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International audience; We propose a new first-order splitting algorithm for solving jointly the primal and dual formulations of large-scale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitting approach, in the sense that the gradient and the linear operators involved are applied explicitly without any inversion, while the nonsmooth functions are processed individually via their proximity operators. This work brings together and notably extends several classical splitting schemes, like the forward-backward and Douglas-Rachford methods, as well as the recent primal-dual method of Chambolle and Pock designed for problems with linear composite terms.

These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures endowed with the distances $W_p$ induced by optimal transport are detailed. The key tools to put in relation optimal transport and PDEs are provided.

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. A sufficient second order optimality condition and uniqueness of the dual variables are assumed for that problem. Sufficient second order optimality conditions are shown for regularized problems with small regularization parameter. Moreover, error estimates with respect to the regularization parameter are derived.

This is the first of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [20] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems.

This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations.

We modify the definition of lopsided convergence of bivariate functionals to obtain stability results for the min/sup points of some control problems. In particular, we develop a scheme of finite dimensional approximations to a large class of non-convex control problems.

We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if~$\bar u$ minimizes the functional $$ \int_\Omega |\nabla_{\H^n}u|^2$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle~$\psi$, then $$ 0\leq \Delta_{\H^n} \bar u\leq \Big(\Delta_{\H^n}\psi\Big)^{+}. $$ Moreover, we discuss how it could be possible to generalize the previous bound to a quasilinear setting once some regularity issues for the equation $$ \div_{\H^n}\Big(|\nabla_{\H^n}u|^{p-2}\nabla_{\H^n}u\Big)=f $$ are satisfied.}