Proceedings of the International Conference on Theory and Application of Mathematics and Informatics ICTAMI 2005 - Alba Iulia , Romania TRACKING PROBLEM FOR LINEAR PERIODIC , DISCRETE - TIME STOCHASTIC SYSTEMS IN HILBERT SPACES

The aim of this paper is to solve the tracking problem for linear periodic discrete-time systems with independent random perturbations, in Hilbert spaces. Under stabilizability conditions, we will find an optimal control, which minimize the cost function associated to this problem, in the case when the control weight cost is only nonnegative and not necessarily uniformly positive. 2000 Mathematics Subject Classification: 93E20, 93C55 1.Notations and the statementof the problem Throughout this paper the spaces H, V , U are separable real Hilbert spaces. We will denote by L(H, V ) (respectively L(H)) the Banach space of all bounded linear operators which transform H into V (respectively H). We write 〈., .〉 for the inner product and ‖.‖ for norms of elements and operators. If A ∈ L(H) then A∗ is the adjoint operator of A. The operator A ∈ L(H) is said to be nonnegative and we write A ≥ 0, if A is self-adjoint and 〈Ax, x〉 ≥ 0 for all x ∈ H. For every Hilbert space H we will denote by H the Banach subspace of L(H) formed by all self-adjoint operators, by H the cone of all nonnegative operators of H and by I the identity operator on H. The operator A ∈ H is positive (and we write A > 0) if A is invertible. The sequence Ln ∈ L(H, V ), n ∈ Z is bounded on Z if sup n∈Z ‖Ln‖ <∞ and is τ -periodic if Ln = Ln+τ for all n ∈ N.We say that the sequence Ln ∈ H, n ∈ N is uniformly positive if there exists a > 0 such that Ln ≥ aI for all n ∈ N.