<div>Degerli Liste Uyeleri, <br><br>Romanya Constantin Brancusi Universitesinden Viorica Ungureanu<br>Abant Izzet Baysal Universitesi Matematik Bolumunu 10-11 Mayis tarihlerinde<br>Erasmus kapsaminda ziyaret edecektir. <br>
<br>Profesor Ungureanu bu donemde asagida detaylari verilmis konusmalari <br>verecektir. <br><br>İlgilerinize saygilarimla arz ederim. <br><br><br>Cenap Ozel <br></div>
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<p style="TEXT-ALIGN: center; MARGIN: 0cm 0cm 0pt; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto" class="MsoNormal" align="center"><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 24pt">"</span><b style="mso-bidi-font-weight: normal"><span style="FONT-FAMILY: 'Courier New'; COLOR: black; FONT-SIZE: 14pt">Stochastic Differential Equations</span></b><span style="FONT-FAMILY: 'Courier New'; COLOR: black; FONT-SIZE: 14pt"> </span><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 14pt; mso-bidi-font-family: 'Courier New'">"</span><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 24pt; mso-bidi-font-family: 'Courier New'"></span></p>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">In this talk we present a set of prerequisites concerning linear stochastic differential </span></pre><pre style="TEXT-ALIGN: justify">
<span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">equations (LSDE for short) with multiplicative noise in Hilbert spaces. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">First, we brie.y recall basic facts concerning stochastic integral, </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">Ito.s formula and existence and approximation problems for the LSDEs.solutions. Two representation results, </span></pre><pre style="TEXT-ALIGN: justify">
<span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">involving the covariance operators associated with the solutions of LSDEs, are then discussed. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">One of them essentially states that the covariance operator is the unique solution of an initial </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">value problem described by a linear di¤erential equation defined on a space of </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">nuclear operators. The other establishes a connection between the nuclear norm of the </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">covariance operator and the solution of certain backward Lyapunov equation. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">Finally, we discuss stability problems for LSDEs.solutions by using two approaches. </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">The first one is classical (see e.g [1]) and provides necessary and sufcient conditions for </span></pre><pre style="TEXT-ALIGN: justify">
<span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">the uniform exponential stability of Datko type, or equivalently, of Lyapunov type. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">The second ([2], [3]) is new and based on the new representation results mentioned above. </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">This approach is useful not only in solving stability problems for LSDEs but also in treating many </span></pre><pre style="TEXT-ALIGN: justify">
<span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">other problems which need a computation of the mean square of the solutions. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">[1] G. Da.Prato, A. Ichikawa, Lyapunov equations for time-varying linear systems , </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">Systems and Control Letters 9(1987), pp. 165-172. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">[2] V. M. Ungureanu, Stochastic uniform observability of linear differential equations with </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">multiplicative noise,J. Math. Anal. Appl. 343 (2008), no. 1, 446.463. </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">[3] V. M. Ungureanu, Representations of mild solutions of time-varying linear stochastic </span></pre>
<pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">equations and the exponential stability of periodic systems, </span></pre><pre style="TEXT-ALIGN: justify"><span style="COLOR: black; FONT-SIZE: 11pt" lang="EN-US">Electronic Journal of Qualitative Theory of Differential equations, nr. 4, 2004, pp. 1-22.</span><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 11pt" lang="EN-US"></span></pre>
<p style="TEXT-ALIGN: center" align="center"><span style="FONT-FAMILY: 'Courier New'; COLOR: black" lang="EN-US"><font size="3">Viorica Ungureanu, <br>Constantin Brancusi University, Romania</font></span></p>
<p style="MARGIN: 0cm 0cm 0pt; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto" class="MsoNormal"><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 22pt">Tarih: </span><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 22pt; mso-bidi-font-family: 'Courier New'">10 Mayıs 2010 Pazartesi</span></p>
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<p style="TEXT-ALIGN: center; MARGIN: 0cm 0cm 0pt; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto" class="MsoNormal" align="center"><b style="mso-bidi-font-weight: normal"><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 14pt">"</span></b><b style="mso-bidi-font-weight: normal"><span style="FONT-FAMILY: 'Courier New'; COLOR: black; FONT-SIZE: 14pt"> Optimal control of linear stochastic evolution <br>
equations in Hilbert spaces and uniform observability</span></b><span style="FONT-FAMILY: 'Courier New'; COLOR: black; FONT-SIZE: 8.5pt"> </span><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 14pt; mso-bidi-font-family: 'Courier New'">"</span><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 24pt; mso-bidi-font-family: 'Courier New'"></span></p>
<p style="TEXT-ALIGN: justify"><span style="FONT-FAMILY: 'Courier New'; COLOR: black" lang="EN-US"><font size="3">This talk concerns a linear quadratic (LQ) problem associated with linear stochastic evolution equations(LSEE) with unbounded coe¢ cients in the drift. It is well known that the existence of some global nonnegative solutions of the di¤erential Riccati equations (DRE), arrisen in connection with LSEE, plays an important role in solving LQ problems. Since stochastic observability property </font></span><span style="FONT-FAMILY: 'Courier New'; COLOR: black" lang="EN-US"><font size="3">is closely related to this subject, we begin by giving a deterministic characterization of this notion. Then we show that, under stabilizability and stochastic uniform observability or detectability conditions, the Riccati equations of stochastic control have nonnegative, bounded on R+ and stabilizing solutions [1]. Based on this result, the LQ problems can be solved [1]. </font></span></p>
<p style="TEXT-ALIGN: justify"><span style="FONT-FAMILY: 'Courier New'; COLOR: black" lang="EN-US"><font size="3">[1] V. M. Ungureanu, Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability, to appear in Czechoslovak <br>
Mathematical Journal. </font></span></p><pre style="TEXT-ALIGN: center"><span style="FONT-SIZE: 14pt" lang="EN-US">Viorica Ungureanu, <br>
Constantin Brancusi University, Romania</span></pre>
<p style="MARGIN: 0cm 0cm 0pt; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto" class="MsoNormal"><span style="FONT-FAMILY: 'Book Antiqua'; COLOR: black; FONT-SIZE: 22pt"> </span></p>
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