# Introduction

Chapter
Part of the Atlantis Studies in Mathematics for Engineering and Science book series (ASMES, volume 8)

## Abstract

It is observed, [280], that there are at least three scientific communities, which contribute to theory of hybrid systems: computer science, modeling and simulation, systems and control communities. The results of our book are about hybrid systems, which are obtained through modeling activity, and they are initiated in analysis of systems which operate in different modes, continuous and discrete. That is, we consider a narrow class of hybrid systems. Let us describe it more precisely. Denote by $$\mathbb{N},\;\mathbb{Z}$$ and $$\mathbb{R}$$ the sets of all natural numbers, integers and real numbers, respectively. The state variable in our book is finite dimensional, $$x \in \mathbb{R}^{n} ,\;n \in \mathbb{N}.$$ Denote by x(t) the position of the state variable, where $$t \in \mathbb{R}$$ is the time argument. Suppose that there is a discrete set of moments θk, $$k \in \mathbb{Z},$$ which one calls, switching moments [279]. We shall call also θk, an event time [280]. If t is in the continuous part, then x(t) satisfies a differential equation, otherwise the value of jumps of x is evaluated by jumps equation [20]. With each event time we associate a switch and a jump. That is, at each event time an event occurs, such that variables x and t jump, and the right-hand-side of the differential equation and the jumps equation switch at the event time. If one specify a hybrid system, then events and switching moments have to be determined. If events are externally induced, then the switching and jumps are controlled, otherwise they are autonomous [280]. In chapters 2–8 we analyze hybrid systems without jumps equations. In the last two chapters jumps equations are considered to model blood pressure distribution and biological integrate-and-fire oscillators.

## Keywords

Periodic Solution Hybrid System Model Predictive Control Functional Differential Equation Systemic Arterial Pressure
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