In this paper, we propose a new Newton method for minimizing convex optimization problems with singular Hessian matrices including the special case that the Hessian matrix of the objective function is singular at any iteration point. The new method we proposed has some updates in the regularized parameter and the search direction. The step size of our method can be obtained by using Armijo backtracking line search. We also prove that the new method has global convergence. Some numerical experimental results show that the new method performs well for solving convex optimization problems whose Hessian matrices of the objective functions are singular everywhere.

In this paper, two different discrete schemes of the second-order linear time-varying system represented by the linearized satellite magnetic attitude control motion equation are obtained by Euler method. Then, the controllability and observability conditions of a new discrete second-order linear time-varying system are proposed and the validity of these conditions is further verified by some numerical examples. Next, the theoretical results are applied to investigate the controllability and observability of the discretized satellite magnetic control system. Different periods $\tau$ are chosen to investigate the effect on the controllability and observability of the resulting discrete system. The corresponding conclusions are obtained.