The value function of an asymptotic exit-time optimal control problem

Abstract

We consider a class of exit–time control problems for nonlinear systems with a nonnegative vanishing Lagrangian. In general, the associated PDE may have multiple solutions, and known regularity and stability properties do not hold. In this paper we obtain such properties and a uniqueness result under some explicit sufficient conditions. We briefly investigate also the infinite horizon problem

Stabilizability in optimal control

“…follows by known optimality principles (see e.g. [22,25,33]). The main contribution of [19,23] was to prove that the existence of a p 0 -Minimum Restraint Function W for some p 0 > 0 allows to produce a pair (x, u) that meets both of these properties.…”

“…is clearly bounded above by any p 0 -Minimum Restraint Function divided by p 0 . Hence our approach could be useful to design approximated optimal closedloop strategies, when there exists a sequence of p 0 -Minimum Restraint Functions approaching V , as in [25], or at least to obtain "safe" performances, keeping the cost under the value W . Moreover, when…”

“…is continuous on the target's boundary and this is crucial to establish comparison, uniqueness, and robustness properties for the associated Hamilton-Jacobi-Bellman equation [22,24,25] and to study associated asymptotic and ergodic problems [26]. From this PDE point of view, problem (1.1)-(1.2) has been widely investigated; a likely incomplete bibliography, also containing applications (for instance, the Füller and shape-from-shading problems), includes [8,16,20,33] and the references therein.…”

On asymptotic exit-time control problems lacking coercivity

“…(see for more details (BVP) in (4.1)), has not a unique solution even among the continuous and nonnegative functions and in the compact control case studied in [16]. Indeed we can expect that V is the maximal subsolution to (BVP), but not the minimal nonnegative supersolution, which in general turns out to be a different value function, V m , where also controls that do not steer to the target are admissible.…”

Boubaker hybrid functions and their application to solve fractional optimal control and fractional variational problems

“…Optimal control and variational problems are two classes of these kinds of problems and the importance of these fields motivated many researchers to consider them. The problems of integer order dynamic system have occurred in engineering, science, geometry and many other fields and the researchers have widely worked on this topic [36], [43], [64], however they have considered the area of fractional problems during last few decades. The application of fractional optimal control problems can be found in engineering and physics.…”