Document Zbl 1031.93032

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Math. Anal. Appl.</a> <a href="/?q=in%3A77486" title="Articles in this Issue">259, No. 1, 64-82 (2001)</a>. </div> <div class="abstract">Linear stochastic control systems defined in infinite-dimensional Hilbert spaces are considered. Three types of stochastic controllability are studied: approximate, complete, and $S$-controllability. The relationships between different types of controllability are explained using methods of stochastic differential equations and the theory of stochastic processes. Moreover, several conditions for these types of controllability are formulated and proved. As an application of the theoretical results, the controllability of the wave equation with a distributed control and disturbances described by a Wiener process is studied. The relationships to deterministic controllability are also discussed.<br class="zbmathjax-paragraph">Finally, it should be stressed that similar stochastic controllability problems have been considered in the papers [<span class="zbmathjax-textit">J. Klamka</span> and <span class="zbmathjax-textit">L. Socha</span>, IEEE Trans. Autom. Control. 22, 880-881 (1997; <a href="/0363.93048">Zbl 0363.93048</a>)] and [<span class="zbmathjax-textit">N. I. Mahmudov</span> and <span class="zbmathjax-textit">A. Denker</span>, Int. J. Control 73, 144-151 (2000; <a href="/1031.93033">Zbl 1031.93033</a>)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4981">J.Klamka (Katowice)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1657540">Cited in <strong>83</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B05" title="MSC2020">93B05</a> </td> <td class="space"> Controllability </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E03" title="MSC2020">93E03</a> </td> <td class="space"> Stochastic systems in control theory (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C25" title="MSC2020">93C25</a> </td> <td class="space"> Control/observation systems in abstract spaces </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Alinear+stochastic+control+systems">linear stochastic control systems</a>; <a href="/?q=ut%3A%5C%28S%5C%29-controllability">$S$-controllability</a>; <a href="/?q=ut%3Ainfinite-dimensional+Hilbert+spaces">infinite-dimensional Hilbert spaces</a>; <a href="/?q=ut%3Astochastic+controllability">stochastic controllability</a>; <a href="/?q=ut%3Awave+equation">wave equation</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0363.93048">Zbl 0363.93048</a>; <a href="/1031.93033">Zbl 1031.93033</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1031.93032" data-ciurl="/ci/01657540" data-biburl="/bibtex/01657540.bib" data-amsurl="/amsrefs/01657540.bib" data-xmlurl="/xml/01657540.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/01657540.pdf" title="Zbl 1031.93032 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1006/jmaa.2000.7386" aria-label="DOI for “Controllability of linear stochastic systems in Hilbert spaces”" title="10.1006/jmaa.2000.7386">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bashirov, A. E.; Kerimov, K. R., On controllability conception for stochastic systems, SIAM J. Control Optim., 35, 384-398 (1997) · <a href="/0873.93076" class="nowrap">Zbl 0873.93076</a></td> </tr><tr> <td>[2]</td> <td class="space">Bashirov, A. E., On weakening of the controllability concepts, Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, December 11-13 (1996), p. 640-645</td> </tr><tr> <td>[3]</td> <td class="space">Bashirov, A. E.; Mahmudov, N. I., On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37, 1808-1821 (1999) · <a href="/0940.93013" class="nowrap">Zbl 0940.93013</a></td> </tr><tr> <td>[4]</td> <td class="space">Bashirov, A. E.; Mahmudov, N. I., Controllability of linear deterministic and stochastic systems, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, December 7-10 (1999), p. 3196-3202</td> </tr><tr> <td>[5]</td> <td class="space">Bensoussan, A.; Da Prato, G.; Delfour, M. C.; Mitter, S. K., Representation and Control of Infinite Dimensional Systems. Representation and Control of Infinite Dimensional Systems, Systems & Control: Foundations & Applications, 2 (1993), Birkhauser: Birkhauser Boston · <a href="/0790.93016" class="nowrap">Zbl 0790.93016</a></td> </tr><tr> <td>[6]</td> <td class="space">Bensoussan, A.; Viot, M., Optimal control of stochastic linear distributed parameter systems, SIAM J. Control, 13, 904-926 (1975) · <a href="/0276.93058" class="nowrap">Zbl 0276.93058</a></td> </tr><tr> <td>[7]</td> <td class="space">Curtain, R. F.; Ichikawa, A., The separation principle for stochastic evolution equations, SIAM J. Control Optim., 15, 367-383 (1977) · <a href="/0359.60076" class="nowrap">Zbl 0359.60076</a></td> </tr><tr> <td>[8]</td> <td class="space">Curtain, R. F.; Zwart, H. 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L., Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. Appl., 18, 542-560 (1967) · <a href="/0158.10201" class="nowrap">Zbl 0158.10201</a></td> </tr><tr> <td>[18]</td> <td class="space">Sunahara, Y.; Kabeuchi, T.; Asada, S.; Aihara, S.; Kishino, K., On stochastic controllability for nonlinear systems, IEEE Trans. Automat. Control, 19, 49-54 (1974) · <a href="/0276.93011" class="nowrap">Zbl 0276.93011</a></td> </tr><tr> <td>[19]</td> <td class="space">Zabczyk, J., Controllability of stochastic linear systems, System Control. Lett., 1, 25-31 (1991) · <a href="/0481.93054" class="nowrap">Zbl 0481.93054</a></td> </tr><tr> <td>[20]</td> <td class="space">Zabczyk, J., Mathematical Control Theory (1992), Birkhauser: Birkhauser Boston/Basel/Berlin · <a href="/1071.93500" class="nowrap">Zbl 1071.93500</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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