## Switched systems as models of dynamic treatment scheduling

##### Abstract

One interesting problem in treating a disease with multiple drug therapies is the
timing of the drug treatment. In this thesis, we study the switching time of the drug
therapies by using the theories of switching system developed by Dayawansa and
Martin [2]. The model we are interested in is developed in the following manner. We
assume some 2 by 1 vector, x, to represent the state of the patient in the treatment,
then the length of the vector is being measured and that is scaled so that the length 0
represent death and 1 represent remission. For each treatment we assume the model
of the form xn+1 = Axn where A is 2 by 2 matrix with |A| 6= 0. Then for two
treatments, we have the combined model
xn+1 = (nA + (1 âˆ’ n)B)xn
where |A||B| 6= 0 and n 2 {0, 1}. We consider n to be a control variable which
represents the treatment to be applied. Then for each sequence of n, the switched
linear system can be written as the product of random matrices A and B. In this
thesis we give conditions that a discrete time switched linear system must satisfy if
it is stable. We do this by calculating the mean and covariance of the set of matrices
obtained by using all possible switches. The theory of switched linear systems has
received considerable attention in the systems theory literature in the last two decades.
However, for discrete time switched systems the literature is much older going back to
at least the early 1960ï¿½ï¿½s with the publication of the paper of Furstenberg and Kesten
in the area of products of random matrices, or if you like the random products of
matrices. The way that we have approached this problem is to consider the switched
linear system as evolving on a partially ordered network that is, in fact, a tree. This
allows us to make use of the developments of 50 years of study on random products
that exists in the statistics literature. A nice byproduct of this research is that we
use Kï¿½ï¿½onigï¿½ï¿½s theorem of finatary trees. This may be the first use of this theorem in
systems and control. In this thesis, based on the proof of several theories, we apply
a 2 by 2 matrices example, to the timing of drug treatment problem and we can find
a particular switching treatments regime to drive the system away from the origin,
which means the patients could be cured.