Piermarco Cannarsa Cecilia Cavaterra Angelo Favini Alfredo Lorenzi Elisabetta Rocca
Science, engineering and economics are full of situations in which one observes the evolution of a given system in time. The systems of interest can differ a lot in nature and their description may require finitely many, as well as infinitely many, variables. Nevertheless, the above models can be formulated in terms of evolution equations, a mathematical structure where the dependence on time plays an essential role. Such equations have long been the object of intensive theoretical study as well as the source of an enormous number of applications.
   A typical class of problems that have been addressed over the years is concerned with the well-posedness of an evolution equation with given initial and boundary conditions (the so-called direct problems). In several applied situations, however, initial conditions are hard to know exactly while measurements of the solution at different stages of its evolution might be available. Different techniques have been developed to recover, from such pieces of information, specific parameters governing the evolution such as forcing terms or diffusion coefficients. The whole body of results in this direction is usually referred to as inverse problems. A third way to approach the subject is to try to influence the evolution of a given system through some kind of external action called control. Control problems may be of very different nature: one may aim at bringing a given system to a desired configuration in finite or infinite time (positional control), or rather try to optimize a performance criterion (optimal control).

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A constructive proof of Gibson's stability theorem
Fatiha Alabau-Boussouira Piermarco Cannarsa
A useful stability result due to Gibson [SIAM J. Control Optim., 18 (1980), 311--316] ensures that, perturbing the generator of an exponentially stable semigroup by a compact operator, one obtains an exponentially stable semigroup again, provided the perturbed semigroup is strongly stable. In this paper we give a new proof of Gibson's theorem based on constructive reasoning, extend the analysis to Banach spaces, and relax the above compactness assumption. Moreover, we discuss some applications of such an abstract result to equations and systems of evolution.
keywords: Poincaré recurrences multifractal analysis. Dimension theory
Semiconcavity of the value function for a class of differential inclusions
Piermarco Cannarsa Peter R. Wolenski
We provide intrinsic sufficient conditions on a multifunction $F$ and endpoint data φ so that the value function associated to the Mayer problem is semiconcave.
keywords: differential inclusions Semiconcave functions optimal control Mayer problems.
On a class of nonlinear time optimal control problems
Piermarco Cannarsa Carlo Sinestrari
We consider the minimum time optimal control problem for systems of the form

$ y'(t)=f(y(t),u(t))\,\quad y(t) \in \mathbb{R}^n,\ u(t)\in U \subset \mathbb{R}^d. $

We assume $f(x,U)$ to be a convex set with $C^1$ boundary for all $x\in\mathbb{R}^n$ and the target $\kappa$ to satisfy an interior sphere condition. For such problems we prove necessary and sufficient optimality conditions using the properties of the minimum time function $T(x)$. Moreover, we give a local description of the singular set of $T$.

keywords: optimal control problem calculus of variations interior sphere condition.
Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign
Piermarco Cannarsa Alexander Khapalov
We study the global approximate controllability properties of a one dimensional reaction-diffusion equation governed via the coefficient of the reaction term. The traditional (linear operator) controllability methods based on the duality pairing do not apply to such a problem. Instead, we focus on the qualitative study of the diffusion and reaction parts of the evolution process at hand. We consider the case when both the initial and target states admit no more than finitely many changes of sign.
keywords: Multiplicative controllability parabolic equation nonlinearity.
Invariance for stochastic reaction-diffusion equations
Piermarco Cannarsa Giuseppe Da Prato
Given a stochastic reaction-diffusion equation on a bounded open subset $\mathcal O$ of $\mathbb{R}^n$, we discuss conditions for the invariance of a nonempty closed convex subset $K$ of $L^2(\mathcal O)$ under the corresponding flow. We obtain two general results under the assumption that the fourth power of the distance from $K$ is of class $C^2$, providing, respectively, a necessary and a sufficient condition for invariance. We also study the example where $K$ is the cone of all nonnegative functions in $L^2(\mathcal O)$.
keywords: invariance of space domains. reaction-diffusion equations Stochastic partial differential equations
Persistent regional null contrillability for a class of degenerate parabolic equations
Piermarco Cannarsa Patrick Martinez Judith Vancostenoble
Motivated by physical models and the so-called Crocco equation, we study the controllability properties of a class of degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold for this problem in general.
First, we prove that we can drive the solution to rest at time $T$ in a suitable subset of the space domain (regional null controllability). However, unlike for nondegenerate parabolic equations, this property is no more automatically preserved with time. Then, we prove that, given a time interval $(T,T')$, we can control the equation up to $T'$ and remain at rest during all the time interval $(T,T')$ on the same subset of the space domain (persistent regional null controllability). The proofs of these results are obtained via new observability inequalities derived from classical Carleman estimates by an appropriate use of cut-off functions.
With the same method, we also derive results of regional controllability for a Crocco type linearized equation and for the nondegenerate heat equation in unbounded domains.
keywords: null controllability observability inequalities Degenerate parabolic equations
Semiconcavity for optimal control problems with exit time
Piermarco Cannarsa Cristina Pignotti Carlo Sinestrari
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
Indirect stabilization of weakly coupled systems with hybrid boundary conditions
Fatiha Alabau-Boussouira Piermarco Cannarsa Roberto Guglielmi
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 hybrid boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.
keywords: Indirect stabilization hyperbolic systems. interpolation spaces evolution equations energy estimates
Null controllability of degenerate parabolic operators with drift
Piermarco Cannarsa Genni Fragnelli Dario Rocchetti
We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.
keywords: Carleman estimates Degenerate parabolic equations null controllability Hardy type inequality

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