Approximate controllability of second-order neutral stochastic evolution equations. (English) Zbl 1121.34082

The authors consider the following problem, \[ d[x'(t)-f_1(t,x(t))]=[Ax(t)+Bu(t)+f_2(t,x(t))]\,dt+g(t,x(t)\,dW(t), \;\;t\in [0,T], \]
\[ x(0)=\xi,\;x'(0)=\varsigma, \]
in a real separable Hilbert space \(H\), \(A: D(A)\subset H\to H\) is the infinitesimal generator of a strongly continuous cosine family on \(H,\) \(W\) is a \(K\)-valued Wiener process with incremental covariance given by the nuclear operator \(Q\) defined on a complete probability space \((\Omega,{\mathcal F}, P)\) equipped with a normal filtration \(({\mathcal F}_t)_{t\geq0},\) \(f_i: [0,T]\times H\to H,\;i=1,2\) and \(g: [0,T]\times H\to L_2(Q^{1/2}K,H),\) where \(K\) is another separable Hilbert space and \( L_2(Q^{1/2}K,H)\) be the space of all Hilbert-Schmidt operators from \( Q^{1/2}K\) to \(H,\) \(u\) is an appropriate control function and \(\xi\), \(\varsigma\) are \({\mathcal F}_0\)-measurable \(H\)-value random variable independent of \(W.\) By using suitable fixed point theorems the authors prove the approximate and weak approximate controllability of mild solution for second order neutral stochastic evolution equations.


34K35 Control problems for functional-differential equations
93B05 Controllability
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34G20 Nonlinear differential equations in abstract spaces
34K40 Neutral functional-differential equations
34K30 Functional-differential equations in abstract spaces