Control Homotopy of Trajectories

Abstract

The main purpose of this paper is to establish the machinery for doing homotopy of (regular) trajectories of control systems. In a mildly different setting than our earlier work in Colonius et al. (J Differ Equ. 2005; 216:324–53), we require this time two trajectories of a (conic) control system to be homotopic by means of their control parameters and simply call them control homotopic. More precisely, let p be a fixed inial point of the state space manifold and let ep denote the end-point mapping that associates to a given control the terminal point of the corresponding trajectory. Then, we say two trajectories α and β are control homotopic if their corresponding controls u and v belong to the same path component of the fiber (ep)− 1(m) for m = ep(u) = ep(v). Due to this point of view, we constrain in the present work our attention to the study of the set $\mathcal {U}$ of addmissible control as an open subset of a certain Banach space. Control homotopy may hence be viewed as an equivalence relation on $\mathcal {U}$ for which the equivalence classes are the path components of the sets (ep)− 1(m), where m belongs to the (regular) accessible set from p. This interpretation also motivates to deal with notions such as control homotopically trivial andcontrol homotopy chain.