View source for Matrix differential equation

An equation in which the unknown is a matrix of functions appearing in the equation together with its derivative.

Consider a linear matrix differential equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628002.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628003.png" />-dimensional matrix function with locally Lebesgue-integrable entries, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628004.png" /> be an absolutely-continuous solution of equation (1) satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628005.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628006.png" /> is the identity matrix. Then the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628008.png" />, is a solution of the linear system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m0628009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280010.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280012.png" /> is a solution of the system (2) satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280014.png" />, then the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280015.png" /> with as columns the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280016.png" /> is a solution of the matrix differential equation (1). If, in addition, the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280017.png" /> are linearly independent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280019.png" />.

Equation (1) is a particular case of the following matrix differential equation (arising in the theory of stability)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

The solution of (3) with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280021.png" /> is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280023.png" /> is the solution of (1) with the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280025.png" /> is the solution of the matrix differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280026.png" /> with the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280027.png" />.

In various applied problems (theories of stabilization, optimal control, filtration of control system, and others) an important role is played by the so-called matrix Riccati differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280028.png" /></td> </tr></table>

Thus, if the matrix Riccati equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280029.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280030.png" /> stands for transposition, has for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280031.png" /> a bounded solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280032.png" /> on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280033.png" />, and if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280034.png" />, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280035.png" />, and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280036.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280037.png" /> holds, then every solution of the controllable system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280038.png" /></td> </tr></table>

closed by the feedback law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280039.png" />, satisfies the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280040.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280041.png" /> is the Euclidean norm and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280042.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062800/m06280043.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.A. Lappo-Danilevsky,   "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Daletskii,   M.G. Krein,   "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.V. Atkinson,   "Discrete and continuous boundary problems" , Acad. Press  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.T. Reid,   "Riccati differential equations" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.Kh. Zakhar-Itkin,   "The matrix Riccati differential equation and the semi-group of linear fractional transformations"  ''Russian Math. Surveys'' , '''28''' :  3  (1973)  pp. 89–131  ''Uspekhi Mat. Nauk'' , '''28''' :  3  (1973)  pp. 83–120</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR></table>