Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints
The authors consider the following differential inclusion: \[ \dot{y} \in F(s,y), \tag{1} \] where \(F:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}^n\) is a multifunction, and they study the problem of finding suitable conditions in order to obtain the lower semicontinuity of the solution map, which is the multifunction that to each \((t,x) \in \mathbb{R} \times \mathbb{R}^n\) associates all trajectories (solutions to equation (1)) issuing from \(x\) at time \(t\). This property has applications in the theory of optimal control problems as the same authors show in the final part of their paper. The authors introduce a condition and they prove that it is sufficient to assure the lower semicontinuity of the solution map. This condition is an extension of some introduced before and it is proved that it can be applied, for example, to the case of a control system with an integral constraint.
scientific article; zbMATH DE number 1512499