On the controllability of the Burger equation
We present here a return method to describe some attainable sets on an interval of the classical Burger equation by means of the variation of the domain.
On the controllability of the fifth-order Korteweg-de Vries equation
On the controllability of the Laplace equation observed on an interior curve.
The boundary approximate controllability of the Laplace equation observed on an interior curve is studied in this paper. First we consider the Laplace equation with a bounded potential. The Lp (1 < p < ∞) approximate controllability is established and controls of Lp-minimal norm are built by duality. At this point, a general result which clarifies the relationship between this duality approach and the classical optimal control theory is given. The results are extended to the Lp (1≤...
On the controllability under constraints on the control for hyperbolic equations.
On the cost of null-control of an artificial advection-diffusion problem
In this paper we study the null-controllability of an artificial advection-diffusion system in dimension n. Using a spectral method, we prove that the control cost goes to zero exponentially when the viscosity vanishes and the control time is large enough. On the other hand, we prove that the control cost tends to infinity exponentially when the viscosity vanishes and the control time is small enough.
On the exact controllability of a nonlinear stochastic heat equation.
On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients
Let be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke’s generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov...
On the existence of nonsmooth control-Lyapunov functions in the sense of generalized gradients
Let be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov...
On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems
In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate...
On the -instability and -controllability of steady flows of an ideal incompressible fluid
In the existing stability theory of steady flows of an ideal incompressible fluid, formulated by V. Arnold, the stability is understood as a stability with respect to perturbations with small in vorticity. Nothing has been known about the stability under perturbation with small energy, without any restrictions on vorticity; it was clear that existing methods do not work for this (the most physically reasonable) class of perturbations. We prove that in fact, every nontrivial steady flow is unstable...
On the lack of null-controllability of the heat equation on the half space.
On the Lyapunov equation in Banach spaces and applications to control problems.
On the null-controllability of diffusion equations
This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check...
On the null-controllability of diffusion equations
This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check...
On the number of control sets on compact homogeneous spaces.
On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials
On the parametric solution to the second-order Sylvester matrix equation .
On the perturbations in a time-optimal closed-loop system
On the properties of reachability, observability, controllability, and constructibility of discrete-time positive time-invariant linear systems with aperiodic choice of the sampling instants.
On the properties of the Aumann integral with applications to differential inclusions and control systems