Khai Tien Nguyen

The paper is concerned with an optimal control problem on [ ], where the dynamics is linear w.r.t. the control functions. For a terminal cost [ ] in a dense [ ] set of [ ] (i.e., in a countable intersection of open dense subsets), two main results are proved. Namely: the set [ ] of conjugate points is closed, with locally bounded [ ]-dimensional Hausdorff measure. Moreover, the set of initial points [ ], which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of [ ].

The paper is concerned with a scalar balance law, where the source term depends on a control function [ ]. Given a control [ ], it is proved that, for generic initial data [ ], the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time [ ] no shock interaction occurs and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time [ ]. Such behavior persists under any suitably small perturbation of the flux, source and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.

This paper studies the existence of minimal solutions to two-point boundary value problems for quasi-monotone dynamical systems. Specifically, the pointwise infimum of all supersolutions is shown to coincide with the minimal solution. This result is then applied to establish a non-uniqueness result for strong solutions to a class of mean field games with a continuum of players.

The present paper establishes a local well-posed result for piecewise regular solutions with single shock of scalar balance laws with singular integral of convolution type kernels. In a neighborhood of the shock curve, a detailed description of the solution is provided for a general class of initial data.