U\) are separable Hilbert spaces and \(d\) is the stochastic differentiation. The results are worked out based on the comparison of the associated linear systems. An application to the stochastic nonlinear wave equation with infinite delay is given.


93B05 Controllability
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93C25 Control/observation systems in abstract spaces
93C20 Control/observation systems governed by partial differential equations
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[1] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) · Zbl 0761.60052
[2] Grecksch, W., Tudor, C.: Stochastic Evolution Equations: A Hilbert Space Approach. Akademic Verlag, Berlin (1995) · Zbl 0831.60069
[3] Tsokos, C.P., Padjett, W.J.: Random Integral Equations with Applications to Life Sciences and Engineering. Academic Press, New York (1974) · Zbl 0287.60065
[4] Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997) · Zbl 0892.60057
[5] Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, London (1991) · Zbl 0762.60050
[6] Fitzgibbon, W.E.: Global existence and boundedness of solutions to the extensible beam equation. SIAM J. Math. Anal. 13, 739–745 (1982) · Zbl 0506.73057
[7] Mahmudov, N.I., McKibben, M.A.: Abstract second order damped Mckean-Vlasov stochastic evolution equations. Stoch. Anal. Appl. 24, 303–328 (2006) · Zbl 1102.35044
[8] McKibben, M.A.: Second order damped functional stochastic evolution equation in Hilbert spaces. Dyn. Syst. Appl. 12, 467–488 (2003) · Zbl 1057.34056
[9] Balasubramaniam, P., Park, J.Y.: Nonlocal Cauchy problem for second order stochastic evolution equations in Hilbert spaces. Dyn. Syst. Appl. 16, 713–728 (2007) · Zbl 1151.34046
[10] Astrom, K.J.: Introduction to Stochastic Control Theory. Academic Press, New York (1970)
[11] Dauer, J.P., Mahmudov, N.I.: Controllability of stochastic semilinear functional differential equations in Hilbert spaces. J. Math. Anal. Appl. 290, 373–394 (2004) · Zbl 1038.60056
[12] Ehrhard, M., Kliemann, W.: Controllability of stochastic linear systems. Syst. Control Lett. 2, 145–153 (1982) · Zbl 0493.93009
[13] Mahmudov, N.I.: Controllability of linear stochastic systems. IEEE Trans. Autom. Control 46, 724–731 (2001) · Zbl 1031.93034
[14] Balasubramaniam, P., Dauer, J.P.: Controllability of semilinear stochastic delay evolutions in Hilbert spaces. Int. J. Math. Math. Sci. 31, 157–166 (2002) · Zbl 1031.93023
[15] Balasubramaniam, P., Dauer, J.P.: Controllability of semilinear stochastic evolutions with time delays. Publ. Math. Debercen 63, 279–291 (2003) · Zbl 1051.34065
[16] Park, J.Y., Balasubramaniam, P., Kumaresan, N.: Controllability for neutral stochastic functional integrodifferential infinite delay systems in abstract space. Numer. Funct. Anal. Optim. 28, 1–18 (2007) · Zbl 1130.93018
[17] Balasubramaniam, P., Park, J.Y., Muthukumar, P.: Approximate controllability of neutral stochastic functional differential systems with infinite delay. Stoch. Anal. Appl., accepted (2008) · Zbl 1186.93014
[18] Henriquez, H.R., Hernandez, M.E.: Approximate controllability of second order distributed implicit functional systems. Nonlinear Anal.: Theory Methods Appl. 70, 1023–1039 (2009) · Zbl 1168.34050
[19] Gard, T.C.: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988) · Zbl 0628.60064
[20] Fattorini, H.O.: Second order linear differential equations in Banach spaces. In: North-Holland Mathematics Studies, vol. 108. North-Holland, Amsterdam (1985) · Zbl 0564.34063
[21] Haase, M.: The Functional Calculus for Sectorial Operators. Birkhauser, Basel (2006) · Zbl 1101.47010
[22] Travis, C.C., Webb, G.F.: Compactness, regularity and uniform continuity properties of strongly continuous cosine families. Houston J. Math. 3(4), 555–567 (1977) · Zbl 0386.47024
[23] Travis, C.C., Webb, G.F.: Second order differential equations in Banach space. In: Proceedings International Symposium on Nonlinear Equations in Abstract Spaces, pp. 331–361. Academic Press, New York (1987)
[24] Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkc. Ekvacioj Ser. Int. 21, 11–41 (1978) · Zbl 0383.34055
[25] Hernandez, E., Henriquez, H.R.: Existence results for second order partial neutral functional differential equations. Dyn. Contin., Discrete Impuls. Syst. 15, 645–670 (2008)
[26] Hale, J.K., Verduyn Lunel, S.M.: Introduction to functional differential equations. In: Applied Mathematical Sciences, vol. 99. Springer, New York (1993) · Zbl 0787.34002
[27] Sadovskii, B.N.: On a fixed point principle. Funct. Anal. Appl. 1, 74–76 (1967) · Zbl 0165.49102
[28] Sukavanam, N.: Approximate controllability of semilinear control systems with growing nonlinearity. In: Lect. Notes in Pure and Applied Maths., vol. 142, pp. 353–357. Marcel Dekker, New York (1993) · Zbl 0790.93020
[29] Henriquez, H.R.: On non-exact controllable systems. Int. J. Control 42, 71–83 (1985) · Zbl 0569.93008
[30] Zhou, H.X.: A note on approximate controllability for semilinear one-dimensional heat equations. Appl. Math. Optim. 8, 275–285 (1982) · Zbl 0503.49023
[31] Naito, K.: Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25, 715–722 (1987) · Zbl 0617.93004
[32] Hino, Y., Murakami, S., Naito, T.: Functional-Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991) · Zbl 0732.34051
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