Bi-objective finite horizon optimal control problems with Bolza and maximum running cost

Publication type:
Paper in peer-reviewed journals
Keywords :
Optimal control problem; Hamilton-Jacobi-Bellman approach; Multi-objective programming; $\varepsilon$-Pareto solutions;
In this paper, we investigate optimal control problems with two objective functions of different nature that need to be minimized simultaneously. One objective is in the classical Bolza form and the other one is defined as a maximum running cost. Our approach is based on the Hamilton-Jacobi-Bellman framework. In the problem considered here the existence of Pareto solutions is not guaranteed. So first, we consider the bi-objective problem to be minimized over the convexified dynamical system. We show that if a vector is (weak) Pareto optimal solution for the convexified problem, then there exists an (weak) ε-Pareto optimal solution of the original problem that is in the neighborhood of this vector. After we define an auxiliary optimal control problem and show that the weak Pareto front of the convexified problem is a subset of the zero level set of the corresponding value function. Moreover, with a geometrical approach we establish a characterization of the Pareto front. It is also proved that the (weak) ε-Pareto front is contained in the negative level set of the auxiliary optimal control problem that is less or equal ε. Some numerical examples are considered to show the relevance of our approach.
    author={Ana Chorobura and Hasnaa Zidani },
    title={Bi-objective finite horizon optimal control problems with 
           Bolza and maximum running cost },
    year={submitted },