Course Control and System Theory of Rational Systems Motivated by
the Life Sciences
Homeworkset 5
Date issued: 9 October 2018.
Date due: 18 October 2018.
1. Prove that the steady state of the following polynomial system is globally asymptotically
stable in the indicated state set using the candidate Lyapunov function provided below.
dx(t)
dt
= −(x(t) − xs)3
, x(0) = x0,
nx = 1, X = R+, xs ∈ Rs+;
V (x) = (x − xs)2
.
If you use a theorem of the lecture notes then quote the theorem and argue that the conditions
of the theorem are met.
2. Explore the concept of local asymptotic stability. Example 6.4.2 provides a system with this
concept. Argue that the steady states xs,1 = 1 and xs,3 = 4 listed in that example are locally
asymptotically stable by verification of the conditions.
Construct an example of a polynomial or rational system of state-space dimension nx = 2 or
larger, or borrow one from the lecture notes, for which you can prove that a steady state is
locally asymptotically stable and not globally asymptotically stable.
Reading advice for Lecture 5
Please read of the lecture notes the Sections 6.1, 6.2, 6.3, and lightly 6.4.
Reading advice for the future Lecture 6
Please read of the lecture notes the Sections 6.5, 6.6, and 6.7. As mentioned before, this is a
recommendation only.
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