<p>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</p><p><i>f</i>'+<i>f</i>² =<i>r</i>², <i>f</i>(0)=0, (*)</p><p>and we want to optimise</p><p><i>F</i>(<i>T</i>)= ∫<sub>0</sub><sup>T</sup> <i>f</i>(<i>t</i>) <i>dt</i></p><p>when <i>r</i> is allowed to vary over the set <i>R</i>(φ ) of all <i>equimeasurable</i> rearrangements of a decreasing function φ and its convex hull <i>CR</i>(φ). </p><p>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 <i>r</i> is increasing. We also generalize the lemma to a more general equation.</p><p>It was proved by Essén that the infimum of <i>F</i>(<i>T</i>) over <i>R</i>(φ) and <i>RC</i>(φ) is attained by the solution <i>f</i> of (*) associated to the increasing rearrangement of an element in <i>R</i>(φ). The supremum of <i>F</i>(<i>T</i>) over <i>RC</i>(φ) is obtained for the solution associated to a decreasing function <i>p</i>, though not necessarily the decreasing rearrangement φ, of an element in <i>R</i>(φ). By changing the perspective we determine the function <i>p </i>that solves the supremum problem.</p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:uu-4288 |
| Date | January 2004 |
| Creators | Sigstam, Kibret |
| Publisher | Uppsala University, Department of Mathematics, Uppsala : Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, text |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 36 |
Page generated in 0.0022 seconds