Transmission conditions on interfaces for Hamilton–Jacobi–Bellman equations

We establish a comparison principle for a Hamilton–Jacobi–Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of ε-partition and minimal ε-partition for intervals of definition of an integral trajectory.     

State-Constrained Optimal Control Problems of Impulsive Differential Equations

The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption.     

The Mayer and Minimum Time Problems with Stratified State Constraints

This paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectories can be analyzed directly. The value functions associated with the state constrained Mayer and minimal time problems are characterized as solutions to a pair of Hamilton-Jacobi inequalities with appropriate boundary conditions. The novel feature of these inequalities lies in the choice of the Hamiltonian.     

Preface of the Special Issue New Horizons in Optimal Control- a Special Tribute to Helmut Maurer,Urszula Ledzewicz and Heinz Schättler

Set-Valued and Variational Analysis -     

Error estimates for stochastic differential games: the adverse stopping case

^** Email: frederic.bonnans{at}inria.fr^*** Email: stefania.maroso{at}inria.fr^**** Email: zidani{at}ensta.fr We obtain error bounds for monotone approximation schemes ofa particular Isaacs equation. This is an extension of the theoryfor estimating errors for the Hamilton–Jacobi–Bellmanequation. To obtain the upper error bound, we consider the ‘Krylovregularization’ of the Isaacs equation to build an approximatesub-solution of the scheme. To get the lower error bound, weextend the method of Barles & Jakobsen (2005, SIAM J. Numer.Anal.) which consists in introducing a switching system whosesolutions are local super-solutions of the Isaacs equation.     

Pontryagin's Principle for Time-Optimal Problems

We consider time-optimal control problems for semilinear parabolic equations with pointwise state constraints and unbounded controls. A Pontryagin's principle is obtained in nonqualified form without any qualification condition. The terminal time, which is a control variable, satisfies an optimality condition, which seems to be new in the context of control problems for partial differential equations.     

Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v _t + max _α (f(x, α)v _x ) = 0, v(0, x) = v ₀(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L ¹-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.