Variational perturbations of the linear-quadratic problem

A sequence of perturbations $$\begin{gathered} \dot x = A_n x + B_n u, \parallel u\parallel _{L^2 } \leqslant 1, (P_n ) \hfill \\ x\left( o \right) = x^o , n = 0, 1, 2, 3,..., \hfill \\ \end{gathered} $$ is given of the linear-quadratic optimal control problem consisting of minimizing $$\int_0^1 {((u - \tilde u)^T (u - \tilde u) + (x - \tilde x)^T (x - \tilde x))dt,} $$ subject to (P₀). We assume that {A _n} bounded inL ¹ and {B _n} is bounded inL ². Then, a necessary and sufficient condition so that, for every?, \(\tilde u\) , \(\tilde x\) ∈L ², and for everyx ⁰, the optimal control for (P_n) converges strongly inL ² to the optimal control for (P₀) and the optimal state converges uniformly is thatA _n →A ₀ weakly inL ¹ andB _n →B ₀ strongly inL ².