Page 75 - Textos de Matemática Vol. 36

Chapter 3
Numeric algorithms for Nash algebraic Riccati equations
In the last decade there have been developed many numerical methods to solve HARE because of its importance in optimal control theory and related field (see [Mehr91] and the references therein).
The terminal value problem for RDE can be solved with standard numer- ical algorithm for systems of differential equations.
In this last section we want to present some recent approaches to non- -symmetric algebraic matrix Riccati equations as they appear in Nash games on infinite time horizon.
3.1 Positive systems
In the following we study dynamical systems originating from linear- -quadratic differential games, which are of the form:
x˙ =Ax+B1u1 +B2u2, x(0)=x0, (3.1)
with matrices A ∈ Rn×n , Bi ∈ Rn×mi for i = 1, 2, and given dimensions n,m1,m2. The variable x is called the state of the game and the input function ui is called the control or strategy of player i.
A system is called positive, if for nonnegative input functions and non- negative initial values the state takes only nonnegative values. Positive systems arise naturally in applications as populations in ecological systems or concen- tration of chemical reactants in chemical processes to name a few. Although this notion of positivity relies on the underlying coordinate system (a change of coordinates can transform a non-positive system into a positive one and vice versa) in practical applications the proper coordinates follow from the problem formulation.
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