Dynamic method of multipliers in terminal control

A method for solving the terminal control problem with a fixed time interval and fixed initial conditions is proposed. The solution to the boundary value problem posed at the right end of the time interval determines the terminal conditions. This boundary value problem is a finite-dimensional convex programming problem. The dynamics of the terminal control problem is described by a linear controllable system of differential equations. This system is interpreted as a conventional system of linear equality constraints. Then the terminal control problem can be regarded as a dynamic convex programming problem posed in an infinite-dimensional functional Hilbert space. In this paper, the functional problem is treated as a saddle-point problem rather than optimization problem. Accordingly, a saddle-point approach to solving the problem is proposed. This approach is based on maximizing the dual function generated by the modified Lagrangian function of the convex programming problem posed in the functional space. The convergence of the proposed methods is also proved in the functional space. This convergence has the additional property of being monotone in norm with respect to controls, phase trajectories, adjoint functions, as well as finite-dimensional terminal variables.