Discussion on Lechicki and Spakowski's counterexample

It is well-known that intersection of continuous correspondences can lost the continuity property. Lechicki and Spakowski's theorem says that intersection of H-lsc functions remains H-lsc if the intersection is a bounded subset of a normed space and its interior is nonempty. Lechicki and Spakowski pointed to the importance of the boundedness assumption in the case of infinite dimensional range giving a counterexample. Even though the counterexample works properly and is one of the most cited patterns of discontinuity, it has no detailed discussion in the literature of economics and optimization theory. What is more, some misleading interpretation of this very important counterexample can be observed. Our technical note clarifies the exact role of Lechicki and Spakowski's counterexample, computing each of the important properties of the correspondences rigorously

Semiflow selection and Markov selection theorems

The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.Comment: In this revised version we have added a new abstract result in Sec. 2. It is used to correct the Navier-Stokes example in application

Effective computability of solutions of differential inclusions-the ten thousand monkeys approach

In this note we consider the computability of the solution of the initial- value problem for ordinary di erential equations with continuous right- hand side. We present algorithms for the computation of the solution using the \thousand monkeys" approach, in which we generate all possi- ble solution tubes, and then check which are valid. In this way, we show that the solution of a di erential equation de ned by a locally Lipschitz function is computable even if the function is not e ectively locally Lips- chitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be e ectively enumerable open, and give an example of a computable locally Lipschitz function which is not e ectively locally Lipschitz

Hybrid trajectory spaces

In this paper, we present a general framework for describing and studying hybrid systems. We represent the trajectories of the system as functions on a hybrid time domain, and the system itself by its trajectory space, which is the set of all possible trajectories. The trajectory space is given a natural topology, the compact-open hybrid Skorohod topology, and we prove the existence of limiting trajectories under uniform equicontinuity assumptions. We give a compactness result for the trajectory space of impulse differential inclusions, a class of nondeterministic hybrid system, and discuss how to describe hybrid automata, a widely-used class of hybrid system, as impulse differential inclusions. For systems with compact trajectory space, we obtain results on Zeno properties, symbolic dynamics and invariant measures. We give examples showing the application of the results obtained using the trajectory space approac

A Robust Stabilization using State Feedback with Feedforward

Abstract-In a general nonlinear control system a stabilizing control strategy is often possible if complete information on external inputs affecting the system is available. Assuming that measurements of persistent disturbances are available it is shown that the existence of a smooth uniform control Lyapunov function implies the existence of a stabilizing state feedback with feedforward control which is robust with respect to measurement errors and external disturbances. Conversely, using differential inclusions parameterized as nonlinear systems with state and disturbance measurement errors, it is shown that there exists a smooth uniform control Lyapunov function if there is a robustly stabilizing state feedback with feedforward. This paper demonstrates that if there exists a smooth control Lyapunov function for a general nonlinear system with persistent disturbances for which one has previously designed a feedback controller, a feedforward always exists to be augmented for stability

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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