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2.2. Riccati Differential equation 15 2.2 Riccati Differential equation
We shall consider the 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.
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)
where generally W is a complex m × n-matrix and where M11, M12, M21, M22 are constant complex matrices of dimensions n × n, n × m, m × n and m × m, respectively. As we have seen by the introductory example of a scalar equation, in order to obtain all information about stationary points (i.e., about solutions of ARE) it is necessary to deal with complex solutions even if the coefficients M11,M12,M21,M22 arereal.
In some discussions in this section we shall allow the coefficient matrices M11,M12,M21, M22 of RDE to be time dependent instead of being constant; in this case it does not make sense to speak of the associated algebraic Riccati equation ARE.
There is an interesting and important relationship between linear systems of differential equations and Riccati differential equations which we shall use in this chapter to obtain an explicit representation formula for all solutions of RDE and ARE as well. This relation has been known at least since the work of Radon [Rado27],[Rado28].
Theorem 2.2.1 (Radon’s Lemma). Let M11 ∈ Rn×n , M12 ∈ Rn×m , M21 ∈ Rm×n, M22 ∈ Rm×m, then the following holds:
i) Let W(t) ∈ Rm×n be a solution of RDE in the interval [t0,tf] ⊂ R. If Q, Q(t) ∈ Rn×n is a solution of the initial value problem
Q˙=(M11+M12W)Q, Q(t0)=In Q
and P(t) := W(t)Q(t), then P is a solution of the associated linear system (of differential equations)
d Q M11 M12Q
dt P = M21 M22 P . (L)