Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback
Serge Nicaise Cristina Pignotti
Discrete & Continuous Dynamical Systems - S 2016, 9(3): 791-813 doi: 10.3934/dcdss.2016029
We study the stabilization problem for the wave equation with localized Kelvin--Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the result of [19] where, in a more general setting, the case of distributed structural damping is considered.
keywords: delay feedbacks Wave equation stabilization.
Asymptotic analysis of a simple model of fluid-structure interaction
Serge Nicaise Cristina Pignotti
Networks & Heterogeneous Media 2008, 3(4): 787-813 doi: 10.3934/nhm.2008.3.787
This paper is devoted to the asymptotic analysis of simple models of fluid-structure interaction, namely a system between the heat and wave equations coupled via some transmission conditions at the interface. The heat part induces the dissipation of the full system. Here we are interested in the behavior of the model when the thickness of the heat part and/or the heat diffusion coefficient go to zero or to infinity. The limit problem is a wave equation with a boundary condition at the interface, this boundary condition being different according to the limit of the above mentioned parameters. It turns out that some limit problems are dissipative but some of them are non dissipative or their behavior is unknown.
keywords: Fluid-structure system asympotic behavior
Semiconcavity for optimal control problems with exit time
Piermarco Cannarsa Cristina Pignotti Carlo Sinestrari
Discrete & Continuous Dynamical Systems - A 2000, 6(4): 975-997 doi: 10.3934/dcds.2000.6.975
In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of general form

$\dot y(t)=f(y(t),u(t))$,  $y(t)\in\mathbb R^n$, $u(t)\in U\subset \mathbb R^m$.

However, suitable assumptions are needed relating $f$ with the running and exit costs.
The semiconcavity property is then applied to obtain necessary optimality conditions, through the formulation of a suitable version of the Maximum Principle, and to study the singular set of the value function.

keywords: exit time problems Optimal control problems dynamic programming semiconcavity. optimality conditions
On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$
Giorgio Fusco Francesco Leonetti Cristina Pignotti
Discrete & Continuous Dynamical Systems - A 2017, 37(2): 725-742 doi: 10.3934/dcds.2017030

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $Ω\subset\mathbb{R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension $n=2$ an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.

keywords: Allen-Cahn equation symmetric solutions minimization pointwise estimates asymptotic behavior
Exponential stability of the wave equation with boundary time-varying delay
Serge Nicaise Cristina Pignotti Julie Valein
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 693-722 doi: 10.3934/dcdss.2011.4.693
We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain $\Omega\subset\RR^n.$ Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version of the model.
keywords: stabilization. Wave equation delay feedbacks

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