Approximation of the Pareto optimal set for multiobjective optimal control problems using viability kernels.

*(English)*Zbl 1307.90159Many real-world applications lead to a general finite-horizon multiobjective optimal control problem where several objective functions need to be optimized simultaneously. The main objective of this paper is to derive a convergent numerical approximation of the Pareto optimal set. The proposed approach is based on a viability theory.

The first contribution is to reformulate the problem of finding the Pareto optimal set as the problem of determining a viability kernel which is done by considering the epigraph of a set-valued return function \(V\). The epigraph of this function coincides with the viability kernel of a certain related augmented dynamical system. The result is that the Pareto optimal set for any time \(t\) and state \(x\) can be obtained just by evaluating \(V\) at \((t,x)\).

The second contribution is the use of existing numerical schemes to derive a convergent approximation of \(V\) and to introduce a so-called approximate set-valued return function as the solution of a multiobjective dynamic programming equation. The epigraph of this function converges in the sense of PainlevĂ©-Kuratowski to the epigraph of \(V\). The result is that an approximation of the Pareto optimal set \(V(t,x)\) can be obtained just by evaluating the approximate set-valued return function at \((t,x)\).

The first contribution is to reformulate the problem of finding the Pareto optimal set as the problem of determining a viability kernel which is done by considering the epigraph of a set-valued return function \(V\). The epigraph of this function coincides with the viability kernel of a certain related augmented dynamical system. The result is that the Pareto optimal set for any time \(t\) and state \(x\) can be obtained just by evaluating \(V\) at \((t,x)\).

The second contribution is the use of existing numerical schemes to derive a convergent approximation of \(V\) and to introduce a so-called approximate set-valued return function as the solution of a multiobjective dynamic programming equation. The epigraph of this function converges in the sense of PainlevĂ©-Kuratowski to the epigraph of \(V\). The result is that an approximation of the Pareto optimal set \(V(t,x)\) can be obtained just by evaluating the approximate set-valued return function at \((t,x)\).

Reviewer: Ctirad Matonoha (Prague)

##### MSC:

90C29 | Multi-objective and goal programming |

49L20 | Dynamic programming in optimal control and differential games |

49M20 | Numerical methods of relaxation type |

54C60 | Set-valued maps in general topology |