## Section: New Results

### Hamilton-Jacobi-Bellman approach

#### Numerical methods for HJB equations

Participants : H. Zidani, E. Cristiani, N. Forcadel, O. Bokanowski [ U. Lab JLL, U. Paris 7 ] .

We aim in developping anti-diffusive numerical schemes for HJB equations with possibly discontinuous initial data.

We investigate in [16] two anti-diffusive numerical schemes, the first one is based
on the Ultra-Bee scheme and the second one is based on the Fast Marching Method.
We prove the convergence and derive L^{1} -error estimates for both schemes.
We also provide numerical examples to validate their accuracy in solving smooth
and discontinuous solutions.

In [18] , we prove the convergence of a non-monotonous scheme
for a one-dimensional first order Hamilton-Jacobi-Bellman equation
of the form , v(0, x) = v_{0}(x) .
The scheme is related to the HJB-UltraBee scheme suggested in [66] .
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.

Finally in [15] , we are interested in some front propagation problems coming from control problems in d -dimensional spaces, with d2 . As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an Hamilton-Jacobi-Bellman equation with discontinuous data, discretized by means of the antidissipative Ultra Bee scheme.

We develop an efficient dynamic storage technique suitable for handling front evolutions in large dimension.
Then we propose a fast algorithm, showing its relevance on several challenging tests in dimension d = 2, 3, 4 .
We also compare our method with the techniques usually used in level set methods.
Our approach leads to a computational cost as well as a memory allocation scaling as O(N_{nb}) in most situations, where N_{nb} is the number of grid nodes around the front.
Finally, let us point out that the approximation in a rough grid gives qualitatively good results.
This study leads also to a very fast numerical code in C++ for solving HJB equations in 4d.

#### Minimum time problems

Participants : O. Bokanowski, A. Briani, H. Zidani.

We have investigated a minimum time problem for controlled non-autonomous differential systems, with a dynamics depending on the final time. The minimal time function associated to this problem does not satisfy the dynamic programming principle. However, we have proved in [13] , that it is related to a standard front propagation problem via the reachability function.

#### Reachability of state-constrained nonlinear control problems lacking controllability

Participants : O. Bokanowski, N. Forcadel, H. Zidani.

We consider a target problem for a nonlinear system under state constraints. In [31] , we give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent state constraints, as well as a target problem for a two-player game with state constraints. Our method gives a good framework for numerical approximations, and some numerical illustrations are included in the paper.

#### Control problems for BV trajectories

Participants : A. Briani, H. Zidani.

This paper aims to investigate a control problem governed by differential equations with Random measure as data and with final state constraints. This problem is motivated by several real applications. For instance, in space navigation area, when steering a multi-stage launcher, the separation of the boosters (once they are empty) lead to discontinuities in the mass variable. In resource management, discontinuous trajectories are also used to modelize the problem of sequential batch reactors.

By using a known reparametrization method (by Dal Maso and Rampazzo, 1991), we obtain that the value function can be characterized by means of an auxiliary control problem involving absolutely continuous trajectories. We study the characterization of the value function of this auxiliary problem and discuss its discrete approximations.

#### Coupling the HJB and shooting method approaches

Participants : E. Cristiani, P. Martinon, H. Zidani.

In optimal control, there is a well-known link between the Hamilton-Jacobi-Bellman (HJB) equation and Pontryagin's Minimum Principle (PMP). Namely, the costate (or adjoint state) in PMP corresponds to the gradient of the value function in HJB. We investigate from the numerical point of view the possibility of coupling these two approaches to solve control problems. Firstm a rough approximation of the value function is computed by the HJB method, and then used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests have been conducted over simple problems involving multiple minima, discontinuous control, singular arcs and state constraints. The application to realistic space launcher problems is currently in progress..

The first results will appear in [35] .