Optimization and Estimation of Solutions of Riccati Equations

Abstract: This thesis consists of three papers on topics related to optimization and estimation of solutions of Riccati equations. We are concerned with the initial value problemf'+f² =r², f(0)=0, (')and we want to optimiseF(T)= ?0T f(t) dtwhen r is allowed to vary over the set R(? ) of all equimeasurable rearrangements of a decreasing function ? and its convex hull CR(?). In the second paper we give a new proof of a lemma of Essén giving lower and upper bounds for the solution to the above equation, when r is increasing. We also generalize the lemma to a more general equation.It was proved by Essén that the infimum of F(T) over R(?) and RC(?) is attained by the solution f of (') associated to the increasing rearrangement of an element in R(?). The supremum of F(T) over RC(?) is obtained for the solution associated to a decreasing function p, though not necessarily the decreasing rearrangement ?, of an element in R(?). By changing the perspective we determine the function p that solves the supremum problem.