Controllability of a Family of Nonlinear Population Dynamics Models

Figure 3

The backward characteristics starting from with (green lines) hits the line , gets renewed by the renewal condition , and then enters the observation domain. More precisely, in addition to the observation time of the pair , the backward characteristics need at most time to hit the line , get renewed by the renewal condition , and take maximum time to enter the observation domain. Thus, we need in all, at least, time to obtain the observability inequality. So, with the conditions , , and , all the characteristics starting at with get renewed by the renewal condition in and enter the observation domain.