Dynamics and Stability of Motion of Shock and Hybrid Systems

For the description of modern technical and technological devices and systems,
mathematical models are used that incorporate different types of differential
equations, including ordinary differential equations, partial differential equations,
operator equations, etc. The stability analysis of the solutions to such equations is
successfully carried out by the classical methods developed in the stability theory
of motion. These general methods are the direct Lyapunov method, the method
of integral inequalities and the comparison technique.
One of the first models that described the complex structure of a real system
was a hybrid Wittenenhausen system (see [27] Chapter 1). In this model, the
system state is determined by two components: continuous and discrete time.
The continuous system state is described by a system of ordinary differential
equations the right-side part of which depends on the discrete state. The discrete
state is changed when the continuous state gets into some region of the state
space.
"Hybridity" of the mathematical model of a real system occurs when its
behaviour is described by different types of equations. The examples of such
physical systems are:
- continuous systems with phase changes (bouncing ball, walking robot, the
growth of biological cells and their division);
- continuous systems controlled by discrete automation devices (thermostat,
chemical production with discretely introduced catalysts, autopilot);
- coordinated processes (aircraft takeoff and landing in a large airport,
control of car streams on autobahns)…