The Linear Quadratic Control Problem With Unconstrained Terminal Condition

Abstract Linear Quadratic Control Problems are control problems with a quadratic cost function and linear dynamic sytem and a linear terminal constraint. This work looks at linear quadratic control problems without terminal conditions. We will first look at the controllability and observability of linear dynamical systems and then establish the necessary conditons for the variation in the cost criterion to be nonnegative for strong perturbations in the control. These conditions are the first order necessary conditions for optimality. We shall also consider the necessary and sufficient conditon for positivity of the quadratic cost criterion. Moreover, necessary and sufficient conditions for strong positivity are derived and we shall show that these conditions are based on the existence of solution to a Riccati matrix differential equation. The symplectic property of the Hamiltonian system helps us to derive the Riccati matrix differential equation. We also look at some of the properties of the Riccati variable. Using these ideas, as an illustration, we consider an application in the control of disease, by considering a variant of the SIR model.