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 α and β 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 \(\mathcal {U}\) 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 (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 andcontrol homotopy chain.

本文的主要目的是建立一种用于对控制系统的（规则）轨迹进行同构的机制。与我们先前在Colonius等人的工作稍有不同。（J Differ Equ。2005; 216：324–53），我们这次需要（圆锥形）控制系统的两条轨迹通过其控制参数来实现同位，并简单地称为控制同位。更精确地，令p为状态空间歧管的固定初始点，并令e _p表示与给定控件关联的终点轨迹，该轨迹对应于相应轨迹。然后，如果两个轨迹α和β对应的控制u当m = e _p（u）= e _p（v）时，和v属于光纤（e _p）^{− 1}（m）的相同路径分量。由于这种观点，我们在当前工作中将注意力集中在将可允许控制集\（\ mathcal {U} \）作为某个Banach空间的开放子集的研究上。因此，控制同伦可以视为\（\ mathcal {U} \）上的等价关系，其等价类是集合（e _p）^{− 1}（m），其中m属于p中的（常规）可访问集。这种解释也促使人们处理诸如控制同位琐碎的和控制同构链的概念。