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3.3. Newton’s Method 69
Hence, the uniqueness of Nash strategies is entirely translated to properties of certain algebraic Riccati equations. Solvability theory for those equations (3.4) appearing in optimal control has reached a mature state (c.f. [AFIJ03]) whereas for non symmetric Riccati equations and especially for (3.3) only few results are available. In [GuLa] a certain class of non symmetric Riccati equations coming from transport theory is considered, which does not fit in our framework.
3.3 Newton’s Method
In view of Theorem 3.2.3 it seems to be the most demanding challenge to find the left-right stabilizing solution to the Nash Riccati equation since this equation is not a standard type Riccati equation as e.g. in optimal control. As the solution of this equation is of importance to synthesize the Nash strategy we really need to calculate the solution. This will be done by application of the Newton method to the Riccati function, which we define to be:
AT0 Q1 R(X)=− 0 AT X−XA− Q +X S1 S2 X,
2
(3.6)
(3.7)

=: S
=: Q with X ∈ R2n×n. The Fr´echet derivative of R(X) is :
=: D
[R′(X)](Z) = −Z(A − SX) − (D − XS)Z. The Newton iteration can be defined by
[R′(Ki)](Ki+1) = [R′(Ki)](Ki) − R(Ki), which, together with some initial value yields the iteration
−Ki+1(A−SKi)−(D−KiS)Ki+1 = Q+KiSKi, i=0,1,2,.... (3.8) 3.4 Main result
If we use the elementwise ordering from the first section, we can manage to make the Newton sequence monotonically increasing.
Theorem 3.4.1. Suppose additionally for the positive system (3.1) that the matrix −A is an M-matrix. Assume for the matrices in the cost functionals (3.2) that Qi ≥ 0, i = 1,2, and S ≤ 0. Assume further that there exists a P ≥ 0, such that R(P) > 0, then the Newton sequence (Ki)i∈N initialized with K0 = 0 is well defined and converges monotonically to a solution K ≥ 0. K is the smallest solution in the set of all nonnegative solutions.