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

$$X'=A(t)X,\quad t\in\mathbf R,\label{1}\tag{1}$$

where $A(t)$ is an $(n\times n)$-dimensional matrix function with locally Lebesgue-integrable entries, and let $X(t)$ be an absolutely-continuous solution of equation \eqref{1} satisfying the condition $X(t_0)=I$, where $I$ is the identity matrix. Then the vector function $x(t)=X(t)h$, $h\in\mathbf R^n$, is a solution of the linear system

$$x'=A(t)x\label{2}\tag{2}$$

satisfying the condition $x(t_0)=h$. Conversely, if $h_1,\dots,h_n\in\mathbf R^n$ and $x_i(t)$ is a solution of the system \eqref{2} satisfying the condition $x_i(t_0)=h_i$, $i=1,\dots,n$, then the matrix $X(t)$ with as columns the solutions $x_i(t)$ is a solution of the matrix differential equation \eqref{1}. If, in addition, the vectors $h_1,\dots,h_n$ are linearly independent, then $\det X(t)\neq0$ for all $t\in\mathbf R$.

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

$$X'=A(t)X-XB(t)+C(t).\label{3}\tag{3}$$

The solution of \eqref{3} with initial condition $X(t_0)=X_0$ is given by the formula

$$X(t)=U(t,t_0)X_0V(t,t_0)+\int\limits_{t_0}^tU(t,s)C(s)V(s,t)ds,$$

where $U(t,s)$ is the solution of \eqref{1} with the condition $X(s,s)=I$, and $V(t,s)$ is the solution of the matrix differential equation $X'=B(t)X$ with the condition $X(s,s)=I$.

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

$$X'=A(t)X-XB(t)+C(t)+XD(t)X.$$

Thus, if the matrix Riccati equation

$$X'=-(F(t)+\lambda I)^TX-X(F(t)+\lambda I)-I+XG(t)G^T(t)X,$$

where $^T$ stands for transposition, has for $\lambda\geq0$ a bounded solution $X(t)$ on the line $\mathbf R$, and if for all $h\in\mathbf R^n$, all $t\in\mathbf R$, and some $\epsilon>0$, the inequality $h^TX(t)h\geq\epsilon h^Th$ holds, then every solution of the controllable system

$$x'=F(t)x+G(t)u,\quad x\in\mathbf R^n,\quad u\in\mathbf R^m,$$

closed by the feedback law $u=-G^T(t)X(t)x/2$, satisfies the inequality

$$|x(t)|\leq M|x(s)|e^{-\lambda(t-s)},\quad s\leq t,$$

where $|\cdot|$ is the Euclidean norm and $M$ does not depend on $s$.

References

[1]	I.A. Lappo-Danilevsky, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint (1953)
[2]	Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)
[3]	F.V. Atkinson, "Discrete and continuous boundary problems" , Acad. Press (1964)
[4]	W.T. Reid, "Riccati differential equations" , Acad. Press (1972)
[5]	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

References

[a1]	J.K. Hale, "Ordinary differential equations" , Wiley (1969)