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Control Theory and Optimization  | enregistrements trouvés : 30

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Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on which one can act by means of a control, like for example a car, a robot, a space shuttle, a chemical reaction or in more general a process that one aims at steering to some desired target state.
Emmanuel Trélat will overview the range of applications of that theory through several examples, sometimes funny, but also historical. He will show you that the study of simple cases of our everyday life, far from insignificant, allow to approach problems like the orbit transfer or interplanetary mission design.
control theory - optimal control - stabilization - optimization - aerospace - Lagrange points - dynamical systems - mission design
Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on ...

49J15 ; 93B40 ; 93B27 ; 93B50 ; 65H20 ; 90C31 ; 37N05 ; 37N35

In this talk we present a inequality obtained with Jérôme Le Rousseau, for sum of eigenfunctions for bi-Laplace operator with clamped boundary condition. These boundary conditions do not allow to reduce the problem for a Laplacian with adapted boundary condition. The proof follow the strategy used for Laplacian, namely we consider a problem with an extra variable and we prove Carleman estimates for this new problem. The main difficulty is to obtain a Carleman estimate up to the boundary. In this talk we present a inequality obtained with Jérôme Le Rousseau, for sum of eigenfunctions for bi-Laplace operator with clamped boundary condition. These boundary conditions do not allow to reduce the problem for a Laplacian with adapted boundary condition. The proof follow the strategy used for Laplacian, namely we consider a problem with an extra variable and we prove Carleman estimates for this new problem. The main difficulty is to ...

35B45 ; 35S15 ; 93B05 ; 93B07

Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality conditions - Hamiltonian system - symplectic flow
Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality ...

93C05 ; 93C15 ; 49K15 ; 34H05

We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for some $y$ such that $(x,y) \in K \}$.
by a hierarchy of inner sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\subset R_f$.
or outer sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\supset D_f$.
for some polynomiales $(J_k)$ of increasing degree :
- for computing convex polynomial underestimators of a given polynomial $f$ on a box $B \subset R^n$.
We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for ...

44A60 ; 90C22

We discuss some new results for the Cheeger constant in dimension two, including:
- a polygonal version of Faber-Krahn inequality;
- a reverse isoperimetric inequality for convex bodies;
- a Mahler-type inequality in the axisymmetric setting;
- asymptotic behaviour of optimal partition problems.
Based on some recent joint works with D.Bucur,
and for the last part also with B.Velichkov and G.Verzini.

49Q10 ; 52B60 ; 35P15 ; 52A40 ; 52A10 ; 35A15

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and ...

49K20 ; 49N70 ; 35K40 ; 35K55 ; 35Q84 ; 65K10 ; 65M06 ; 65M12 ; 91A23 ; 91A15

In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the of the continuous time solution of the BSDE. The nodes of the discrete time processes can be expressed as conditional expectations. As we shall demonstrate, the choice of discrete time process is very important, as its properties will impact the performance of the overall numerical scheme. In the second stage, the conditional expectation is approximated in functional form using least squares regression on synthetically generated data Monte Carlo simulations drawn from a suitable probability distribution. A key feature of the regression step is that the explanatory variables are built on a user chosen finite dimensional linear space of functions, which the user specifies by setting basis functions. The choice of basis functions is made on the hypothesis that it contains the solution, so regularity and boundedness assumptions are used in its construction. The impact of the choice of the basis functions is exposed in error estimates.
In addition to the choice of discrete time approximation and the basis functions, the Markovian structure of the problem gives significant additional freedom with regards to the Monte Carlo simulations. We demonstrate how to use this additional freedom to develop generic stratified sampling approaches that are independent of the underlying transition density function. Moreover, we demonstrate how to leverage the stratification method to develop a HPC algorithm for implementation on GPUs.
Thanks to the Feynmann-Kac relation between the the solution of a BSDE and its associated semilinear PDE, the approximation of the BSDE can be directly used to approximate the solution of the PDE. Moreover, the smoothness properties of the PDE play a crucial role in the selection of the hypothesis space of regressions functions, so this relationship is vitally important for the numerical scheme.
We conclude with some draw backs of the regression approach, notably the curse of dimensionality.
In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the ...

65C05 ; 65C30 ; 93E24 ; 60H35 ; 60H10

The Bossel-Daners is a Faber-Krahn type inequality for the first Laplacian eigenvalue with Robin boundary conditions. We prove a stability result for such inequality.

49Q10 ; 49K20 ; 35P15 ; 35J05 ; 47J30

In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
\[
\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},
\]
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue measure.
We prove that any minimizer $_{opt}$ has a regular part of the topological boundary which is relatively open and
$C^{\infty}$ and that the singular part has Hausdorff dimension smaller than $d-d^*$, where $d^*\geq 5$ is the minimal
dimension allowing the existence of minimal conic solutions to the blow-up problem.

We mainly use techniques from the theory of free boundary problems, which have to be properly extended to the case of
vector-valued functions: nondegeneracy property, Weiss-like monotonicity formulas with area term; finally through the
properties of non tangentially accessible domains we shall be in a position to exploit the ''viscosity'' approach recently proposed by De Silva.

This is a joint work with Dario Mazzoleni and Bozhidar Velichkov.
In this talk we deal with the regularity of optimal sets for a shape optimization problem involving a combination
of eigenvalues, under a fixed volume constraints. As a model problem, consider
\[
\min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\},
\]
where $\langle_i(\cdot)$ denotes the eigenvalues of the Dirichlet Laplacian and $|\cdot|$ the $d$-dimensional Lebesgue m...

49Q10 ; 35R35 ; 47A75 ; 49R05

This will be an introduction to sub-Riemannian geometry from the point of view of control theory. We will define sub-Riemannian structures and prove the Chow Theorem. We will describe normal and abnormal geodesics and discuss the completeness of the Carnot-Carathéodory distance (Hopf-Rinow Theorem). Several examples will be given (Heisenberg group, Martinet distribution, Grusin plane).

53C17 ; 49Jxx

Multi angle  Geometric control and dynamics
Rifford, Ludovic (Auteur de la Conférence) | CIRM (Editeur )

The geometric control theory is concerned with the study of control systems in finite dimension, that is dynamical systems on which one can act by a control. After a brief introduction to controllability properties of control systems, we will see how basic techniques from control theory can be used to obtain for example generic properties in Hamiltonians dynamics.

34H05 ; 93C15 ; 93B27

This talk concerns the concept of dissipativity in the sense of Willems for nonautonomous linear-quadratic (LQ) control systems. A nonautonomous system of Hamiltonian ODEs can be associated with such an LQ system, and the analysis of the corresponding symplectic dynamics provides valuable information on the dissipativity properties. The presence of exponential dichotomy, the occurrence of weak disconjugacy, and the existence of nonnegative solutions of the Riccati equation provided by the Hamiltonian system are closely related to the presence of (normal or strict) dissipativity and to the definition of the (normal or strong) storage functions.
This is a joint work with: Roberta Fabbri, Russell Johnson, Sylvia Novo and Rafael Obaya.
This talk concerns the concept of dissipativity in the sense of Willems for nonautonomous linear-quadratic (LQ) control systems. A nonautonomous system of Hamiltonian ODEs can be associated with such an LQ system, and the analysis of the corresponding symplectic dynamics provides valuable information on the dissipativity properties. The presence of exponential dichotomy, the occurrence of weak disconjugacy, and the existence of nonnegative ...

37B55 ; 49N10 ; 93C15

We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We show the strong convergence of the approximation and then discussed several examples for $N = 1$ and $N = 2$. The reconstruction of both the state and the source term is also discussed, as well as the boundary case. The parabolic case - more delicate as it requires the use of appropriate weights - will be also addressed. Joint works with Nicolae Cîndea and Diego Araujo de Souza. We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic ...

35L10 ; 65M12 ; 93B40

Multi angle  Isoperimetry with density
Morgan, Frank (Auteur de la Conférence) | CIRM (Editeur )

In 2015 Chambers proved the Log-convex Density Conjecture, which says that for a radial density f on $R^n$, spheres about the origin are isoperimetric if and only if log f is convex (the stability condition). We discuss recent progress and open questions for other densities, unequal perimeter and volume densities, and other metrics.

49Q20 ; 53C17 ; 49N60

The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively the evolution of the distribution of the agents' states and the evolution of the value function. Since it comes from a control problem, this PDE system differs in general from the one arising in mean field games.
Recently, this kind of model has been applied to crowd dynamics. More precisely, in this talk we will be interested in modeling congestion effects: the agents move but try to avoid very crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density of agents is large. The cost may depend on the density in a non-local or in a local way. We will present one class of models for each case and study the associated PDE systems. The first one has classical solutions whereas the second one has weak solutions. Numerical results based on the Newton algorithm and the Augmented Lagrangian method will be presented.
This is joint work with Yves Achdou.
The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilto...

35K40 ; 35K55 ; 35K65 ; 35D30 ; 49N70 ; 49K20 ; 65K10 ; 65M06

We consider the problem of lagrangian controllability for two models of fluids. The lagrangian controllability consists in the possibility of prescribing the motion of a set of particle from one place to another in a given time. The two models under view are the Euler equation for incompressible inviscid fluids, and the quasistatic Stokes equation for incompressible viscous fluids. These results were obtained in collaboration with Thierry Horsin (Conservatoire National des Arts et Métiers, Paris) We consider the problem of lagrangian controllability for two models of fluids. The lagrangian controllability consists in the possibility of prescribing the motion of a set of particle from one place to another in a given time. The two models under view are the Euler equation for incompressible inviscid fluids, and the quasistatic Stokes equation for incompressible viscous fluids. These results were obtained in collaboration with Thierry Horsin ...

35Q93 ; 35Q31 ; 76D55 ; 93B05

We introduce a new function which measures the torsional instability of a partially hinged rectangular plate. By exploiting it, we compare the torsional performances of different plates reinforced with stiffening trusses. This naturally leads to a shape optimization problem which can be set up through a minimaxmax procedure.

35Q74 ; 49Q10 ; 74K20

The valuation of American options (a widespread type of financial contract) requires the numerical solution of an optimal stopping problem. Numerical methods for such problems have been widely investigated. Monte-Carlo methods are based on the implementation of dynamic programming principles coupled with regression techniques. In lower dimension, one can choose to tackle the related free boundary PDE with deterministic schemes.
Pricing of American options will therefore be inevitably heavier than the one of European options, which only requires the computation of a (linear) expectation. The calibration (fitting) of a stochastic model to market quotes for American options is therefore an a priori demanding task. Yet, often this cannot be avoided: on exchange markets one is typically provided only with market quotes for American options on single stocks (as opposed to large stock indexes - e.g. S&P500 - for which large amounts of liquid European options are typically available).
In this talk, we show how one can derive (approximate, but accurate enough) explicit formulas - therefore replacing other numerical methods, at least in a low-dimensional case - based on asymptotic calculus for diffusions.
More precisely: based on a suitable representation of the PDE free boundary, we derive an approximation of this boundary close to final time that refines the expansions known so far in the literature. Via the early premium formula, this allows to derive semi-closed expressions for the price of the American put/call. The final product is a calibration recipe of a Dupire's local volatility to American option data.
Based on joint work with Pierre Henry-Labordère.
The valuation of American options (a widespread type of financial contract) requires the numerical solution of an optimal stopping problem. Numerical methods for such problems have been widely investigated. Monte-Carlo methods are based on the implementation of dynamic programming principles coupled with regression techniques. In lower dimension, one can choose to tackle the related free boundary PDE with deterministic schemes.
Pricing of ...

93E20 ; 91G60

We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get rapid stabilization for Korteweg-de Vries equations and how to stabilize in finite time $1-D$ parabolic linear equations by means of periodic time-varying feedback laws. We start by presenting some results on the stabilization, rapid or in finite time, of control systems modeled by means of ordinary differential equations. We study the interest and the limitation of the damping method for the stabilization of control systems. We then describe methods to transform a given linear control system into new ones for which the rapid stabilization is easy to get. As an application of these methods we show how to get ...

35B35 ; 35Q53 ; 93C10 ; 93C20 ; 35K05 ; 93B05 ; 93B17 ; 93B52

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