Family of parametric optimal control problems with weighted $L_1$- and $L_2$-norms in the cost functional

We consider a family of parametric optimal control problems with weighted $L_1$- and $L_2$-norms of the control in the cost functional. The weighted coefficient at the $L_1$-norm plays a role of the parameter. We study the dependence of solutions of the optimal control problem with respect to parameter values in irregular case. We prove the theorem that describes properties of solutions of the problem in a neighbourhood of an irregular parameter value and gives their asymptotic expansions. We obtain simple rules for computing solutions of perturbed problems using only solution of the unperturbed problem.