arXiv subjects

P. Cannarsa

Publications and source records attributed to P. Cannarsa.

3 recordsLinked to original sources

arXiv2025

Carleman estimate with piecewise weight and applications to inverse problems for first-order transport equations

We consider a first-order transport equation $\ppp_tu(x,t) + (H(x)\cdot\nabla u(x,t)) + p(x)u(x,t) = F(x,t)$ for $x \in \OOO \subset \R^d$, where $\OOO$ is a bounded domain and $0<t<T$. We prove a Carleman estimate for more generous condition on the principal coefficients $H(x)$ than in the existing works. The key is the construction of a piecewise smooth weight function in $x$ according to a suitable decomposition of $\OOO$. Our assumptions on $H$ generalize the conditions in the existing articles, and require that a directed graph created by the corresponding stream field has no closed loops. Then, we apply our Carleman estimate to two inverse problems of determinination of an initial value and one of a spatial factor of a source term, so that we establish Lipschitz stability estimates for the inverse problems.

math.AP↗

arXiv2011

Indirect stabilization of weakly coupled systems with hybrid boundary conditions

We investigate stability properties of indirectly damped systems of evolution equations in Hilbert spaces, under new compatibility assumptions. We prove polynomial decay for the energy of solutions and optimize our results by interpolation techniques, obtaining a full range of power-like decay rates. In particular, we give explicit estimates with respect to the initial data. We discuss several applications to hyperbolic systems with {\em hybrid} boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.

math.OC↗

arXiv2011

Null controllability of Grushin-type operators in dimension two

We study the null controllability of the parabolic equation associated with the Grushin-type operator $A=\partial_x^2+|x|^{2γ}\partial_y^2\,, (γ>0),$ in the rectangle $Ω=(-1,1)\times(0,1)$, under an additive control supported in the strip $ω=(a,b)\times(0,1)\,, (0 1$. In the transition regime $γ=1$, we show that both behaviors live together: a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular geometric configuration, null controllability is equivalent to the observability of the Fourier components of the solution of the adjoint system uniformly with respect to the frequency.

math.AP↗