Page 11 - Textos de Matemática Vol. 36

Chapter 1 Introduction
Riccati differential equations are among the simplest non-linear differential equations and, clearly, then initial value problems can be solved locally. But, in contrast to linear systems of differential equations, their solutions may show the phenomenon of finite escape time . This generally means that after a finite time interval the solution ceases to exist. One of the simplest examples showing these features is the scalar Riccati differential equation
with the real solutions
w′ =1+w2, w(x)=tan(x+c), c∈R.
The complex solutions w± = ±i (where i is the imaginary unit) represent the stationary points of the differential equation. In general we consider the matrix Riccati differential equation
W˙ = M21(t)+M22(t)W −WM11(t)−WM12(t)W, (t ∈ C or t ∈ R) (RDE)
where generally W(t) and M11(t),M12(t),M21(t),M22(t) may be complex ma- trices of dimensions m×n and n×n,n×m,m×n and m×m, respectively. As long as the coefficients in RDE are non-constant, for our purposes here, it suffices to consider the subclass of RDE with real coefficient matrices Mij , 1 ≤ i, j ≤ 2, and real solutions W(t) with a real time parameter t. In particular, we are in- terested to study here piecewise continuous and mainly piecewise continuous periodic coefficients.
The main part of this section will be devoted to the subclass of RDE with constant coefficients, either real or complex. Then together with RDE we also study the corresponding algebraic Riccati equation
0=M21 +M22W −WM11 −WM12W, (ARE) 1