Optimal control problems with non-control regions: necessary optimality conditions

Loss control regions in optimal control problems

“…We point out that, similarly to Theorem 2.1 that is an extension of the HMP that can be found in our previous paper [4], Theorem 3.1 is an extension of the PMP that can be found in our previous work [5] in the simpler context of a fixed initial condition and no final state constraint.…”

“…Theorem 2.1 is an extension of a HMP that can be found in our previous work [4] in the simpler context of a fixed initial condition and no final state constraint. Furthermore our methodology in [4] consists in a thorough sensitivity analysis of the hybrid control system, involving an inductive reasoning to handle the consecutive crossing times thanks to implicit function theorems. In the present work, Theorem 2.1, not only handles mixed terminal state constraints, but moreover is proved with a simpler method based on an augmentation technique.…”

“…It means that the trajectory does not cross the boundary ∂E k ∩ E k+1 tangentially (see Figure 1). Note that, in the present work, we consider a transverse condition that is (slightly) weaker than the one examined in previous works such as [4,30] and given by    u admits left and right limits at τ k denoted by u − (τ k ) and u…”

“…To address optimal control problems with loss control regions, we pursue the approach initiated in our previous works [4,5], by considering the control function as a permanent control in control regions and as a constant parameter in loss control regions. With this point of view, our framework falls into the domain of hybrid optimal control theory which extends the classical theory to discontinuous dynamics.…”

“…a parameter that can change its value when the trajectory moves from one region to another). To the best of our knowledge, this last concept has never been considered in the literature until our previous work [4].…”