P. Cannarsa

• Laurea in Mathematics
• Professor at University of Rome Tor Vergata

We consider a nonlinear parabolic equation with a nonlocal term which preserves the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm of the so...

We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and st...

We study a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being partly captured by the other layer, and the effect of all non-...

A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $\lambda$ tends to zero. In the literature, this problem has been addressed under various controllability or ergodicity conditions ensuring that the rescaled...

The aim of this work is to provide a systemic study and generalization of the celebrated weak KAM theory and Aubry-Mather theory in sub-Riemannian setting, or equivalently, on a Carnot-Caratheodory metric space. In this framework we consider an optimal control problem with state equation of sub-Riemannian type, namely, admissible trajectories are s...

For solution $u(x,t)$ to degenearte parabolic equations in a bounded domain $\Omega$ with homogenous boundary condition, we consider backward problems in time: determine $u(\cdot,t_0)$ in $\Omega$ by $u(\cdot,T)$, where $t$ is the time variable and $0\le t_0 < T$. Our main results are conditional stability under boundedness assumptions on $u(\cdot,...

We study the exact controllability of the evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, mea...

We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiat...

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the L^2-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in H^1. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a...

In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying H\"ormander's bracket generating condition, where the target is a nonempty closed set. We show that, in dimension 2, the minimum time function is locally Lipschitz continuous while, in dimension 3, it is Lipschi...

The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or ergodic) Hamilton-Jacobi equation. The goal of this paper is to addres...

In a separable Hilbert space X, we study the controlled evolution equation u (t) + Au(t) + p(t)Bu(t) = 0, where A ≥ −σI (σ ≥ 0) is a self-adjoint linear operator, B is a bounded linear operator on X, and p ∈ L 2 loc (0, +∞) is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllabl...

In this poster I describe a result of bilinear controllability of a parabolic equation on a compact network structure. Following the procedure of [F. Alabau-Boussouira, P. Cannarsa, C. Urbani “Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, NoDEA, vol. 29(38) (2022)], we were able to prove a c...

In the paper [P. Cannarsa, C. Mendico, Asymptotic analysis for Hamilton-Jacobi- Bellman equations on Euclidean space, (2021) Arxiv], we proved the existence of the limit as the time horizon goes to infinity of the averaged value function of an optimal control problem. For the classical Tonelli case such a limit is called the critical constant of th...

Consider a locally Lipschitz function u on the closure of a possibly unbounded open subset Ω of Rn with nonempty boundary. Suppose u is (locally) semiconcave on Ω¯ with a fractional semiconcavity modulus. Is it possible to extend u in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case...

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euc...

Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life...

We consider the linear degenerate wave equation, on the interval $(0, 1)$ $$ w_{tt} - (x^\alpha w_x)_x = p(t) \mu (x) w, $$ with bilinear control $p$ and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the ground state. We prove that, generically with respec...

Partial differential equation on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrodinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life s...

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solut...

We consider the inverse problem of identification of degenerate diffusion coefficient of the form x α a(x) in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient a and the power α by knowing interior data at some tim...

Despite the importance of control systems governed by bilinear controls for the description of phenomena that could not be modeled by additive controls, such as for instance the vibration of a beam composed by smart materials, or the process of increasing the speed of a chemical reaction by adding catalysts, the action of multiplicative controls is...

If $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in...

We consider the inverse problem of identification of degenerate diffusion coefficient of the form $x^\alpha a(x)$ in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient $a$ and the power $\alpha$ by knowing an interi...

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the va...

This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton–Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.

We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$\Omega = (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $\omega=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $\omega = (a,b) \times \omega_y \times \mathbb{T}$, with $-1<a<b<1$. We prove...

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euc...

We study the stabilizability of a class of abstract parabolic equations of the form $$\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\ge 0 \end{aligned}$$where the control \(p(\cdot )\) is a scalar function, A is a self-adjoint operator on a Hilbert space X that satisfies \(A\ge -\sigma I\), with \(\sigma >0\), and B is a bounded linear operator o...

The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for s...

This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton-Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.

The goal of this paper is to study the long-time average behavior of the value function of optimal control problems with dynamics associated with a family of vector fields satisfying the Lie Algebra rank condition (H\"ormander condition). For such problems the Hamiltonian fails to be coervice in the momentum variable. Nevertheless, by a dynamical a...

Consider a locally Lipschitz function $u$ on the closure of a possibly unbounded open subset $\Omega$ of $\mathbb{R}^n$ with $C^{1,1}$ boundary. Suppose $u$ is semiconcave on $\overline \Omega$ with a fractional semiconcavity modulus. Is it possible to extend $u$ in a neighborhood of any boundary point retaining the same semiconcavity modulus? We s...

The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for s...

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in the paper (P. Cannarsa, W. Cheng, K. Wang, J. Yan, 2019 [17]) in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using...

The aim of this paper is to study the long-time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus \({\mathbb {T}}^n\) in Cardaliaguet (Dyn Games Appl 3:473–488, 2013), where solutions are shown to converge to the solution of a certain ergodic mean field games system on \({\ma...

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon $T$ goes to infinity. For this purpose, we analyze first Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a sol...

A closed set $K$ of a Hilbert space $H$ is said to be invariant under the evolution equation $$ X'(t)=AX(t)+f\big(t,X(t)\big)\qquad ( t>0) $$ whenever all solutions starting from a point of $K$, at any time $t_0\geqslant 0$, remain in $K$ as long as they exist. For a self-adjoint strictly dissipative operator $A$, perturbed by a (possibly unbounde...

In this paper, we study two Energy Balance Models with Memory arising in climate dynamics, which consist in a 1D degenerate nonlinear parabolic equation involving a memory term, and possibly a set-valued reaction term (of Sellers type and of Budyko type, in the usual terminology). We provide existence and regularity results, and obtain uniqueness a...

This book presents important recent applied mathematics research on environmental problems and impacts due to climate change. Although there are inherent difficulties in addressing phenomena that are part of such a complex system, exploration of the subject using mathematical modelling is especially suited to tackling poorly understood issues in th...

If U : [0, +∞[×M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂ t U + H(x, ∂ x U) = 0, where M is a not necessarily compact manifold, and H is a Tonelli Hamiltonian, we prove the set Σ(U), of points where U is not differen-tiable, is locally contractible. Moreover, we study the homotopy type of Σ(U). We als...

In a separable Hilbert space X, we study the linear evolution equation u ′ (t) + Au(t) + p(t)Bu(t) = 0, where A is an accretive self-adjoint linear operator, B is a bounded linear operator on X, and p ∈ L 2 loc (0, +∞) is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground...

The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the form \begin{equation*} u_t(t,x)+(x^{\alpha}u_x(t,x))_x+p(t)x^{2-\alpha}u(t,x)=0,\qquad t\geq0,x\in(0,1) \end{equation*} via bilinear control $p\in L_{loc}^2(0,+\infty)$. More precisely, we provide a control f...

We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(\cdot)$ is the control function. The proof is based on a linearization argu...

We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form u ′ (t) + Au(t) + p(t)Bu(t) = 0, t ≥ 0 where the operator −A is a self-adjoint accretive operator on a Hilbert space and p(·) is the control function. The proof is based on a linearization argument. We prove that the linearized system...

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the...

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \cite{CCWY2018} in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an ap...

We consider the transport equation \(\partial _t u(x,t) + H(t)\cdot \nabla u(x,t) = 0\) in \(\varOmega \times (0,T),\) where \(T>0\) and \(\varOmega \subset \mathbb R^d \) is a bounded domain with smooth boundary \(\partial \varOmega \). First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. T...

We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the associated minimizers.

The aim of this paper is to study first order Mean field games subject to a linear controlled dynamics on $\mathbb{R}^{d}$. For this kind of problems, we define Nash equilibria (called Mean Field Games equilibria) as Borel probability measures on the space of admissible trajectories and we prove the existence and uniqueness of such equilibria. More...

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the...

We consider the transport equation in where is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function or a real-valued function by initial values and data on a subboundary of . Our results are conditional stability of Hölder type in a subdomain D provided that the outward normal component of is posi...

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristic semiflows associated with the Hamilton-Jacobi equations, and build the relation between the ω-limit sets of the semiflows and the projected Aubry sets.

Regularity properties are investigated for the value function of the Bolza optimal control problem with affine dynamic and end-point constraints. In the absence of singular geodesics, we prove the local semiconcavity of the sub-Riemannian distance from a compact set Γ⊂Rn. Such a regularity result was obtained by the second author and L. Rifford in...

We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function $H(x)$ or a real-valued function $p(x)$ by initial values and data on a subboundary of $\OOO$. Our...

For α ϵ (0,2) we study the null controllability of the parabolic operator Pu = u t (|x| α u x ) x (1 < x < 1), which degenerates at the interior point x = 0 for locally distributed controls acting only on one side of the origin (that is, on some interval (a, b) with 0 < a < b < 1). Our main results guarantee that P is null controllable if and only...

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the va...

In this paper, we study deterministic mean field games for agents who operate in a bounded domain. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of the solution to the associated minimization problem is no longer guarante...

We prove a result of bilinear controllability for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\in [0,T] \end{equation*} where the operator $-A$ is the infinitesimal generator of an analytic semigroup of bounded linear operators on a Hilbert space and $p(\cdot)$ is the control function. The p...

We derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. In the literature, this classical problem is widely investigated. The novelty of our result lies in the fact that the presence of state constraints enters the Euler-Lagrange equations as a local feedback, whi...

The aim of this paper is to study the long time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus ${\mathbb T}^n$ in [P. Cardaliaguet, {\it Long time average of first order mean field games and weak KAM theory}, Dyn. Games Appl. 3 (2013), 473-488], where solutions are shown t...

We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a bounded domain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition on $H$ which gu...

For $\alpha\in (0,2)$ we study the null controllability of the parabolic operator $$Pu= u_t - (\vert x \vert ^\alpha u_x)_x\qquad (1<x<1),$$ which degenerates at the interior point $x=0$, for locally distributed controls acting only one side of the origin (that is, on some interval $(a,b)$ with $0<a<b<1$). Our main results guarantees that $P$ is nu...

We derive necessary optimality conditions for minimizers of regular functionals in the calculus of variations under smooth state constraints. In the literature, this classical problem is widely investigated. The novelty of our result lies in the fact that the presence of state constraints enters the Euler-Lagrange equations as a local feedback, whi...

In this paper, we study two Energy Balance Models with Memory arising in climatology, which consist in a 1D degenerate nonlinear parabolic equation involving a memory term, and possibly a set-valued reaction term (of Sellers type and of Budyko type, in the usual terminology). We provide existence and regularity results, and obtain uniqueness and st...

This is a survey paper for the recent results on and beyond propagation of singularities of viscosity solutions. We also collect some open problems in this topic.

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristic semiflows associated with the Hamilton-Jacobi equations, and build the relation between the $\omega$-limit sets of the semiflows and the projected Aubry sets.

We study the local controllability properties of generic 2-D and 3-D bio-mimetic swimmers employing the change of their geometric shape to propel themselves in an incompressible fluid described by the Navier–Stokes equations. It is assumed that swimmers' bodies consist of finitely many parts, identified with the fluid they occupy, that are subseque...

We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the associated minimizers.

We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $\Omega$ of $\R^n$ assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in $\Omega$. Then,...

We study the nonhomogeneous Dirichlet problem for first order Hamilton-Jacobi equations associated with Tonelli Hamiltonians on a bounded domain Ω of R n assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in Ω. Then, we analyse the si...

On a bounded domain $\Omega$ in euclidean space $\mathbb{R}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies H\"ormander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.

We consider the typical one-dimensional strongly degenerate parabolic operator $Pu= u_t - (x^\alpha u_x)_x$ with $0<x<\ell$ and $\alpha\in(0,2)$, controlled either by a boundary control acting at $x=\ell$, or by a locally distributed control. Our main goal is to study the dependence of the so-called controllability cost needed to drive an initial c...

Partial and full sensitivity relations are obtained for nonauto-nomous optimal control problems with infinite horizon subject to state constraints, assuming the associated value function to be locally Lipschitz in the state. Sufficient structural conditions are given to ensure such a Lipschitz regularity in presence of a positive discount factor, a...

In this paper, we study deterministic mean field games for agents who operate in a bounded domain. In this case, the existence and uniqueness of Nash equilibria cannot be deduced as for unrestricted state space because, for a large set of initial conditions, the uniqueness of the solution to the associated minimization problem is no longer guarante...

We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state $ u^*\in H_0^1 (0,1)$, with as man...

For autonomous Tonelli systems on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}, we develop an intrinsic proof of the existence of gen...

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell, based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behaviour: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic on...

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to H\"ormander's bracket generating condition. Due to the presence of characteristic boundary points, si...

We investigate the value function V:R+×Rn→R+∪(+∞) of the infinite horizon problem in optimal control for a general-not necessarily discounted-running cost and provide sufficient conditions for its lower semicontinuity, continuity, and local Lipschitz regularity. Then we use the continuity of V(t,(dot operator)) to prove a relaxation theorem and to...

We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domainΩ=(−1,1)×T×T taking as observation regions slices of the form ω=(a,b)×T×T or tubes ω=(a,b)×ωy×T, with −1<a<b<1. We prove that observability fails for an arbitrary time T>0 but both observability and Lipschitz stability hold true after a po...

In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we a...

où est la constante critique de Mañé. Nous désignons par l'ensemble des points où u n'est pas différentiable, et par l'ensemble d'Aubry de u. Nous introduisons aussi l'ensemble des points de coupure de u comme étant l'ensemble des points où aucune courbe caractéristique en temps négatif de u aboutissant en x ne peut être étendue au-delà de x en une...

We consider a system of two inhomogeneous wave equations coupled in cascade. The source terms are of the form σ 1(t)f(x), and σ 2(t)g(x), where the σ i ’s are known functions whereas the sources f and g are unknown and have to be reconstructed. We investigate the reconstruction of these two space-dependent sources from a single boundary measurement...

For certain age-structured population models, the cone of positive functions is preserved when the dynamics is perturbed by white noise. Solutions can be forced to assume negative values, even when initial conditions are strictly positive. Necessary and sufficient conditions are expressed under which the solutions are nonnegative.

Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics. This manuscript has been conceived to introduce the reader to global Carleman es...

We consider the one-dimensional degenerate parabolic equation $$ u_t - (x^\alpha u_x)_x =0 \qquad x\in(0,1),\ t \in (0,T) ,$$ controlled by a boundary force acting at the degeneracy point $x=0$. First we study the reachable targets at some given time $T$ using $H^1$ controls, extending the moment method developed by Fattorini and Russell to this cl...

We study the local controllability properties of a 2-D bio-mimetic swimmer employing the change of its geometric shape to move itself in an incompressible fluid governed by Navier-Stokes equations. It is assumed that the swimmer's body consists of finitely many parts, identified with the fluid they occupy, that are subsequently linked by the rotati...

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $\mu_a>0$. We establish observability inequalities for weakly (when $\mu_a \in [0,1[$) as well as strongly (when $\mu_a \in [1,2[$) degenerate equations. We also prove a negative...

We interpret the close link between the critical points of Mather's barrier functions and minimal homoclinic orbits with respect to the Aubry sets on $\mathbb{T}^n$. We also prove a critical point theorem for barrier functions, and the existence of such homoclinic orbits on $\mathbb{T}^2$ as an application.

We study quantitative estimates of compactness in W 1,1 loc for the map S t , t > 0 that associates to every given initial data u 0 ∈ Lip(R N) the corresponding solution S t u 0 of a Hamilton-Jacobi equation u t + H x, ∇ x u = 0 , t ≥ 0, x ∈ R N , with a convex and coercive Hamiltonian H = H(x, p). We provide upper and lower bounds of order 1/ε N o...

The concept of measure of a set originates from the classical notion of volume of an interval in \(\mathbb R^N\). Starting from such an intuitive idea, by a covering process one can assign to any set a nonnegative number which “quantifies its extent”. Such an association leads to the introduction of a set function called exterior measure, which is...