The Fokker–Planck Framework in the Modeling of Pedestrians’ Motion

Second-order analysis of Fokker–Planck ensemble optimal control problems

“…This section is devoted to the analysis of solutions to our FP problem (3) -( 4). The results presented in this section are mainly extensions of well-known results for different FP problems [4,11] and parabolic problems [22] to our case given by ( 8) - (9).…”

“…The two properties of the bilinear form F given in Lemma 2.2 yield existence of a unique solution to (9) in the following way.…”

“…Next, we investigate further regularity properties of weak solutions to (9). For this purpose, let us recall some continuous Sobolev-embedings, cf.…”

“…Furthermore, we analyze the case of approximate closed-loop controls. We remark that our results in this section apply to the FP ensemble control problems considered in [9,[14][15][16][33][34][35].…”

“…In the field of epidemic problems, stochastic models are of great interest because they take into account the fact that biological actions and human behaviour are subject to random fluctuations [25,26,42]. Stochastic systems also appear in the modelling and control of collective motion in biological systems -such as the movement of groups of bacteria or herds of animals -as well as in pedestrian motion and traffic flows [9,[33][34][35]. In these and many other works, the focus is on drift-diffusion models as follows…”