Section: New Results

Hamilton-Jacobi approach for deterministic control problems

Participants : Albert Altarovici, Olivier Bokanowski, Yingda Cheng [University of Texas] , Anna Desilles, Nicolas Forcadel, Zhiping Rao, Chi-Wang Shu [Brown University] , Hasnaa Zidani.

The paper [30] deals with deterministic optimal control problem with state constraints and non-linear dynamics. It is known for such a problem that the value function is in general discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypass the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

In [34] , [19] we study an optimal control problem governed by measure driven differential systems and in presence of state constraints. First, under some weak invariance assumptions, we study in [19] the properties of the value function and obtain its characterization by means of an auxiliary control problem of absolutely continuous trajectories. For this, we use some known techniques of reparametrization and graph completion. Then we give a characterization of the value function as the unique constrained viscosity solution of a Hamilton-Jacobi equation with measurable time dependant Hamiltonians.

The general case without assuming any controllability assumption is considered in [34] . We prove that the optimal solutions can still be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints.

The paper [17] deals with minimal time problems governed by nonlinear systems under general time dependent state constraints and in the two-player games setting. In general, it is known that the characterization of the minimal time function, as well as the study of its regularity properties, is a difficult task in particular when no controllability assumption is made. In addition to these difficulties, we are interested here to the case when the target, the state constraints and the dynamics are allowed to be time-dependent.s We introduce a particular reachability control problem, which has a supremum cost function but is free of state constraints. This auxiliary control problem allows to characterize easily the backward reachable sets, and then, the minimal time function, without assuming any controllability assumption. These techniques are linked to the well known level-set approaches. Our results can be used to deal with motion planning problems with obstacle avoidance, see [16] .

Several works have been also carried out in the domain of numerical methods of HJB equations. The paper [31] aims at studing a discontinuous Galerkin scheme for front propagation with obstacles. We extend a first work published in [11] , to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of [12] , leading to a level set formulation driven by a Hamilton-Jacobi variational inequality. The DG scheme is motivated by the variational formulation when the equation corresponds to linear convection problems in presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization. Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.