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Register a new userObservability inequalities on measurable sets for the Stokes system and applications
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In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as an observability inequalities on space-time measurable sets of positive measure for non-stationary Stokes system. Furthermore, we provide their applications in the theory of shape optimization and time optimal control problems.
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Observability inequalities on lattice points are established for non-negative solutions of the heat equation with potentials in the whole space. As applications, some controllability results of heat equations are derived by the above-mentioned observability inequalities.
The regularity theory for variational inequalities over polyhedral sets developed in a series of papers by Robinson, Ralph and Dontchev-Rockafellar in the 90s has long become classics of variational analysis. But in the available proofs of almost all main results, fairly nontrivial as they are, techniques of variational analysis do not play a significant part. In the paper we develop a new approach that allows to obtain some generalizations of the the mentioned results without invoking anything beyond elementary geometry of convex polyhedra and some basic facts of the theory of metric regularity.
Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of $alpha$-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize them.
Let $mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $tau$. A closed densely defined operator $x$ affiliated with $mathfrak{M}$ is called $tau$-measurable if there exists a number $lambda geq 0$ such that $tau left(e^{|x|}(lambda,infty)right)<infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $tau$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $tau$-measurable operators. A general parallelogram law for $tau$-measurable operators are given as well.
We develop a data-driven approach to the computation of a-posteriori feasibility certificates to the solution sets of variational inequalities affected by uncertainty. Specifically, we focus on instances of variational inequalities with a deterministic mapping and an uncertain feasibility set, and represent uncertainty by means of scenarios. Building upon recent advances in the scenario approach literature, we quantify the robustness properties of the entire set of solutions of a variational inequality, with feasibility set constructed using the scenario approach, against a new unseen realization of the uncertainty. Our results extend existing results that typically impose an assumption that the solution set is a singleton and require certain non-degeneracy properties, and thereby offer probabilistic feasibility guarantees to any feasible solution. We show that assessing the violation probability of an entire set of solutions, rather than of a singleton, requires enumeration of the support constraints that shape this set. Additionally, we propose a general procedure to enumerate the support constraints that does not require a closed form description of the solution set, which is unlikely to be available. We show that robust game theory problems can be modelling via uncertain variational inequalities, and illustrate our theoretical results through extensive numerical simulations on a case study involving an electric vehicle charging coordination problem.
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