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 problem f'+f² =r², f(0)=0, (*) and we want to optimise F(T)= ∫0T f(t) dt when r is allowed to vary over the set R(φ ) of all equimeasurable rearrangements of a decreasing function φ and its convex hull CR(φ). 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(φ) and RC(φ) is attained by the solution f of (*) associated to the increasing rearrangement of an element in R(φ). The supremum of F(T) over RC(φ) is obtained for the solution associated to a decreasing function p, though not necessarily the decreasing rearrangement φ, of an element in R(φ). By changing the perspective we determine the function p that solves the supremum problem.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-4288 |
| Date | January 2004 |
| Creators | Sigstam, Kibret |
| Publisher | Uppsala universitet, Matematiska institutionen, Uppsala : Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 36 |
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