Antisymmetric Extremal Mapping and Linear Dynamics

An optimal control problem is considered on a fixed time interval. Choosing a control generates a phase trajectory of this problem. The left end of the trajectory is fixed, while a finite-dimensional problem of calculating a fixed point of an extremal mapping is set up at the right end. In the optimal situation, the right end of the phase trajectory coincides with the fixed point of the mapping. In other words, the task is, by choosing a suitable control, to construct a phase trajectory in a Hilbert space that leaves the initial position at the left end of the time interval and arrives at the fixed point of the extremal mapping at the right end of the time interval. To solve the problem within the framework of the Lagrangian formalism, we propose a new approach based on saddle point sufficient optimality conditions. An iterative computational process of saddle point gradient type is investigated. The process is proved to converge strongly in phase and dual trajectories, as well as in terminal variables of the finite-dimensional boundary value problem of linear programming, and to converge weakly in controls. The emphasis is placed on the fact that only proof-based computational techniques transform a mathematical model into a tool for making a guaranteed decision.