A relaxation result for state constrained inclusions in infinite dimension
Helene Frankowska Elsa M. Marchini Marco Mazzola
In this paper we consider a state constrained differential inclusion $\dot x\in \mathbb A x+ F(t,x)$, with $\mathbb A$ generator of a strongly continuous semigroup in an infinite dimensional separable Banach space. Under an ``inward pointing condition'' we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion $\dot x\in \mathbb A x+ \overline{\textrm{co}}F(t,x)$. Some applications to control problems involving PDEs are given.
keywords: relaxation state constraints mild solution Differential inclusion inward pointing condition.
On Bolza optimal control problems with constraints
Piermarco Cannarsa Hélène Frankowska Elsa M. Marchini
We provide sufficient conditions for the existence and Lipschitz continuity of solutions to the constrained Bolza optimal control problem

$\text{minimize}\quad \int_0^T L(x(t),u(t))\dt + l(x(T))$

over all trajectory / control pairs $(x,u)$, subject to the state equation

x'(t)=$f(x(t),u(t)) $ for a.e. $t\in [0,T]$
$u(t)\in U $ for a.e. $t\in [0,T]$
$x(t)\in K $ for every $t\in [0,T]$
$x(0)\in Q_0\.$

The main feature of our problem is the unboundedness of $f(x,U)$ and the absence of superlinear growth conditions for $L$. Such classical assumptions are here replaced by conditions on the Hamiltonian that can be satisfied, for instance, by some Lagrangians with no growth. This paper extends our previous results in Existence and Lipschitz regularity of solutions to Bolza problems in optimal control to the state constrained case.

keywords: existence of minimizers interior approximation of constrained trajectories. Bolza problem state contraints optimal control Lipschitz regularity of optimal trajectories
Exponential stability for a class of linear hyperbolic equations with hereditary memory
Monica Conti Elsa M. Marchini Vittorino Pata
We establish a necessary and sufficient condition of exponential stability for the contraction semigroup generated by an abstract version of the linear differential equation $$∂_t u(t)-\int_0^\infty k(s)\Delta u(t-s)ds = 0 $$ modeling hereditary heat conduction of Gurtin-Pipkin type.
keywords: semigroups of linear contractions exponential stability. memory kernels Hereditary heat conduction
Semilinear wave equations of viscoelasticity in the minimal state framework
Monica Conti Elsa M. Marchini Vittorino Pata
A semilinear integrodifferential equation of hyperbolic type is studied, where the dissipation is entirely contributed by the convolution term accounting for the past history of the variable. Within a novel abstract framework, based on the notion of minimal state, the existence of a regular global attractor is proved.
keywords: Hyperbolic equation with memory minimal state global attractor. viscoelasticity
Global attractors for nonlinear viscoelastic equations with memory
Monica Conti Elsa M. Marchini V. Pata
We study the asymptotic properties of the semigroup $S(t)$ arising from the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain \begin{eqnarray} |\partial_t u|^\rho \partial_{t t} u-\Delta \partial_{t t} u-\Delta \partial_t u\\ -\Big(1+\int_0^\infty \mu(s)\Delta s \Big)\Delta u +\int_0^\infty \mu(s)\Delta u(t-s)\Delta s +f(u)=h \end{eqnarray} written in the past history framework of Dafermos [10]. We establish the existence of the global attractor of optimal regularity for $S(t)$ when $\rho\in [0,4)$ and $f$ has polynomial growth of (at most) critical order 5.
keywords: solution semigroup global attractor. memory kernel Nonlinear viscoelastic equations
On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity
Veronica Felli Elsa M. Marchini Susanna Terracini
Asymptotics of solutions to Schrödinger equations with singular dipole-type potentials are investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.
keywords: Hardy's inequality dipole moment Singular potentials Schrödinger operators.

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