Chapter 3
Analysis of the Dynamics
For the analysis of the optimal control problem, it is crucial to understand the properties of
the dynamics for constant values of the control u 2 [0;a]. In this analysis, we will assume
that the values of p and q belong to the following domain S = f(p;q) : p > 0;q > 0g of the
system. The dynamical behavior of the system in [6] was analyzed and some of its properties
can be easily derived.
Proposition 1 ( [4] )The uncontrolled system (u=0) has a unique asymptotically stable
equilibrium point at (peq;qeq) given by:
3 b
2
peq = qeq = : (3.1)d
Proof. The analysis of the stability of an equilibrium of a non-linear system requires the
evaluation of the Jacobian. The uncontrolled system (u = 0); is described by the following
non-linear system of equations:
 

p = pln pq p(0) = p0
(3.2) 

2q = bp  + dp
3 q Guq q(0) = q0:
9
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