Stability of controllable elastic distributed systems

Simple criteria for the observability of elastic systems are established. Theorems are proved that enable one to determine whether distributed controllable systems, whether linear or non-linear, are asymptotically stable, by examining a model not involving elasticity. The results are obtained without truncation of elastic modes. Elasticity in the structure of controlled objects may modify the characteristics of the system to such a degree that a control system developed without allowing for elasticity, or allowing for only a few modes of elastic vibration, does not guarantee stability of the real physical system, since it may lead to instability in the omitted modes. A further complication is the approximate nature of dynamical schemes for real controlled objects; the only more or less reliable parameters are those relating to the lowest elastic modes. In addition, the output characteristics of the various sensors and actuating elements are usually non-linear and governed by differential equations. Hence the importance of developing methods for the synthesis and analysis of non-linear control systems for objects with inaccurately specified characteristics, in such a way as to guarantee asymptotic stability of the state of equilibrium of the full system without truncation of elastic vibratory modes. This problem will be solved in the present paper.