Finite-rank perturbation of the linear-quadratic control problem

Let (X, <,>) be a Hilbert space, M^T = M : X→ X linear self-adjoint map, u₀, u₁,......u_m∈ X, a ∈ X, Z be a closed vector subspace in X. Consider the following optimization problem: f(x) = (((< x, Mx>)/2)+(0,x>)+((1/2)Σ(_i,j=1,^m)l_iji,x>)) → min (1) x ∈ a + Z (2). We assume that the matrix L = (l_ij) is symmetric and f is convex. Suppose that we can solve (numerically or analytically) the problem of the form f_v(x) = ((()/2) + ()) (3) x∈ a + Z (4) for any v ∈ X. The question we address in this paper is whether it is possible to obtain the optimal solution to (1), (2) based on the information obtained by solving problems (3), (4).