Dissipative periodic processes

Abstract : The objective of this paper is to develop a general and meaningful theory of dissipative periodic systems. For ordinary periodic differential equations one studies the iterates of a map T of a state space into itself where the map T is topological and the space is locally compact (n-dimensional Euclidean space). However, for the applications the authors have in mind, the solutions will be unique only in the forward direction of time and the state spaces are not locally compact. Because of this generalization of the results for ordinary differential equations is by no means trivial. The basic theory of dissipative periodic processes on Banach spaces are developed in Sections 2 and 3 of the paper. How this applies to retarded functional differential equations of retarded type is discussed in the fourth section. Two sufficient conditions for dissipativeness are given in terms of Liapunov functions. They formalize the intuitive notion that many systems for large displacements dissipate energy. Application of these results is illustrated. (Author)

[1]  Marshall Slemrod,et al.  Nonexistence of oscillations in a nonlinear distributed network , 1971 .

[2]  Constantine M. Dafermos,et al.  An invariance principle for compact processes , 1971 .

[3]  M. Slemrod.  Asymptotic behavior of periodic dynamical systems on banach spaces , 1970 .

[4]  J. Hale,et al.  Stability of functional differential equations of neutral type , 1970 .

[5]  Jack K. Hale,et al.  Dynamical systems and stability , 1969 .

[6]  F. Browder.  A new generalization of the Schauder fixed point theorem , 1967 .

[7]  T. Yoshizawa.  Asymptotic stability of solutions of an almost periodic system of functional-differential equations , 1964 .

[8]  J. P. LaSalle,et al.  A Study of Synchronous Asymptotic Stability , 1957 .

[9]  N. Levinson,et al.  Transformation Theory of Non-Linear Differential Equations of the Second Order , 1944 .

[10]  M. Slemrod,et al.  An invariance principle for dynamical systems on Banach space - Application to the general problem of thermoelastic stability , 1971 .

[11]  C. Dafermos.  Asymptotic stability in viscoelasticity , 1970 .

[12]  J. P. Lasalle.  Stability theory for ordinary differential equations. , 1968 .

[13]  Jack K. Hale,et al.  Geometric Theory of Functional Differential Equations , 1967 .

[14]  V. A. Pliss.  Nonlocal Problems of the Theory of Oscillations , 1966 .

[15]  吉沢 太郎.  Stability theory by Liapunov's second method , 1966 .

[16]  Taro Yoshizawa,et al.  Extreme stability and almost periodic solutions of functional-differential equations , 1964 .

[17]  G. S. Jones.  Periodic motions in banach space and applications to functional-differential equations. , 1962 .

[18]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .