Monday, 12 February 2024
We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.
The goal of this course is to introduce the basic tools of microlocal analysis and to show how these techniques can be used for the study of wave control. We start with the wave front set and pseudo-differential calculus. Then, we present the propagation of singularities theorem of L.H ̈ormander. Finally, we present the notion of microlocal defect measures with their microlocal properties ( elliptic regularity and support propagation ). All these facts will be illustrated by various applications, in particular the observability of the wave equation.
Our focus is on exploring the controllability and asymptotic stabilization of nonlinear partial differential equations in which the quadratic term plays a pivotal role. In these cases, both controllability and asymptotic stabilization are not achievable without the inclusion of these quadratic terms. This includes Schr ̈odinger equations, the inviscid Burgers equation, Kortewegde Vries equations and a Saint-Venant (shallow-water) equation. Regarding controllability, we demonstrate that controlling such systems may require more time than anticipated based on the speed of propagation. For the stabilization aspect, we introduce methodologies for devising stabilizing feedback laws. Additionally, we present unresolved challenges in these two domains.
Some classical nonlinear semigroups arising in mechanics induce unilateral bounds on solutions. Hamilton-Jacobi equations and 1-d scalar conservation laws are classical examples of such nonlinear effects: solutions spontaneously develop one sided Lipschitz or semi-concavity conditions.When this occurs the range of the semigroup is unilaterally bounded by a threshold. On the other hand, in practical applications, one is often led to consider the problem of time-inversion, so to identify the initial sources that originated the observed dynamics at the final time. In these lectures we shall discuss this problem and address the following specific questions:
– Identification of the range of the semigroup.
– Identification of the class of initial data leading to a target.
– Least-square approaches versus backward-forward resolution.
– Numerical reconstruction.
We shall also present a number of open problems arising in this area and the possible link with reinforcement learning.
Tuesday, 13 February 2024
We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.
The goal of this course is to introduce the basic tools of microlocal analysis and to show how these techniques can be used for the study of wave control. We start with the wave front set and pseudo-differential calculus. Then, we present the propagation of singularities theorem of L.H ̈ormander. Finally, we present the notion of microlocal defect measures with their microlocal properties ( elliptic regularity and support propagation ). All these facts will be illustrated by various applications, in particular the observability of the wave equation.
In this lecture, I will present Carleman Estimates for the Laplace operator and some of its applications. In particular, I will discuss the Cauchy problem for the Laplace operator, its uniqueness and stability properties, and then introduce general Carleman Estimates for the Laplace operator. As an application, I will give a quantification of the unique continuation property for the Laplace operator with respect to a potential, and discuss the link with the Landis conjecture, and applications to the optimal control problem for a semilinear elliptic PDE.
Some classical nonlinear semigroups arising in mechanics induce unilateral bounds on solutions. Hamilton-Jacobi equations and 1-d scalar conservation laws are classical examples of such nonlinear effects: solutions spontaneously develop one sided Lipschitz or semi-concavity conditions.When this occurs the range of the semigroup is unilaterally bounded by a threshold. On the other hand, in practical applications, one is often led to consider the problem of time-inversion, so to identify the initial sources that originated the observed dynamics at the final time. In these lectures we shall discuss this problem and address the following specific questions:
– Identification of the range of the semigroup.
– Identification of the class of initial data leading to a target.
– Least-square approaches versus backward-forward resolution.
– Numerical reconstruction.
We shall also present a number of open problems arising in this area and the possible link with reinforcement learning.
Wednesday, 14 February 2024
We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.
The goal of this course is to introduce the basic tools of microlocal analysis and to show how these techniques can be used for the study of wave control. We start with the wave front set and pseudo-differential calculus. Then, we present the propagation of singularities theorem of L.H ̈ormander. Finally, we present the notion of microlocal defect measures with their microlocal properties ( elliptic regularity and support propagation ). All these facts will be illustrated by various applications, in particular the observability of the wave equation.
In this lecture, I will present Carleman Estimates for the Laplace operator and some of its applications. In particular, I will discuss the Cauchy problem for the Laplace operator, its uniqueness and stability properties, and then introduce general Carleman Estimates for the Laplace operator. As an application, I will give a quantification of the unique continuation property for the Laplace operator with respect to a potential, and discuss the link with the Landis conjecture, and applications to the optimal control problem for a semilinear elliptic PDE.
Some classical nonlinear semigroups arising in mechanics induce unilateral bounds on solutions. Hamilton-Jacobi equations and 1-d scalar conservation laws are classical examples of such nonlinear effects: solutions spontaneously develop one sided Lipschitz or semi-concavity conditions.When this occurs the range of the semigroup is unilaterally bounded by a threshold. On the other hand, in practical applications, one is often led to consider the problem of time-inversion, so to identify the initial sources that originated the observed dynamics at the final time. In these lectures we shall discuss this problem and address the following specific questions:
– Identification of the range of the semigroup.
– Identification of the class of initial data leading to a target.
– Least-square approaches versus backward-forward resolution.
– Numerical reconstruction.
We shall also present a number of open problems arising in this area and the possible link with reinforcement learning.
Thursday, 15 February 2024
We introduce in an abstract framework the notion of infinite dimensional time invariant well posed linear system (with inputs and outputs). We provide examples of such systems described by some classical evolution PDEs with distributed control and observation. We next define the fundamental notion of reachable space for this type of systems and we relate its properties with various controllability and observability types. We illustrate this theory by studying the reachability and controllability properties examples of various systems described by linear evolution partial differential equations.
The goal of this course is to introduce the basic tools of microlocal analysis and to show how these techniques can be used for the study of wave control. We start with the wave front set and pseudo-differential calculus. Then, we present the propagation of singularities theorem of L.H ̈ormander. Finally, we present the notion of microlocal defect measures with their microlocal properties ( elliptic regularity and support propagation ). All these facts will be illustrated by various applications, in particular the observability of the wave equation.
A control system is differentially flat if every trajectory can be expressed in terms of a function (called a flat output) and its derivatives. The flatness approach, introduced by Michel Fliess, Jean L ́evine, Philippe Martin and Pierre Rouchon in 1995 for ODEs, was next applied to PDEs by B ́eatrice Laroche, Philippe Martin and Pierre Rouchon in 2000 to derive the approximate controllability of the heat equation.
In this course, we will show how the flatness approach can be used to derive null or exact controllability results for many evolution equations with some boundary controls.
We first show how to obtain the null controllability of the 1D heat equation, or of the ND heat equation in cylinders, providing both the control input and the trajectory as series. We shall see how to compute easily a lot of derivatives of the flat output to compute the partial sums in the series. Next, we show how to apply the flatness approach to the linear 1D Schr ̈odinger equation and to the linear Korteweg-de Vries equation.
We proceed to the null controllability of the heat equation with variable coefficients, namely
\[ (a(x)u_x)_x+ b(x)u_x+c(x)u=\rho (x)u_t.\]
Here the coefficients a, b, c, ρ are merely assumed to be measurable, and we stress that Carleman estimates (if any) are not yet available. We mainly assume first that a ∈ L1 and 1/a ∈ L1 (weak degeneracy). Next, we pass to the case of a strong degeneracy at one point of the boundary. The flatness approach is combined with Pr ̈ufer method to obtain the asymptotics of the eigenvalues. Next, we are concerned with the determination of the reachable states of the heat equation, first for the boundary control, and next for the distributed control.
Finally, we address the issue of the exact controllability of a semilinear heat equation, and show how a Cauchy problem in the variable x can be solved by using scales of Banach spaces constituted of Gevrey functions. We obtain a local exact controllability in a space of holomorphic functions. The method is next extended to any anisotropic 1D equation of the form
\[ \partial _t^N y=\zeta _M \partial _x^M y +f(x,y,...,\partial _x^{M-1}y), \quad 1\le N<M,\]
that may be severely ill-posed. We obtain again some local exact controllability result in some space of holomorphic functions. The result can be applied to
-
the Korteweg-de Vries equation: $\partial _t y+\partial _x^3 y + \partial _xy+y\partial _xy=0$;
-
the ``good'' and ``bad'' Boussinesq equation: $\partial _t^2 y =\pm \partial_x^4 y +\partial _x ^2 y -\partial _x^2 (y^2)$;
-
the Ginzburg-Landau equation: $\partial _t y = e^{i\theta} \partial _x ^2 y + e^{i\varphi} |y|^2y$ where $\theta,\varphi\in\R$;
-
the Kuramoto-Sivashinsky equation: $\partial _ty +\partial _x ^4 y +\partial _x^2 y +y\partial _x y=0$.
Some classical nonlinear semigroups arising in mechanics induce unilateral bounds on solutions. Hamilton-Jacobi equations and 1-d scalar conservation laws are classical examples of such nonlinear effects: solutions spontaneously develop one sided Lipschitz or semi-concavity conditions.When this occurs the range of the semigroup is unilaterally bounded by a threshold. On the other hand, in practical applications, one is often led to consider the problem of time-inversion, so to identify the initial sources that originated the observed dynamics at the final time. In these lectures we shall discuss this problem and address the following specific questions:
– Identification of the range of the semigroup.
– Identification of the class of initial data leading to a target.
– Least-square approaches versus backward-forward resolution.
– Numerical reconstruction.
We shall also present a number of open problems arising in this area and the possible link with reinforcement learning.
Friday, 16 February 2024
Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this course we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.
The stabilizability of the control system (A,B), where A generates a continuous semigroup on a Hilbert space and B is a bounded control operator, has been characterized via some weak observability inequalities by Trélat, Wang and Xu. We analyse this in the context of certain coupled systems.
In this talk I will present some results for a nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. To derive the first order optimality conditions one needs to prove differentiability of the control to state map which in turn requires higher regularity of the solutions. We will consider the cases of regular as well as singular potential. In both cases we study the tracking type optimal control problem with control in the velocity equation. The results like the existence of optimal control and characterization of optimal control in terms of the solution of the corresponding adjoint system will be discussed.
A control system is differentially flat if every trajectory can be expressed in terms of a function (called a flat output) and its derivatives. The flatness approach, introduced by Michel Fliess, Jean L ́evine, Philippe Martin and Pierre Rouchon in 1995 for ODEs, was next applied to PDEs by B ́eatrice Laroche, Philippe Martin and Pierre Rouchon in 2000 to derive the approximate controllability of the heat equation.
In this course, we will show how the flatness approach can be used to derive null or exact controllability results for many evolution equations with some boundary controls.
We first show how to obtain the null controllability of the 1D heat equation, or of the ND heat equation in cylinders, providing both the control input and the trajectory as series. We shall see how to compute easily a lot of derivatives of the flat output to compute the partial sums in the series. Next, we show how to apply the flatness approach to the linear 1D Schr ̈odinger equation and to the linear Korteweg-de Vries equation.
We proceed to the null controllability of the heat equation with variable coefficients, namely
\[ (a(x)u_x)_x+ b(x)u_x+c(x)u=\rho (x)u_t.\]
Here the coefficients a, b, c, ρ are merely assumed to be measurable, and we stress that Carleman estimates (if any) are not yet available. We mainly assume first that a ∈ L1 and 1/a ∈ L1 (weak degeneracy). Next, we pass to the case of a strong degeneracy at one point of the boundary. The flatness approach is combined with Pr ̈ufer method to obtain the asymptotics of the eigenvalues. Next, we are concerned with the determination of the reachable states of the heat equation, first for the boundary control, and next for the distributed control.
Finally, we address the issue of the exact controllability of a semilinear heat equation, and show how a Cauchy problem in the variable x can be solved by using scales of Banach spaces constituted of Gevrey functions. We obtain a local exact controllability in a space of holomorphic functions. The method is next extended to any anisotropic 1D equation of the form
\[ \partial _t^N y=\zeta _M \partial _x^M y +f(x,y,...,\partial _x^{M-1}y), \quad 1\le N<M,\]
that may be severely ill-posed. We obtain again some local exact controllability result in some space of holomorphic functions. The result can be applied to
-
the Korteweg-de Vries equation: $\partial _t y+\partial _x^3 y + \partial _xy+y\partial _xy=0$;
-
the ``good'' and ``bad'' Boussinesq equation: $\partial _t^2 y =\pm \partial_x^4 y +\partial _x ^2 y -\partial _x^2 (y^2)$;
-
the Ginzburg-Landau equation: $\partial _t y = e^{i\theta} \partial _x ^2 y + e^{i\varphi} |y|^2y$ where $\theta,\varphi\in\R$;
-
the Kuramoto-Sivashinsky equation: $\partial _ty +\partial _x ^4 y +\partial _x^2 y +y\partial _x y=0$.
Monday, 19 February 2024
Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this course we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.
In PDEs, quite often it is become necessary to approximate the PDEs by a family of PDEs involving a parameter which may go to 0 or ∞. It can also happen the other way that the physical systems produces a family of PDEs and we may need to find the limiting PDE. It can be due to several reasons, for example due to the presence of multi-scales. Multi scales arise in many physical and industrial problems, and industrial constructions which lead to very complicated structures. Homogenization is a branch of science where one try to understand the microscopic structures via a macroscopic medium by taking care of the various scales involved in the problem which appears through the solutions.
In this talk, we consider a high oscillating domain where the oscillatory part consists of two high contrast materials which leads to a family of non-uniform (with respect to the parameter) elliptic operators. We consider an associated optimal control problem in such a domain and we study the asymptotic analysis of the same. We briefly recall two-scale method and method of unfolding operators and obtain the limits using the method of unfolding.
In this talk, we discuss the pointwise control constrained Dirichlet boundary optimal control problem governed by an elliptic problem and its numerical approximation by finite element methods.
This course follows the lectures given by B. Dehman. We will begin by discussing the relationships between controllability, observability and exponential stability (Haraux theorem). Then we will study observability of wave equations on measurable observation sets. Finally, we will try to determine the best possible observation set, that is, we will study the optimal observability problem, for wave equations but also for parabolic equations for which results are strongly different.
The martingale problem for degenerate diffusions in $\mathcal{R}^{d}$ with continuous coefficients may not have unique solution measures. However, the solution set $A_{x}$ of solution measures for any initial condition $x \in \mathcal{R}^{d}$ is compact convex and non-empty. The Markov selection problem is to select a $P_{x} \in A_{x}$ for each $x$ such that the family $\{P_{x} \}$ satisfies the Chapman- Kolmogorov equations, thus forming a Markov process. The Krylov selection procedure is a well known resolution of this problem. In this talk, I shall sketch an alternative approach based on small noise limits. The talk will include an overview of diffusion theory. (Joint work with K. Suresh Kumar and Anugu Sumith Reddy).
Tuesday, 20 February 2024
These lectures are mainly based on different joint works with Mehdi Badra.
We consider systems of the form
$$z' = Az + Bf,\quad z(0)=z_0,$$
where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems
$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$
where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.
We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such feedback gains can be approximated by
feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that
$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.
We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that
$Z \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and
$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.
In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by
feedback gains for $(A_\varepsilon,B_\varepsilon)$.
In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.
We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.
In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.
In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.
Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this course we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.
This course follows the lectures given by B. Dehman. We will begin by discussing the relationships between controllability, observability and exponential stability (Haraux theorem). Then we will study observability of wave equations on measurable observation sets. Finally, we will try to determine the best possible observation set, that is, we will study the optimal observability problem, for wave equations but also for parabolic equations for which results are strongly different.
In this lecture, I will present Carleman Estimates for the Laplace operator and some of its applications. In particular, I will discuss the Cauchy problem for the Laplace operator, its uniqueness and stability properties, and then introduce general Carleman Estimates for the Laplace operator. As an application, I will give a quantification of the unique continuation property for the Laplace operator with respect to a potential, and discuss the link with the Landis conjecture, and applications to the optimal control problem for a semilinear elliptic PDE.
This course focuses on small-time local controllability (STLC) for affine control systems with scalar input x0 = f0(x) + u(t)f1(x). First, we will look at linear theory (Kalman rank condition), STLC proofs by linearization and by power series expansion, and their links with linear or Holderian cost estimates. Going further, we will introduce Lie brackets of vector fields and state the Lie algebra rank condition, which is a necessary condition for STLC (but not sufficient, when drift is active). Thanks to a new representation formula for the state, we will demonstrate STLC necessary conditions, including the most classical ones (Sussmann, Stefani). This will show the crucial role of interpolation inequalities (e.g. Gagliardo-Nirenberg) in the proof of STLC necessary conditions. Finally, using the emblematic example of the 1D Schroodinger equation with bilinear control, we will see how these results can be extended to PDEs.
Wednesday, 21 February 2024
These lectures are mainly based on different joint works with Mehdi Badra.
We consider systems of the form
$$z' = Az + Bf,\quad z(0)=z_0,$$
where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems
$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$
where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.
We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such feedback gains can be approximated by
feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that
$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.
We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that
$Z \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and
$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.
In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by
feedback gains for $(A_\varepsilon,B_\varepsilon)$.
In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.
We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.
In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.
In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.
Coupled systems appear everywhere in complex models and in some cases there are different time scales involved. The coupling and the scales make this kind of system very difficult to study from theoretical and computational viewpoints. One hopes that some particular properties of the system could be studied through simpler uncoupled systems. This is what the singular perturbation method (SPM) does concerning stability properties. The SPM approach has been introduced for ordinary differential equations and can also be applied for partial differential equations but in the latter case there are no general theorems and stability properties have to be obtained for each particular system. In this course we introduce the SPM for infinite-dimensional systems and obtain stability results. We will consider parabolic, hyperbolic and dispersive equations appearing in coupled systems in some cases also involving ordinary differential equations.
This course follows the lectures given by B. Dehman. We will begin by discussing the relationships between controllability, observability and exponential stability (Haraux theorem). Then we will study observability of wave equations on measurable observation sets. Finally, we will try to determine the best possible observation set, that is, we will study the optimal observability problem, for wave equations but also for parabolic equations for which results are strongly different.
In this lecture, I will present Carleman Estimates for the Laplace operator and some of its applications. In particular, I will discuss the Cauchy problem for the Laplace operator, its uniqueness and stability properties, and then introduce general Carleman Estimates for the Laplace operator. As an application, I will give a quantification of the unique continuation property for the Laplace operator with respect to a potential, and discuss the link with the Landis conjecture, and applications to the optimal control problem for a semilinear elliptic PDE.
This course focuses on small-time local controllability (STLC) for affine control systems with scalar input $x'=f_{0}(x)+u(t)f_{1}(x).$. First, we will look at linear theory (Kalman rank condition), STLC proofs by linearization and by power series expansion, and their links with linear or Holderian cost estimates. Going further, we will introduce Lie brackets of vector fields and state the Lie algebra rank condition, which is a necessary condition for STLC (but not sufficient, when drift is active). Thanks to a new representation formula for the state, we will demonstrate STLC necessary conditions, including the most classical ones (Sussmann, Stefani). This will show the crucial role of interpolation inequalities (e.g. Gagliardo-Nirenberg) in the proof of STLC necessary conditions. Finally, using the emblematic example of the 1D Schroodinger equation with bilinear control, we will see how these results can be extended to PDEs.
Thursday, 22 February 2024
These lectures are mainly based on different joint works with Mehdi Badra.
We consider systems of the form
$$z' = Az + Bf,\quad z(0)=z_0,$$
where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems
$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$
where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.
We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such feedback gains can be approximated by
feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that
$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.
We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that
$Z \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and
$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.
In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by
feedback gains for $(A_\varepsilon,B_\varepsilon)$.
In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.
We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.
In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.
In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.
This course follows the lectures given by B. Dehman. We will begin by discussing the relationships between controllability, observability and exponential stability (Haraux theorem). Then we will study observability of wave equations on measurable observation sets. Finally, we will try to determine the best possible observation set, that is, we will study the optimal observability problem, for wave equations but also for parabolic equations for which results are strongly different.
A control system is differentially flat if every trajectory can be expressed in terms of a function (called a flat output) and its derivatives. The flatness approach, introduced by Michel Fliess, Jean L ́evine, Philippe Martin and Pierre Rouchon in 1995 for ODEs, was next applied to PDEs by B ́eatrice Laroche, Philippe Martin and Pierre Rouchon in 2000 to derive the approximate controllability of the heat equation.
In this course, we will show how the flatness approach can be used to derive null or exact controllability results for many evolution equations with some boundary controls.
We first show how to obtain the null controllability of the 1D heat equation, or of the ND heat equation in cylinders, providing both the control input and the trajectory as series. We shall see how to compute easily a lot of derivatives of the flat output to compute the partial sums in the series. Next, we show how to apply the flatness approach to the linear 1D Schr ̈odinger equation and to the linear Korteweg-de Vries equation.
We proceed to the null controllability of the heat equation with variable coefficients, namely
\[ (a(x)u_x)_x+ b(x)u_x+c(x)u=\rho (x)u_t.\]
Here the coefficients a, b, c, ρ are merely assumed to be measurable, and we stress that Carleman estimates (if any) are not yet available. We mainly assume first that a ∈ L1 and 1/a ∈ L1 (weak degeneracy). Next, we pass to the case of a strong degeneracy at one point of the boundary. The flatness approach is combined with Pr ̈ufer method to obtain the asymptotics of the eigenvalues. Next, we are concerned with the determination of the reachable states of the heat equation, first for the boundary control, and next for the distributed control.
Finally, we address the issue of the exact controllability of a semilinear heat equation, and show how a Cauchy problem in the variable x can be solved by using scales of Banach spaces constituted of Gevrey functions. We obtain a local exact controllability in a space of holomorphic functions. The method is next extended to any anisotropic 1D equation of the form
\[ \partial _t^N y=\zeta _M \partial _x^M y +f(x,y,...,\partial _x^{M-1}y), \quad 1\le N<M,\]
that may be severely ill-posed. We obtain again some local exact controllability result in some space of holomorphic functions. The result can be applied to
-
the Korteweg-de Vries equation: $\partial _t y+\partial _x^3 y + \partial _xy+y\partial _xy=0$;
-
the ``good'' and ``bad'' Boussinesq equation: $\partial _t^2 y =\pm \partial_x^4 y +\partial _x ^2 y -\partial _x^2 (y^2)$;
-
the Ginzburg-Landau equation: $\partial _t y = e^{i\theta} \partial _x ^2 y + e^{i\varphi} |y|^2y$ where $\theta,\varphi\in\R$;
-
the Kuramoto-Sivashinsky equation: $\partial _ty +\partial _x ^4 y +\partial _x^2 y +y\partial _x y=0$.
In this talk, I will discuss a constructive proof for the null controllability of a linear boundary controlled 1D parabolic PDE with spatially-varying coefficients. In our approach, we first discretize the spatial derivatives in the parabolic PDE using the finite-difference approximation to obtain a linear ODE in time. We also discretize the initial state of the PDE to get an initial state for the ODE. We let the ODE evolve in response to the initial state, without any input, over a small time interval to reach an intermediate state. Then using the flatness approach we construct an input signal that transfers the ODE from the intermediate state to the zero state. We prove that, as the discretization step size converges to zero, this input signal converges to a limiting input signal and this limit signal together with the zero input-signal over the small time interval transfers the parabolic PDE from the given initial state to the origin.
Friday, 23 February 2024
These lectures are mainly based on different joint works with Mehdi Badra.
We consider systems of the form
$$z' = Az + Bf,\quad z(0)=z_0,$$
where A is the infinitesimal generator of an analytic semigroup on a Hilbert space Z, B is a control operator, f ∈ U is the control variable, and U is another Hilbert space. We consider approximate systems
$$z_\varepsilon' = A_\varepsilon z_\varepsilon + B_\varepsilon f,\quad z_\varepsilon(0)=z_{0,\varepsilon},$$
where $A_\varepsilon$ is the infinitesimal generator of an analytic semigroup on a Hilbert space $Z_\varepsilon$, $B_\varepsilon$ is a control operator. For numerical approximations by Finite Element Methods, $\varepsilon =h$ is the mesh size of the triangulation. For approximations by a penalty method, $\varepsilon $ is the penalty parameter.
We are interested in finding feedback gains $K\in{\mathcal L}(Z,U)$ such that $(e^{t(A+BK)})_{t\geq 0}$ is exponentially stable on $Z$. We would like to know if such feedback gains can be approximated by
feedback gains $K_\varepsilon\in{\mathcal L}(Z_\varepsilon,U)$, where $K_\varepsilon$ is chosen so that
$(e^{t(A_\varepsilon+B_\varepsilon K_\varepsilon)})_{t\geq 0}$ is exponentially stable on $Z_\varepsilon$.
We consider the case of nonconforming approximations, that is the case where $Z_\varepsilon \not \subset Z$. We assume that
$Z \subset H$, $Z_\varepsilon \subset H$, $H$ is a Hilbert space, and that there exist projectors $P\in{\mathcal L}(H)$ and
$P_\varepsilon\in{\mathcal L}(H)$ such that $P H = Z$ and $P_\varepsilon H = Z_\varepsilon$.
In Lecture 1, we give sufficient conditions on the triplets $(A,B,P)$ and $(A_\varepsilon,B_\varepsilon,P_\varepsilon)$ and some approximations assumptions with which we prove that Riccati based feedback gains for $(A,B)$ can be approximated by
feedback gains for $(A_\varepsilon,B_\varepsilon)$.
In Lecture 2, we show that the stabilization of the Oseen system (the linearized Navier-Stokes equations around an unstable stationary solution) by a boundary control, and its approximation by a F.E.M. fits into the functional setting introduced in Lecture 1.
We obtain new error estimates for the F.E. approximation of the stationary Oseen system with nonhomogeneous boundary conditions, in convex or non-convex polyhedral domains in ${\mathbb R}^3$.
In Lecture 3, we improve the convergence rates obtained in Lecture 1, by using Reduced Order Model based on spectral projections. We give some applications.
In Lecture 4, we study Fluid-Structure-Interaction systems in the case where the structure is a damped elastic beam or shell located at the boundary of the fluid domain. We recall some existence results of maximal in time strong solutions to these systems. We study the local stabilization of these systems around unstable stationary solutions. We show that the feedback gains can be determined by the method introduced in Lecture 3. Some hints will be given on the numerical approximation of a linearized model.
A control system is differentially flat if every trajectory can be expressed in terms of a function (called a flat output) and its derivatives. The flatness approach, introduced by Michel Fliess, Jean L ́evine, Philippe Martin and Pierre Rouchon in 1995 for ODEs, was next applied to PDEs by B ́eatrice Laroche, Philippe Martin and Pierre Rouchon in 2000 to derive the approximate controllability of the heat equation.
In this course, we will show how the flatness approach can be used to derive null or exact controllability results for many evolution equations with some boundary controls.
We first show how to obtain the null controllability of the 1D heat equation, or of the ND heat equation in cylinders, providing both the control input and the trajectory as series. We shall see how to compute easily a lot of derivatives of the flat output to compute the partial sums in the series. Next, we show how to apply the flatness approach to the linear 1D Schr ̈odinger equation and to the linear Korteweg-de Vries equation.
We proceed to the null controllability of the heat equation with variable coefficients, namely
\[ (a(x)u_x)_x+ b(x)u_x+c(x)u=\rho (x)u_t.\]
Here the coefficients a, b, c, ρ are merely assumed to be measurable, and we stress that Carleman estimates (if any) are not yet available. We mainly assume first that a ∈ L1 and 1/a ∈ L1 (weak degeneracy). Next, we pass to the case of a strong degeneracy at one point of the boundary. The flatness approach is combined with Pr ̈ufer method to obtain the asymptotics of the eigenvalues. Next, we are concerned with the determination of the reachable states of the heat equation, first for the boundary control, and next for the distributed control.
Finally, we address the issue of the exact controllability of a semilinear heat equation, and show how a Cauchy problem in the variable x can be solved by using scales of Banach spaces constituted of Gevrey functions. We obtain a local exact controllability in a space of holomorphic functions. The method is next extended to any anisotropic 1D equation of the form
\[ \partial _t^N y=\zeta _M \partial _x^M y +f(x,y,...,\partial _x^{M-1}y), \quad 1\le N<M,\]
that may be severely ill-posed. We obtain again some local exact controllability result in some space of holomorphic functions. The result can be applied to
-
the Korteweg-de Vries equation: $\partial _t y+\partial _x^3 y + \partial _xy+y\partial _xy=0$;
-
the ``good'' and ``bad'' Boussinesq equation: $\partial _t^2 y =\pm \partial_x^4 y +\partial _x ^2 y -\partial _x^2 (y^2)$;
-
the Ginzburg-Landau equation: $\partial _t y = e^{i\theta} \partial _x ^2 y + e^{i\varphi} |y|^2y$ where $\theta,\varphi\in\R$;
-
the Kuramoto-Sivashinsky equation: $\partial _ty +\partial _x ^4 y +\partial _x^2 y +y\partial _x y=0$.