Jean-Pierre Raymond

Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations

We study the exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution, by means of a feedback boundary control, in dimension 2 or 3. The feedback law is determined by solving a Linear-Quadratic control problem. We do not assume that the normal component of the control is equal to zero. In that case the state equation, satisfied by the velocity field y, is decoupled into an evolution equation satisfied by Py, where P is the so-called Helmholtz projection operator, and a quasi-stationary elliptic equation satisfied by (I − P )y. Using this decomposition we show that the feedback law can be expressed only in function of Py. In the two dimensional case we show that the linear feedback law provides a local exponential stabilization of the Navier-Stokes equations.

Error Estimates for the Numerical Approximation of Dirichlet Boundary Control for Semilinear Elliptic Equations

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in

Pontryagin's Principle for State-Constrained Control Problems Governed by Parabolic Equations with Unbounded Controls

{\mathbb R}^2

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order

Pontryagin's Principle For Local Solutions of Control Problems with Mixed Control-State Constraints

O(h^{1 - 1/p})

Error Estimates for the Numerical Approximation of a Distributed Control Problem for the Steady-State Navier-Stokes Equations

for some

Optimal Control Problems with Mixed Control-State Constraints

p > 2

Pontryagin's principle for time-optimal problems

, which is consistent with the

Regional Null Controllability of a Linearized Crocco-Type Equation

W^{1 - 1/p,p}(\Gamma)

Neumann Boundary Control of Hyperbolic Equations with Pointwise State Constraints

-regularity of the optimal control.