Lagrange Duality Theory for Convex Control Problems

The Lagrange dual of control problems with linear dynamics, convex cost and convex inequality state and control constraints is analyzed. If an interior point assumption is satisfied, then the existence of a solution to the dual problem is proved; if there exists a solution to the primal problem, then a complementary slackness condition is satisfied. A necessary and sufficient condition for feasible solutions in the primal and dual problems to be optimal is also given. The dual variables p and v corresponding to the system dynamics and state constraints are proved to be of bounded variation while the multiplier corresponding to the control constraints is proved to lie in $\mathcal{L}^1 $. Finally, a control and state minimum principle is proved. If the cost function is differentiable and the state constraints have two derivatives, then the state minimum principle implies that a linear combination of p and v satisfy the conventional adjoint condition for state constrained control problems.