The Riccati equation and its asymptotic behavior












0












$begingroup$


Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
Consider $G_{t}$ as solution of a discrete-time Riccati equation
as follows



begin{align*}
G_{N}= & Q,\
G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
end{align*}

Can we determine conditions under which the solution $G_{t}$ decreases,
i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
$succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
symmetric matrix. Thanks.










share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
    a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
    and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
    Consider $G_{t}$ as solution of a discrete-time Riccati equation
    as follows



    begin{align*}
    G_{N}= & Q,\
    G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
    end{align*}

    Can we determine conditions under which the solution $G_{t}$ decreases,
    i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
    $succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
    symmetric matrix. Thanks.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
      a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
      and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
      Consider $G_{t}$ as solution of a discrete-time Riccati equation
      as follows



      begin{align*}
      G_{N}= & Q,\
      G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
      end{align*}

      Can we determine conditions under which the solution $G_{t}$ decreases,
      i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
      $succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
      symmetric matrix. Thanks.










      share|cite|improve this question









      $endgroup$




      Consider matrices $Ainmathbb{R}^{ntimes n},Binmathbb{R}^{ntimes m}$,
      a positive semidefinite symmetric matrix $Qinmathbb{R}^{ntimes n}$
      and a positive definite symmetric matrix $Rinmathbb{R}^{mtimes m}$.
      Consider $G_{t}$ as solution of a discrete-time Riccati equation
      as follows



      begin{align*}
      G_{N}= & Q,\
      G_{t}= & A^{T}G_{t+1}A+Q-A^{T}G_{t+1}B(R+B^{T}G_{t+1}B)^{-1}B^{T}G_{t+1}A,,0leq tleq N-1.
      end{align*}

      Can we determine conditions under which the solution $G_{t}$ decreases,
      i.e. conditions under which $G_{t}-G_{t-1}succeq0$ please? The notation
      $succeq0$ means that the matrix $G_{t}-G_{t-1}$ is a positive semidefinite
      symmetric matrix. Thanks.







      linear-algebra optimal-control positive-semidefinite kalman-filter






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 26 '18 at 8:47









      G. TravG. Trav

      1529




      1529






















          0






          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014056%2fthe-riccati-equation-and-its-asymptotic-behavior%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014056%2fthe-riccati-equation-and-its-asymptotic-behavior%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Biblatex bibliography style without URLs when DOI exists (in Overleaf with Zotero bibliography)

          ComboBox Display Member on multiple fields

          Is it possible to collect Nectar points via Trainline?