This paper introduces a formal definition of the concepts of Deadlock and Livelock for a general class of Hybrid Control Systems (HCS). Such a characterization hinges on three important aspects: firstly, the concept of composition of HCS. Secondly, the general concept of specifications and their composition for HCS. Finally, the dynamical structure and behaviors for HCS. The first aspect is introduced in a novel manner, including both aspects from the literature of discrete transition systems, and accounting for concepts such as feedback inteconnections of dynamical systems. The second point accounts for general properties that are of interest from a system theory and control perspective. The third part encompasses and categorizes between the diverse and possibly pathological behaviors that are characteristic to HCS. We investigate the problems of Deadlock and Livelock Verification.