We consider three monads on $\mathsf{Top}$ , the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads $V \to H$ . In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$ -smooth probability measures on $\mathsf{Top}$ . By composing these morphisms of monads, we obtain that taking the supports of $\tau$ -smooth probability measures is also a morphism of monads.

In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error in L2(Ω) of the control variable is h | log h | in dimensions 2 and 3.

Optimal control problems for semilinear elliptic equationswith control constraints and pointwise state constraints arestudied. Several theoretical results are derived, which arenecessary to carry out a numerical analysis for this class ofcontrol problems. In particular, sufficient second-order optimalityconditions, some new regularity results on optimal controls and asufficient condition for the uniqueness of the Lagrange multiplierassociated with the state constraints are presented.

Lower semicontinuity results are obtained for multiple integrals of the kind $\int _{\mathbb{R}^n}\!f(x, \!\nabla_\mu u\!){\rm d} \mu$, where μ is a given positive measure on $\mathbb{R}^n$, and the vector-valued function u belongs to the Sobolev space $H ^{1,p}_\mu (\mathbb{R}^n, \mathbb{R}^m)$ associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when μ belongs to a suitable class of rectifiable measures.