On a numerical solution of equations of extremals for a variational problem with constraints

The paper is concerned with the variational problem int_{t_0}^{t_1} L(q,v,t)dt to {rm extr}, q=(q_1, dots, q_n), v=(v_1, dots, v_n), v_i={dot q}_i, subject to the constraints q(t_0)=q^0, q(t_1)=q^1, f_i (q,t)=0, 1 leq i leq m; f_j (q,v,t)=0, m+1 leq j leq r (r leq n). The author points out that the integral manifold related to the classical Euler-Lagrange system for the original problem is usually stable but not asymptotically stable. In this connection he presents a modified system of necessary optimality conditions whose integral manifold is asymptotically stable. The stability of Euler and Runge-Kutta methods when applied to the modified optimality system is also established.