## Constrained optimization of linear systems for infinite horizon problemsSome methods of optimal control theory are extended with a view toward applications to production and inventory control. A linear, discrete time, deterministic system with vector state and decision variables is optimized relative to a quadratic criterion. The optimal control is shown to be piecewise linear in the state vector when the decision is constrained to be nonnegative, and an algorithm is presented for computing optimal controls. The following results are obtained for the infinite horizon unconstrained problem with no discounting of future costs: (1) necessary conditions for convergence of optimal N-period policies. (2) optimal properties of this limit policy. These results are applied to modify the finite horizon algorithm to obtain optimal controls for the infinite horizon constrained problem. Results of some computations are presented. (Author). |

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### Contents

QUADRATIC OPTIMIZATION OF A DYNAMIC LINEAR SYSTEM | 5 |

NONNEGATIVE CONTROL VECTOR | 16 |

CHAPTER k INFINITE HORIZON UNCONSTRAINED CONTROL PROBLEM | 36 |

5 other sections not shown

### Common terms and phrases

affine transformations algorithm assume Chapter completely controllable system completes the proof compute the optimal constrained control contraction mapping control problem control sequence control space control vector convergence convex function criterion function decision rules decision variables defining inequalities degenerate region dynamic programming e-optimal elements exists f(xQ Farkas Lemma fN(xQ functional equation Hence Holt et.al hyperplane Iglehart index set infinite horizon problem initial points initial state space inventory problem Kalman Lemma lim B(N limit policy linear control law linear transformation loss function method minimization minimum value N-period optimal policies non-negative optimal control optimal end point optimal feedback matrix optimal N-period optimal regulator problem optimal trajectories partitioned form period constrained problem period problem planning horizon positive semi-definite production and inventory pseudoinverse quadratic form recursive region of linear result satisfied sequence of affine solution spectral radius stage problem stage region Theorem 3.2 transition matrix unconstrained xt(N xt+1 zero