A general Hamilton-Jacobi framework for non-linear state-constrained control problems

State-Constrained Optimal Control Problems of Impulsive Differential Equations

“…The main idea to treat the time-dependent state constraints is to characterize the epigraph of the value function instead of characterizing the value function directly. Here, we extend the ideas developed in [1] to the case of time-dependent state constraints, and prove that the epigraph of ϑ can be characterized by means of a Lipschitz continuous viscosity solution of a time-measurable HJB equation (this notion of viscosity notion will be made precise in Section 4).…”

“…Under some controllability assumption and when the set of state-constraints is not time-dependent, the value function can be shown to be the unique constrained-viscosity solution on an adequate HJB equation, see in [28,29,18]. We refer also to [4,1] for a discussion on the general case where the control problem is lacking controllability properties.…”

“…The characterization ofv by an HJB equation on a tube K . is not a simple task because the evolution ofv depends also on the evolution of the map K. Here we extend to time-dependent state-constrained control problems an idea developed recently in [1] which allows to compute all the epigraph of the value functionv by solving an appropriate variational HJB equation.…”

“…In particular, according to theorem 3.2, the distance function to the set θ∈[0,1] {θ} × K θ fulfilled the conditions). By using an idea introduced in [1], an equivalent way to characterize the epigraph of ϑ consists of considering the control problem…”

Optimal feedback control for undamped wave equations by solving a HJB equation

“…In the last decades, several theoretical and numerical developments in HJB theory led to powerful and efficient numerical approaches that can be used for control problems up to 6-dimensional problems [2,7,10,8,15,37,40]. For higher dimensional problems, various approaches have been studied in the literature, including model reduction or advanced numerical schemes as, e.g., sparse grids, see [11].…”