A stabilizable switched linear system does not necessarily admit a smooth homogeneous Lyapunov function

The contribution of this paper is twofold. Firstly, an example of a (positive) linear switched system that can be stabilized, via a controlled switching signal, but does not admit a smooth and positively homogeneous control Lyapunov function, is provided. The spectral properties of the subsystem matrices and of the Lyapunov candidates of the convex dierential inclusion associated with the switched system, are thoroughly investigated. Secondly, by taking inspiration from the example, new feedback stabilization techniques for stabilizable positive switched systems are provide

A stabilizable switched linear system does not necessarily admit a smooth homogeneous Lyapunov function

The contribution of this paper is twofold. Firstly, an example of a (positive) linear switched system that can be stabilized, via a controlled switching signal, but does not admit a smooth and positively homogeneous control Lyapunov function, is provided. The spectral properties of the subsystem matrices and of the Lyapunov candidates of the convex differential inclusion associated with the switched system, are thoroughly investigated. Secondly, by taking inspiration from the example, new feedback stabilization techniques for stabilizable positive switched systems are provided

On the stability, stabilizability and control of certain classes of Positive Systems

In this thesis stability, stabilizability and other control issues for certain classes of Positive Systems are investigated. In the first part, the focus is on Compartmental Systems: we start from Compartmental Switched Systems and show that, with respect to the general class of Positive Switched Systems, a much clearer picture of stability under arbitrary switching, stability under persistent switching, and stabilizability (where the control action may either pertain the switching function or involve the design of feedback controllers) can be drawn. Secondly, for the class of Compartmental Multi-Input Systems the problem of designing a state-feedback matrix that preserves the compartmental property of the resulting closed-loop system, meanwhile achieving asymptotic stability is addressed. Such an analysis finally leads to the development of an algorithm that allows to assess problem solvability and provides a possible solution whenever it exists. The second part of the thesis is devoted to the Positive Consensus Problem: for a homogeneous Positive Multi-Agent System we investigate the problem of determining a state-feedback law that can be individually implemented by each agent, preserves the positivity of the overall system, and leads to the achievement of consensus. Finally, for a particular class of Positive Bilinear Systems that arises in drugs concentration design for HIV treatment, we address the problem of determining an optimal constant input that stabilizes the system while maximizing its robustness against the presence of the external disturbance