Time-dependent vector stabilization

Stabilization is one of the central themes of control theory. It was shown in [1–4] that the class of continuous stationary feedbacks is too restrictive for the purposes of stabilization of nonlinear systems. In other words, in order to design a continuous feedback stabilizer one needs to use functions depending on time and state [6, 8]. To stay with the class of stationary feedbacks one has to deal with piecewise continuous functions [5]. The synthesis procedures for both continuous and piecewise continuous stabilizers are developed only for some special types of systems. For example, in [5, 4] it is shown how to construct stationary piecewise continuous stabilizers for generic two-dimensional affine nonlinear systems. On the other hand, the papers [6, 8, 9] show how to design feedbacks for certain types of nonholonomic systems. Both approaches (nonstationary continuous and stationary discontinuous) are quite complicated as far as the feedback synthesis is concerned. The statement of the problem considered in this paper is not new. There are many publications related to the problems that are similar in nature to the stabilization of a single time-dependent vector. Similar equations arise naturally in learning systems, repetitive, and adaptive control systems. That underlines the importance of the approach presented here. Not only the main result but especially the technique employed in this publication might have consequences of significant magnitude for various branches of applied mathematics. Novelty of the approach developed in this paper is due to averaging the time-dependent vector field over a time interval. By its nature this approach does not impose any unnatural restrictions on the vector field in question. That strongly advocates its advantage and power. The time-averaging technique is not new itself. In fact equity traders use it daily comparing dynamics of 200and 50-day moving averages for