PhD thesis proposal : Stability of optimal control with long time horizon

This PhD project is focused on the optimal control of ordinary dierential equations (ODEs). The objective is to identify a control input that minimizes a specic cost functional over a family of system trajectories. The Pontryagin maximum principle establishes necessary conditions for optimality. A central challenge which is a key focus of this project is to analyze the stability of the maximum principle's results under perturbations of the initial data. The proposed research concerns the investigation of the stability of the mentioned necessary conditions when the time interval where the problem is stated is very long. A typical example is the case when the cost is the (Cesaro) mean of a given function on a time interval [0, T] and the horizon T tends to innity. This will be the starting problem of the thesis. Several other interesting questions will be investigated depending on the progress of the PhD student. Beyond the main focus, the project explores two additional areas. First, it investigates optimal control problems involving state constraints, where the regularity of solutions is a little-studied topic. Second, the research extends to non-convex cases; although stability in convex control sets is nowadays well understood, the non-convex situation requires further analysis. The work may also consider connections with Hamilton-Jacobi-Bellman equations. Since the maximum principle is expressed as a generalized equat