Multi-objective infinite horizon optimal control problems: characterization of the Pareto fronts and Pareto solutions

In this paper, multi-objective infinite horizon optimal control problems with state constraints are investigated. First, a mono-objective auxiliary optimal control problem, free of state constraints, is introduced. The weak Pareto front of the multi-objective optimal problem is related to a set contained in the boundary of the zero level set of the value function of the auxiliary control problem. Moreover, a more detailed characterization of the Pareto front for the multi-objective problem is presented. In the infinite horizon context, the value function of the auxiliary optimal control problem satisfies a Hamilton–Jacobi–Bellman equation; however, it is not the unique solution. A semi-Lagrangian scheme, based on the Dynamic Programming Principle, is considered to compute the value function of the auxiliary optimal control problem. Furthermore, optimal Pareto trajectory reconstruction is analyzed.