Marjeta Kramar Fijavž

Positive Linear Systems

We present one important large field of applications to the theory developed so far: control theory. More specifically, we present an elementary introduction to positive linear systems.

Semigroups of max-plus linear operators

We define strongly continuous max-additive and max-plus linear operator semigroups and study their main properties. We present some important examples of such semigroups coming from non-linear evolution equations.

Diffusion in Networks with Time-Dependent Transmission Conditions

We study diffusion in a network which is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. We prove at first a result on existence and uniqueness using form methods. Our main results concern the long-term behavior of the solution. In the case when the conductivity and the diffusion coefficients match (so that mass is conserved) we show that the solution converges exponentially fast to an equilibrium. We also show convergence to a special solution in some other cases.

Positivity and Delay Equations

We have seen that, in general, the growth bound ω0(T) of a C0-semigroup (T(t))t≥0 and the spectral bound s(A) of its generator A do not coincide, even if positivity is assumed. It turns out that in Hilbert spaces a deeper analysis is possible using the boundedness of the resolvent. This has the consequence that for a positive semigroup T(t))t≥0 on a Hilbert space the equality s(A) = ω 0(T) holds. This is the most important result of Section 15.2.

Applications of Positive Matrices

We have now accumulated enough material to pause for a while to discuss its consequences in concrete situations. We have revised linear algebra facts from a functional analytic perspective and obtained a construction to get functions of matrices in a coordinate-free manner, without the use of the Jordan normal form. This was useful when we considered positive matrices, and enabled us to see important and deep spectral consequences of positivity.

The Matrix Exponential Function

We continue our investigation of the asymptotic behavior of dynamical systems described by matrices, which was started in last chapter, now moving to the continuous time case. This means that we investigate the asymptotic properties of the matrix exponential function.

Positive Matrix Semigroups and Applications

Now we investigate positive one-parameter matrix semigroups, or, using a more common name, positive matrix exponentials. As expected, positivity and irreducibility in this case also lead to remarkable spectral and asymptotic properties.

Population Equations with Diffusion

Many applications of positive semigroups occur in mathematical biology or chemistry. In the finite-dimensional part of our text we have already discussed a very simple discrete-time population model, called the Leslie model (see Section 6.3). In this chapter we present a time-continuous age-structured population model with spatial diffusion. We present a rather advanced model in order to show the reader some generalizations and applications.

An Invitation to Positive Matrices

In this chapter we set the stage for our story. We fix our notation and summarize the linear algebraic background that will be needed for the first part of the book. We present some motivating examples of positive matrices at the very beginning and shall return to these examples later on.

Linear Boltzmann Transport Equations with Scattering

In this chapter we give an application of positive semigroup theory to linear transport equations. This is a wonderful piece of mathematics modeling neutron transport in a reactor which uses much of the theory we developed in this text.