Lie brackets and real analyticity in control theory
Limitations on the control of Schrödinger equations
We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control is not controllable...
Linear dynamical systems: an axiomatic approach.
Linear dynamical systems of higher genus.
Local and global null controllability of time varying linear control systems
Local and global null controllability of time varying linear control systems
For linear control systems with coefficients determined by a dynamical system null controllability is discussed. If uniform local null controllability holds, and if the Lyapounov exponents of the homogeneous equation are all non-positive, then the system is globally null controllable for almost all paths of the dynamical system. Even if some Lyapounov exponents are positive, an irreducibility assumption implies that, for a dense set of paths, the system is globally null controllable.
Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.
Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.
Local controllability of odd systems
Local exact controllability of the age-dependent population dynamics with diffusion.
Local exact controllability of the diffusion equation in one dimension.
Local exact controllability to the trajectories of the Boussinesq system
Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions
In this paper we deal with the local exact controllability of the Navier-Stokes system with nonlinear Navier-slip boundary conditions and distributed controls supported in small sets. In a first step, we prove a Carleman inequality for the linearized Navier-Stokes system, which leads to null controllability of this system at any time T>0. Then, fixed point arguments lead to the deduction of a local result concerning the exact controllability to the trajectories of the Navier-Stokes system.
Local exact lagrangian controllability of the Burgers viscous equation
Local null controllability of a fluid-solid interaction problem in dimension 3
We are interested by the three-dimensional coupling between an incompressible fluid and a rigid body. The fluid is modeled by the Navier-Stokes equations, while the solid satisfies the Newton's laws. In the main result of the paper we prove that, with the help of a distributed control, we can drive the fluid and structure velocities to zero and the solid to a reference position provided that the initial velocities are small enough and the initial position of the structure is close to the reference...
Local null controllability of a two-dimensional fluid-structure interaction problem
In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given , the system can be driven at rest and the structure to its reference configuration at time . To show this result, we first consider a linearized system....
Local null controllability of a two-dimensional fluid-structure interaction problem
In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T > 0, the system can be driven at rest and the structure to its reference configuration at time T. To show this result, we first consider a linearized system....
Local small time controllability and attainability of a set for nonlinear control system
In the present paper, we study the problem of small-time local attainability (STLA) of a closed set. For doing this, we introduce a new concept of variations of the reachable set well adapted to a given closed set and prove a new attainability result for a general dynamical system. This provide our main result for nonlinear control systems. Some applications to linear and polynomial systems are discussed and STLA necessary and sufficient conditions are obtained when the considered set is a hyperplane....
Local small time controllability and attainability of a set for nonlinear control system
In the present paper, we study the problem of small-time local attainability (STLA) of a closed set. For doing this, we introduce a new concept of variations of the reachable set well adapted to a given closed set and prove a new attainability result for a general dynamical system. This provide our main result for nonlinear control systems. Some applications to linear and polynomial systems are discussed and STLA necessary and sufficient conditions are obtained when the considered set...