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A.1.
Basic results for linear equations
79
i.e., for A(t) ≡ A one has
x(t) = e(t−t0)Ax0 +
For the remaining part of this section we assume that N:R→Cn×n,R:R→Cm×n and M,B:R→Cm×m
are piecewise continuous and locally bounded. Frequently we drop the time variable t if no cause of confusion exists or if these matrices are constant; thus we write A, B, instead of A(t), B(t), etc.
Linear differential equations of the form
X˙ = XN(t) + M(t)X + R(t) (A.7)
are called Sylvester differential equations. Inthespecialcasewhenn=m,R=R∗ andM =N∗ =:A,equation
(A.7) has the form
X˙ = A(t)X + XA∗(t) + R(t), (A.8) and it is called a Lyapunov differential equation. In the case of constant coeffi-
cients, we consider also the corresponding algebraic Sylvester equation
XN + MX + R = 0 (A.9)
and the continuous-time algebraic Lyapunov equation (CALE)
XA∗ +AX+R=0. (A.10)
Algebraic equations of the form (A.10) were investigated more than 100 years ago by Lyapunov in the context of stability theory of differential equations (see [LaTi85]).
A useful tool in the study of Sylvester and Lyapunov equations is the Kronecker (or tensor) product of two matrices. If A ∈ Cm×m and B ∈ Cn×n, then the nm × nm matrix
a11B a12B ... a1nB
a21B a22B ... a2nB A ⊗ B := . . . ...
an1B an2B . . . annB
is called the (right) Kronecker product of A and B. We define in the sequel a vector-valued linear function associated with a matrix and closely related to the Kronecker product. Given a matrix X ∈ Cm×n, the vector obtained by
t t0
e(t−τ)Ab(τ)dτ. (A.6)
