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 e(p) denote the end-point mapping that associates to a given control the terminal point of the corresponding trajectory. Then, we say two trajectories alpha and beta are control homotopic if their corresponding controls u and v belong to the same path component of the fiber (e(p))(-1)(m) for m = e(p)(u) = e(p)(v). Due to this point of view, we constrain in the present work our attention to the study of the set U of addmissible control as an open subset of a certain Banach space. Control homotopy may hence be viewed as an equivalence relation on U for which the equivalence classes are the path components of the sets (e(p))(-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 and control homotopy chain.