Problems About the Set of Attainability

1. Control systems. In this series of lectures we will consider “control systems” described by differential equations of the type

I.1.1

$${\text{dx/dt}}\,\,{\text{ = }}\,{\text{x'}}\,\,{\text{ = }}\,{\text{f}}\left( {{\text{t,}}\,{\text{x,}}\,{\text{u}}} \right),$$

where t is a real variable representing the time, x is an n-vector determining the instantaneous state of the (physical) system, and u is an m-vector corresponding to the instantaneous action of a certain control mechanism. This control action is supposed to be adjustable as a function of time u(t), or of the state u(x), or of both: u = u(t, x). Once the control action is prescribed, equation (I.1.1) is a differential equation governing the evolution of the system. A typical problem of control theory, is how to choose the control law u(t, x), in order to achieve a certain goal (for example to reach a given point in minimum time).

In these lectures we will consider only problems in which it is assumed that the control action u(t, x), even if not determined from the beginning, can be adjusted in an exact manner. In other words, we will not be concerned about “stochastic systems”, where some “random functions” (which are neither exactly known nor adjustable) influence the evolution of the system.