Finite time stability under perturbing forces and on product spaces

This paper continues the development of a qualitative theory of stability, recently initiated by the authors, for systems operating over finite time intervals. The theory is motivated by 1) the need for a more practical concept of stability than is provided by the classical theory; and 2) the search for methods for investigating stability of a system trajectory (either analytically or numerically given) without the necessity of performing complicated transformations of the differential equations involved. The systems studied in this paper are nonautonomous, i.e., they are under the influence of external forces, and the concept of finite time stability (precisely defined in the paper) in this case involves the bounding of trajectories within specified regions of the state space during a given finite time interval. (The input is assumed to be bounded by a known quantity during this time interval.) Sufficient conditions are given for various types of finite time stability of a system under the influence of perturbing forces which enter the system equations linearly. These conditions take the form of existence of "Liapunov-like" functions whose properties differ significantly from those of classical Liapunov functions. In particular, there is no requirement of definiteness on such functions or their derivative. The remainder of the paper deals with the problem of determining finite time stability properties of a system from knowledge of the finite time stability properties of lower-order subsystems which, when appropriately coupled, form the original system. An example is given which illustrates some of the concepts discussed in the paper.