Inverse observability inequalities for integrodifferential equations in square domains

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality

Reachability problems for a wave-wave system with a memory term

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theor

Control problems for weakly coupled systems with memory

We investigate control problems for wave-Petrovsky coupled systems in the presence of memory terms. By writing the solutions as Fourier series, we are able to prove Ingham type estimates, and hence reachability results. Our findings have applications in viscoelasticity theory and linear acoustic theory

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

We consider an anisotropic hyperbolic equation with memory term: ∂t2u(x,t)=∑i,j=1n∂i(aij(x)∂ju)+∫0t∑|α|≤2bα(x,t,η)∂xαu(x,η)dη+R(x,t)f(x) for $x \in \Omega$ and $t\in (0, T)$ , which is a simplified model equation for viscoelasticity. The main result is a both-sided Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor $f(x)$ of the force term $R(x, t)\,f(x)$ . The proof is based on a Carleman estimate and due to the anisotropy, the existing transformation technique does not work and we introduce a new transformation of u in order to treat the integral terms

Partial observability of a wave-Petrovsky system with memory

Our goal is to show partial observability results for coupled systems with memory terms. To this end, by means of non-harmonic analysis techniques we prove Theorem \ref{th:obs_par1} and Theorem \ref{th:obs_par2} below

Weak solutions for time-fractional evolution equations in Hilbert spaces

We introduce a notion of weak solution for abstract fractional differential equations, motivated by the definition of Caputo derivative. We prove existence results for weak and strong solutions. We also give two examples as application of our results: time-fractional wave equations and time-fractional Petrovsky systems