## Section: New Results

### Hamilton Jacobi Bellman approach

#### Optimal feedback control of undamped wave equations by solving a HJB equation

Participants : Hasnaa Zidani, Axel Kröner.

An optimal finite-time horizon feedback control problem for (semi linear) wave equations is studied in [25] . The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional space which makes infeasible the numerical resolution by HJB approach. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach

#### Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations

Participant : Hasnaa Zidani.

The works [27] , [91] deal with deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ${I\phantom{\rule{-1.70717pt}{0ex}}R}^{N}$. As a consequence, the dynamics and running cost present discontinuities at the interfaces of these domains. This leads to a complex interplay that has to be analyzed among transmission conditions to "glue" the propagation of the value function on the interfaces. Several questions arise: how to define properly the value function(s) and what is (are) the right Bellman Equation(s) associated with this problem?. In the case of a simple geometry (namely when the space ${I\phantom{\rule{-1.70717pt}{0ex}}R}^{N}$ is partitioned into two subdomains separated with an interface which is assumed to be a regular hypersurface without any connectedness requirement), [27] discuss different conditions on the hyperplane where the dynamic and the running cost are discontinuous, and the uniqueness properties of the Bellman problem are studied. In this paper we use a dynamical approach, namely instead of working with test functions, the accent is put on invariance properties of an augmented dynamics related to the integrated control system. The comparison principle is accordingly based, rather than on (semi)continuity of the Hamiltonian appearing in the Hamilton–Jacobi–Bellman equation, on some weak separation properties of this dynamics with respect to the stratification.

#### Control Problems on Stratifiable state-constraints Sets

Participants : Cristopher Hermosilla, Hasnaa Zidani.

This work deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the value function has not enough regularity, or can fail to be the unique constrained viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. Here, we consider the case of a set of constraints having a strati

ed structure. Under this circumstance, the interior of this set may be empty or disconnected, and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints. In such situations, the classical pointing quali

cation hypothesis are not relevant. The discontinuous Value Function is then characterized by means of a system of HJB equations on each stratum that composes the state-constraints. This result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur.

#### Constrained optimization problems in finite and infinite dimensional spaces

Participant : Cristopher Hermosilla.

We investigate in [39] convex constrained nonlinear optimization problems and optimal control with convex state constraints. For this purpose we endow the interior of constraints set with the structure of Riemannian manifold. In particular, we consider a class of Riemannian metric induced by the squared Hessian of a Legendre functions. We describe in details the geodesic curves on this manifolds and we propose a gradient-like algorithm for constrained optimization based on linear search along geodesics. We also use the Legendre change of coordinates to study the Value Function of a Mayer problem with state constraints. We provide a characterization of the Value Function for this problem as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.