Attainable Set Research Articles

Weak solutions of stochastic reaction diffusion equations and their optimal control

In this paper we consider a class of stochastic reaction diffusion equations with polynomial nonlinearities. We prove existence and uniqueness of weak solutions and their regularity properties. We introduce a suitable topology on the space of stochastic relaxed controls and prove continuous dependence of solutions on controls with respect to this topology and the norm topology on the natural space of solutions. Also we prove that the attainable set of measures induced by the weak solutions is weakly compact. Then we consider some optimal control problems, including the Bolza problem, and some target seeking problems in terms of the attainable sets in the space of measures and prove existence of optimal controls. In the concluding section we present briefly some extensions of the results presented here.

On the Set of Points of Smoothness for the Value Function of Affine Optimal Control Problems

We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of the interior of the attainable set. As a byproduct we obtain that it is actually smooth on a possibly smaller set, still open and dense.

A sufficient condition for existence of constantly nondominated trajectories in linear time-invariant systems

The paper considers multicriteria optimal control problem in linear time-invariant systems with a single bounded input. It is proven that under certain assumptions a state trajectory is nondominated throughout the whole control process if and only if it is yielded by a control signal which belongs to a particular class of bang-bang functions. Furthermore, an explicit formula relating hyperplanes tangent to attainable set and control switching times is presented.

ДОСТИЖИМЫЕ ЗНАЧЕНИЯ ЦЕЛЕВЫХ ФУНКЦИОНАЛОВ ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОЙ СИСТЕМЫ С ИМПУЛЬСНЫМ ВОЗДЕЙСТВИЕМ

For a linear functional differential system with aftereffect and impulses, a description of the attainability set is given. The attainability is considered in the term of a given system of on-target functionals in the case of polyhedral constraints with respect to control and impulses.

A step forward on order-α robust nonparametric method: inclusion of weight restrictions, convexity and non-variable returns to scale

Partial frontiers have been recently developed in order to overcome several drawbacks of the traditional nonparametric techniques. These robust frontier (order-\(\alpha\) and order-m) methods avoid the curse of dimensionality, are less sensitive to outliers and extreme data and may include direct environmental information in the model. Nonetheless, the disadvantages of these partial frontier-based methods according to the formulation proposed in the literature are that they do not allow weight restrictions or non-variable returns to scale technology. The procedure here proposed is an extension of the traditional order-\(\alpha\) method, allowing the estimation of an empirical convex \(\alpha\)-level, assuming also some additional constraints, such as the virtual weight restrictions and non-variable returns to scale. In the particular case of nonconvex attainable sets, unrestricted formulations and variable returns to scale assumption, the proposed procedure returns the same results as the standard order-\(\alpha\).

Scheduling of recovery actions in the supply chain with resilience analysis considerations

Supply chain engineering models with resilience considerations have been mostly focused on disruption impact quantification within one analysis layer, such as supply chain design or planning. Performance impact of disruptions has been typically analysed without scheduling of recovery actions. Taking into account schedule recovery actions and their duration times, this study extends the existing literature to supply chain scheduling and resilience analysis by an explicit integration of the optimal schedule recovery policy and supply chain resilience. In particular, we compute a schedule optimal control policy and analyse the performance of this policy by varying the perturbation vector and representing the outcomes of variations in the form of an attainable set. We propose a scheduling model that considers the coordination of recovery actions in the supply chain. Further, we suggest a resilience index by using the notion of attainable sets. The attainable sets are known in control theory; their calculation is based on the schedule control model results and the minimax regret approach for continuous time parameters given by intervals. We show that the proposed indicator can be used to estimate the impact of disruption and recovery dynamics on the achievement of planned performance in the supply chain.

Scale-dependent portfolio effects explain growth inflation and volatility reduction in landscape demography

Population demography is central to fundamental ecology and for predicting range shifts, decline of threatened species, and spread of invasive organisms. There is a mismatch between most demographic work, carried out on few populations and at local scales, and the need to predict dynamics at landscape and regional scales. Inspired by concepts from landscape ecology and Markowitz's portfolio theory, we develop a landscape portfolio platform to quantify and predict the behavior of multiple populations, scaling up the expectation and variance of the dynamics of an ensemble of populations. We illustrate this framework using a 35-y time series on gypsy moth populations. We demonstrate the demography accumulation curve in which the collective growth of the ensemble depends on the number of local populations included, highlighting a minimum but adequate number of populations for both regional-scale persistence and cross-scale inference. The attainable set of landscape portfolios further suggests tools for regional population management for both threatened and invasive species.

Globalization and productivity: A robust nonparametric world frontier analysis

How to exit the economic recession and improve the economic condition in a globalized world is a central policy issue of importance. This paper examines whether countries that pursue outward orientation policies and that are increasingly economically integrated with the rest of the world have seen an increase in economic performance. Increasing globalization and interconnection among countries generates spatial and temporal dependence which will affect the production process of each country. We extend existing methodological tools – robust frontier in non parametric location-scale models - to estimate the world frontier of 44 countries over 1970–2007 and to obtain more reliable measure of productivity and efficiency to better investigate the driven forces behind the catching-up productivity process among countries. We explore the channels under which Foreign Direct Investment (FDI) and time affect the production process and its components: impact on the attainable production set (input-output space), and the impact on the distribution of efficiencies.

On the norm attainment set of a bounded linear operator

In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff–James orthogonality techniques, we give a necessary condition for a nonzero bounded linear operator to attain norm at a particular point of the unit sphere. We prove four corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff–James orthogonality. Restricting our attention to lp2(p∈N∖{1}) spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm.

Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet–Laplacian among open sets of R N of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it is possible to obtain informations on the geometry of the minimizers of convex combinations: we study, in particular, when these minimizers are no longer convex, and the optimality of balls. As an application of our results we study the boundary of the attainable set for the Dirichlet spectrum. Our techniques involve symmetry results a la Serrin, explicit constants in quantitative inequalities, as well as a purely geometrical problem: the minimization of the Fraenkel 2-asymmetry among convex sets of fixed measure.

Computation of the target state and feedback controls for time optimal consensus in multi-agent systems

ABSTRACTN identical agents with bounded inputs aim to reach a common target state (consensus) in the minimum possible time. Algorithms for computing this time-optimal consensus point, the control law to be used by each agent and the time taken for the consensus to occur, are proposed. Two types of multi-agent systems are considered, namely (1) coupled single-integrator agents on a plane and, (2) double-integrator agents on a line. At the initial time instant, each agent is assumed to have access to the state information of all the other agents. An algorithm, using convexity of attainable sets and Helly's theorem, is proposed, to compute the final consensus target state and the minimum time to achieve this consensus. Further, parts of the computation are parallelised amongst the agents such that each agent has to perform computations of O(N2) run time complexity. Finally, local feedback time-optimal control laws are synthesised to drive each agent to the target point in minimum time. During this part of the operation, the controller for each agent uses measurements of only its own states and does not need to communicate with any neighbouring agents.

Pontryagin’s maximum principle for dynamic systems on time scales

In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. We generalize known results to the case when a certain set of admissible values of the control is not necessarily closed (but convex) and the attainable set is not necessarily convex. At the same time, we impose an additional condition on the graininess of the time scale. For linear systems, sufficient conditions in the form of the maximum principle are obtained.

Приближенный метод решения задач оптимального управления для дискретных систем, основанный на локальной аппроксимации множества достижимости

Приближенный метод решения задач оптимального управления для дискретных систем, основанный на локальной аппроксимации множества достижимости

The influence of public subsidies on farm technical efficiency: A robust conditional nonparametric approach

The objective of this paper is to assess the influence of public subsidies on farm technical efficiency using recent advances in nonparametric efficiency analysis. To this end, we use robust conditional frontier techniques as well as insights from recent developments in nonparametric econometrics. The paper contributes to the ongoing methodological discussion on how to model the effect of public subsidies on farmers’ production decisions. The analysis is conducted using an unbalanced panel data of 1604 observations from 313 French farms located in the French region Meuse over the period 2006–2011. The estimates indicate that public subsidies influence negatively the conditional technical efficiency of farms. This suggests that public subsidies affect both the range of the attainable set for the inputs and outputs and the distribution of the efficiency scores inside the attainable set.

On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations with respect to semimartingale integrators. We present new connections between their solutions. In particular, we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set-valued stochastic equations. We also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. The results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases.

Robust Scheduling Practices in the U.S. Airline Industry: Costs, Returns, and Inefficiencies

Airlines use robust scheduling to mitigate the impact of unforeseeable disruptions on profits. We examine how effectively three common practices—flexibility to swap aircraft, flexibility to reassign gates, and scheduled aircraft downtime—accomplish this goal. We first estimate a multiple-input, multiple-outcome production frontier, which defines the attainable set of outcomes from given inputs. We then recover unobserved input costs and calculate how expenditure on inputs affects outcomes and revenues. We find that the per-dollar return from expenditure on gates, or more effective management of existing gate capacity, is three times larger than the per-dollar returns from other inputs. Next, we use the estimated trade-offs faced by carriers along the frontier to measure the value to carriers of reducing delays. Finally, we calculate the improvement in carriers’ outcomes and profits if their operational inefficiencies are eliminated. On average, we estimate that operational inefficiencies cost carriers about $1.7 billion in revenue annually. This paper was accepted by Serguei Netessine, operations management.

Estimation of the attainability set for a linear system based on a linear matrix inequality

The problem under consideration is the construction of an attainability set’s internal approximation for a full controllable linear time-invariant system. This approximation is obtained as an intersection of two domains given by quadratic forms. One of these forms is based on parameters of the original system. The other form is produced by the solution to some linear matrix inequality. The method proposed here is illustrated by a numerical example.

Method of stochastic sensitivity synthesis in a stabilisation problem for nonlinear discrete systems with incomplete information

ABSTRACTA problem of the synthesis of the assigned probabilistic distribution near the equilibrium of stochastic nonlinear discrete-time system with incomplete state information is considered. We construct a static feedback regulator which provides a required stochastic sensitivity of this equilibrium. It is shown that this problem is reduced to the solution of some quadratic matrix equations. The solvability of these quadratic equations is analysed, and attainability sets are described. In the two-dimensional case, two variants of the control input are discussed and compared. These general results are applied to the suppression of large-amplitude oscillations around the equilibria of the stochastically forced Henon model with noisy observations. We show how suggested control technique, by minimising the stochastic sensitivity, allows us to suppress chaos and provide a structural stabilisation.

Perturbed stable systems on a plane. Part 1

The possibility of extending the concept of rough dynamical systems in the presence of a time-varying perturbation defined up to a functional set in the space of piecewise continuous functions is considered. The constructability of this extension is achieved by the construction of limit cycles giving an estimate of attainability sets for oscillatory systems.

Relationship between attainable sets of functional and set-valued functional stochastic inclusions

ABSTRACTIt is proved that attainable sets of stochastic differential inclusions, considered by Kisielewicz and M. Michta (J. Math. Anal. Appl. [submitted]), with a family of ΙF-non-anticipative matrix-valued stochastic processes for n ⩾ 1 such that ∑∞n = 1E|gnt|2 < ∞, are subsets of values of solutions of associated to them set-valued stochastic differential equations. Furthermore, the above type relationships are investigated for functional and set-valued functional stochastic inclusions. In particular, it is proved that for every 0 ⩽ t ⩽ T the closed convex hull of decomposable hull of attainable sets of functional stochastic inclusions covers with a set of all -measurable selectors of value Xt of any solution X of set-valued functional stochastic inclusions.