# Feedback Synthesis for a Terminal Control Problem

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### Abstract

A terminal control problem with linear controlled dynamics on a fixed time interval is considered. A boundary value problem in the form of a linear programming problem is stated in a finite-dimensional terminal space at the right endpoint of the interval. The solution of this problem implicitly determines a terminal condition for the controlled dynamics. A saddle-point approach to solving the problem is proposed, which is reduced to the computation a saddle point of the Lagrangian. The approach is based on saddle-point inequalities in terms of primal and dual variables. These inequalities are sufficient optimality conditions. A method for computing a saddle point of the Lagrangian is described. Its monotone convergence with respect to some of the variables on their direct product is proved. Additionally, weak convergence with respect to controls and strong convergence with respect to phase and adjoint trajectories and with respect to terminal variables of the boundary value problem are proved. The saddle-point approach is used to synthesize a feedback control in the case of control constraints in the form of a convex closed set. This result is new, since, in the classical case of the theory of linear regulators, a similar assertion is proved without constraints imposed on the controls. The theory of linear regulators relies on matrix Riccati equations, while the result obtained is based on the concept of a support function (mapping) for the control set.

## Keywords:

terminal control boundary value problem Lagrangian saddle-point methods synthesis of feedback control convergence## Notes

### ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00312.

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