Peter Stechlinski

Hybrid stabilization and synchronization of nonlinear systems with unbounded delays

The stabilization of nonlinear systems with bounded and unbounded time-delays via hybrid control is studied. We investigate and identify switching rules for which stabilization can be verified a priori. When this approach is inadequate, stabilizing state-dependent switching rules are constructed. This method is based on partitioning the state-space into switching regions. Unwanted physical behavior, such as chattering and Zeno behavior, is avoided. Sufficient conditions are established using Razumikhin-like theorems. The theoretical results provide insight into how hybrid control strategies can be constructed to synchronize a class of nonlinear systems with unbounded delay. The findings are illustrated through numerical simulations.

Stabilization of Impulsive Systems via Open-Loop Switched Control

In this chapter, the stabilization of nonlinear impulsive systems under time-dependent switching control is investigated. In the open-loop approach, the switching rule is programmed in advance and the switched system is composed entirely of unstable subsystems. Sufficient conditions are found that establish the existence of stabilizing time-dependent switching rules using the Campbell–Baker–Hausdorff formula and Lyapunov stability theory.

SIS models with switching and pulse control

Abstract This paper investigates an SIS model with seasonal changes in disease transmission. The model is formulated as a switched system and its long-term behavior is analyzed. A constant treatment strategy is then applied to a switched SIS model which takes media coverage into account. A pulse vaccination control strategy with vaccine failure is also considered. Throughout the paper, the stability of the disease-free solution is analyzed and threshold criteria involving the basic reproduction number are established which guarantee disease eradication. Sufficient conditions are also given for the endemic case.

Robust Synchronization of Distributed-Delay Systems via Hybrid Control

Drive and response systems which exhibit time-delays and uncertainties are synchronized using hybrid control. Classes of dwell-time satisfying switching rules are identified under which synchronization can be achieved in a robust manner. The theoretical results are established using multiple Lyapunov functions and Halanay-like inequalities.

Stabilization of a Class of Nonlinear Systems using State‐dependent Switching Control and Impulsive Control

This paper develops a hybrid control strategy for stabilizing a class of nonlinear systems using switching and impulsive control. More specifically, some easily verifiable conditions are developed which guarantee the exponential stability of a class of switched and impulsive nonlinear systems under a particular state‐dependent switching rule. The main result is proved using a common Lyapunov function. An illustrating example is given with a simulation.

Hybrid and Switched Systems

This chapter reviews the theory of switched systems.Governed by a combination of mode-dependent continuous/discrete dynamics and logic-based switching, and having a wide range of motivating applications, the qualitative behavior of switched systems is highlighted here. Stability theory is emphasized; topics of discussion include stability under arbitrary and constrained switching, as well as switching control.

Switching Control Strategies

This chapter is motivated by the application of control strategies to eradicate epidemics. In part, the previous switched epidemic models are reintroduced with continuous (e.g., vaccination of newborns continuously in time) or switching control (i.e., piecewise continuous application of vaccination or treatment schemes) for evaluation and optimization. As discussed earlier, infectious disease models are a crucial component in designing and implementing detection, prevention, and control programs (e.g., the World Health Organization’s program against smallpox, leading to its global eradication by 1977). The switched SIR model is first returned to for consideration and analysis of vaccination of the susceptible group (e.g., newborns or the entire cohort). Subsequently, the developed theoretical methods are applied to the switched SIR model with a treatment program in effect. Common Lyapunov functions are used to provide controlled eradication of diseases modeled by the so-called SEIR (Susceptible-Exposed-Infected-Recovered) model with seasonal variations captured by switching. A screening process, where traveling individuals are examined for infection, is proposed and studied for the switched multi-city model of the previous chapter. Switching control of diseases such as Dengue and Chikungunya which are spread via mosquito–human interactions, is investigated.

A Case Study: Chikungunya Outbreak in Réunion

This chapter is devoted to analyzing the spread of the Chikungunya virus, a vector-borne disease, modeled here according to interactions between human and mosquito populations. After introducing the full model, the remainder of this chapter focuses on studying the efficacy of different control strategies. Once the model is formulated and analyzed, a case study of the 2005–06 Chikungunya outbreak in Reunion is completed. Here, the mosquito birth rate is modeled as a time-varying switching parameter, to incorporate differences between the rainy season and dry season. Variations in the contact rate between mosquitoes and humans are also considered. Control strategies are analyzed and evaluated for comparison (e.g., destruction of breeding sites, reduced contact rates).

The Switched SIR Model

The modeling of epidemics by hybrid and switched systems is introduced and analyzed. To begin, the classical SIR model is derived and its defining features are detailed. Motivated by variations in the contact rate between members of the population, a switched SIR model is formulated. The flexibility of the switched systems framework and its accompanying theory is highlighted by relaxing some of the population demographics and epidemiological assumptions. A switching incidence rate function form is considered to model abrupt changes in population behavior. The incorporation of stochastic perturbations into the model is also investigated. The findings here focus on the qualitative behavior of the models (i.e., stability theory). More specifically, global attractivity and partial stability are demonstrated, as well as persistence of the disease.

Epidemic Models with Switching

In this chapter, the methods developed thus far are applied to a variety of infectious disease models with different physiological and epidemiological assumptions. Many of the previous results are immediately applicable, thanks to the flexibility of the simple techniques used here. However, some complicating modeling assumptions lead to a need for different switched systems techniques not present in the previous chapter. First, the so-called SIS model is considered, followed by incorporation of media coverage, network epidemic models with interconnected cities (or patches), and diseases spread by vector agents (e.g., mosquitoes) which are modeled using time delays. Straightforward extensions of eradication results are given for models with vertical transmission, disease-induced mortality, waning immunity, passive immunity, and a model with general compartments.