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*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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**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.

$ 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 $T$.

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.

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

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.

*hybrid*boundary conditions, including the coupling of two wave equations subject to Dirichlet and Robin type boundary conditions, respectively.

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