A topological method for finding invariant sets of continuous systems

2020 ◽  
pp. 104581
Author(s):  
L. Fribourg ◽  
E. Goubault ◽  
S. Mohamed ◽  
M. Mrozek ◽  
S. Putot
Author(s):  
Laurent Fribourg ◽  
Eric Goubault ◽  
Sameh Mohamed ◽  
Marian Mrozek ◽  
Sylvie Putot

1995 ◽  
Vol 1 (1) ◽  
pp. 1-10 ◽  
Author(s):  
V. Lakshmikantham ◽  
Z. Drici

A basic feedback control problem is that of obtaining some desired stability property from a system which contains uncertainties due to unknown inputs into the system. Despite such imperfect knowledge in the selected mathematical model, we often seek to devise controllers that will steer the system in a certain required fashion. Various classes of controllers whose design is based on the method of Lyapunov are known for both discrete [4], [10], [15], and continuous [3–9], [11] models described by difference and differential equations, respectively. Recently, a theory for what is known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, namely, time as an arbitrary closed sets of reals, and allows us to handle both systems simultaneously [1], [2], [12], [13]. This theory permits one to get some insight into and better understanding of the subtle differences between discrete and continuous systems. We shall, in this paper, utilize the framework of the theory of dynamic systems on time scales to investigate the stability properties of conditionally invariant sets which are then applied to discuss controlled systems with uncertain elements. For the notion of conditionally invariant set and its stability properties, see [14]. Our results offer a new approach to the problem in question.


Author(s):  
Alfonso Sorrentino

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.


1989 ◽  
Vol 54 (3) ◽  
pp. 566-571 ◽  
Author(s):  
Jiří Pancíř ◽  
Ivana Haslingerová

A semi-empirical topological method was applied to a study of an Ir(112) surface as well as to both a nondissociative and dissociative chemisorption on this surface. In all cases studied an attachment of carbon to the surface is energetically more favorable than an attachment of oxygen. The preferential capture of the CO molecule on atop sites is remarkable. The capture on n-fold hollow positions as well as the dissociative chemisorption of carbon monoxide on the Ir(112) surface are energetically prohibited.


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