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Controllability of stochastic semilinear functional differential equations in Hilbert spaces. (English) Zbl 1038.60056
The authors investigate the following semilinear controlled equation with delay in a Hilbert space: $$ dX(t)=[-AX(t)+Bu(t)+f(t,X_t)]dt+g(t,X_t)dW(t),\quad t\in[0,T], $$ where $W$ is a Hilbert space-valued Wiener process, $A$ generates an analytic semigroup and $f(t,\cdot)$, $g(t,\cdot)$ are functions on the path space, $X_t=\{X(t+s)(\omega): s\in[-r,0]\}$. The basic space is $D(A^\alpha)=$ the domain of the fractional power operator $A^\alpha$. It is proved that under Lipschitz and growth conditions on $f$, $g$, approximate controllability of the deterministic system implies approximate controllability of the stochastic one (i.e., the $L_p$-closure of possible values of $X(T,u)$, as $u$ varies, is the whole $L_p$, $p$ being related to $\alpha$). A criterion of exact controllability is also given (the assumption on the compactness of the semigroup is then dropped). Finally, an application to a stochastic heat equation is shown.

MSC:
60H15Stochastic partial differential equations
93E99Stochastic systems and stochastic control
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References:
[1] Ahmed, N. U.; Ding, X.: A semilinear mckean--Vlasov stochastic evolution equation in Hilbert spaces. Stochastic process. Appl. 60, 65-85 (1995) · Zbl 0846.60060
[2] Balachandran, K.; Dauer, J.: Controllability of nonlinear systems in Banach spaces. J. optim. Theory appl. 115, 7-28 (2002) · Zbl 1023.93010
[3] Balasubramaniam, P.; Dauer, J.: Controllability of semilinear stochastic delay evolution equations in Hilbert spaces. Internat. J. Math. math. Sci., 1-10 (2002) · Zbl 1017.93018
[4] Caraballo, T.: Asymptotic exponential stability of stochastic partial differential equations with delay. Stochastics 33, 27-47 (1990) · Zbl 0723.60074
[5] Da Prato, G.; Kwapien, S.; Sabczyk, J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23, 1-23 (1987)
[6] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions. (1992) · Zbl 0761.60052
[7] Dauer, J.; Mahmudov, N. I.: Approximate controllability of semilinear functional equations in Hilbert spaces. J. math. Anal. appl. 273, 310-327 (2002) · Zbl 1017.93019
[8] Klamka, J.: Schauder’s fixed point theorem in nonlinear controllability problems. Control cybernet. 29, 153-165 (2000) · Zbl 1011.93001
[9] Mahmudov, N. I.: Controllability of linear stochastic systems. IEEE trans. Automat. control 46, 724-731 (2001) · Zbl 1031.93034
[10] Mahmudov, N. I.: Controllability of linear stochastic systems in Hilbert spaces. J. math. Anal. appl. 259, 64-82 (2001) · Zbl 1031.93032
[11] Mahmudov, N. I.: On controllability of semilinear stochastic systems in Hilbert spaces. IMA J. Math. control inform. 19, 363-376 (2002) · Zbl 1138.93313
[12] N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., submitted for publication · Zbl 1084.93006
[13] Mohammed, S. E.: Stochastic functional differential equations. Research notes in mathematics 99 (1984) · Zbl 0584.60066
[14] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023
[15] Taniguchi, T.: Almost sure exponential stability for stochastic partial functional differential equations. Stochastic anal. Appl. 16, 965-975 (1998) · Zbl 0911.60054
[16] Taniguchi, T.; Liu, K.; Truman, A.: Existence, uniqueness and asymptotic behaviour of mild solutions to stochastic functional equations in Hilbert spaces. J. differential equations 181, 72-91 (2002) · Zbl 1009.34074
[17] Triggiani, R.: On the lack of exact controllability for mild solutions in Banach spaces. J. math. Anal. appl. 50, 438-446 (1975) · Zbl 0305.93013
[18] Wu, J.: Theory and applications of partial functional differential equations. Applied mathematical sciences 119 (1996) · Zbl 0870.35116