Attainable Set Research Articles

Lorentzian distance on the Lobachevsky plane *

Left-invariant Lorentzian structures on the 2D solvable non-Abelian Lie group are studied. Sectional curvature, attainable sets, Lorentzian length maximizers, distance, spheres, and infinitesimal isometries are described.

Approximations and inference for envelopment estimators of production frontiers

Nonparametric methods have been commonly used to assess the performance of both private and public organizations. Among them, the most popular ones are envelopment estimators such as Free Disposal Hull (FDH) or Data Envelopment Analysis (DEA), which estimate the attainable sets and their efficient boundaries by enveloping the cloud of observed units in the appropriate input-output space. However, these nonparametric envelopment techniques do not provide estimates of marginal products and other coefficients of economic interest. This paper presents a new approach that provides local estimates of all the desired partial derivatives and economic coefficients, which complement and complete the analysis based on nonparametric envelopment estimators. We improve nonparametric estimators by estimating nonparametrically smoothed efficient boundaries and providing derivatives and other coefficients without having to assume any parametric structure for the frontier and the inefficiency distribution. Our approach offers several advantages, such as a flexible nonparametric adjustment of the efficient frontier based on local linear models; a general multivariate efficiency model based on directional distances where one can choose the desired benchmark direction; the possibility of assessing the impact of external-environmental variables; a bootstrap-based statistical inference for deriving confidence intervals on the estimated coefficients for nonparametric and robust frontier approximations; the possibility of including factors aggregating inputs or outputs and recovering the estimated coefficients in the original units. To demonstrate the usefulness of the proposed approach, we provide an illustration in the field of education, where economic coefficients are important but the parametric assumptions have been questioned.

Initial data identification in space dependent conservation laws and Hamilton-Jacobi equations

Consider a Conservation Law and a Hamilton-Jacobi equation with a flux/Hamiltonian depending also on the space variable. We characterize first the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed profile. An explicit example then shows the deep differences between the cases of x-independent and x-dependent fluxes/Hamiltonians.

Design of Pseudo-Command Restricted Controller for Tailless Unmanned Aerial Vehicles Based on Attainable Moment Set

This work investigates the pseudo-command restricted problem for tailless unmanned aerial vehicles with snake-shaped maneuver flight missions. The main challenge of designing such a pseudo-command restricted controller lies in the fact that the necessity of control allocation means it will be difficult to provide a precise envelope of pseudo-command to the flight controller; designing a compensation system to deal with insufficient capabilities beyond this envelope is another challenge. The envelope of pseudo-command can be expressed by attainable moment sets, which leave some open problems, such as how to obtain the attainable moment sets online and how to reduce the computational complexity of the algorithm, as well as how to ensure independent control allocation and the convexity of attainable moments sets. In this article, an innovative algorithm is proposed for the calculation of attainable moment sets, which can be implemented by fitting wind tunnel data into a function to solve the problems presented above. Furthermore, the algorithm is independent of control allocation and can be obtained online. Moreover, based on the above attainable moment sets algorithm, a flight performance assurance system is designed, which not only guarantees that the command is constrained within the envelope so that its behavior is more predictable, but also supports adaptive compensation for the pseudo-command restricted controller. Finally, the effectiveness of the AMS algorithm and the advantages of the pseudo-command restricted control system are validated through two sets of independent simulations.

Robust fault-tolerant control for dynamic positioning of ships with prescribed performance

This study presents a new fault-tolerant control scheme for ship dynamic positioning. The scheme aims to handle unknown disturbances, imprecise fault estimation, and control input constraints. First, an adaptive sliding mode disturbance observer is introduced. This observer effectively reconstructs disturbances in finite time, even without knowledge of their derivative upper bounds. Next, we design a novel prescribed performance function that sets bounds on tracking errors. This function is combined with an auxiliary intermediate control technique to create a high-level controller. Lastly, a robust fault-tolerant control allocation scheme is proposed to redistribute the generalized forces among faulty actuators. It autonomously finds a feasible force vector in the attainable force set. Simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Traveling along horizontal broken geodesics of a homogeneous Finsler submersion

In this paper, we discuss how to travel along horizontal broken geodesics of a homogeneous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets Aq(C) of the set of analytic vector fields C determined by the family of horizontal unit geodesic vector fields C to the fibers F={ρ−1(c)} of a homogeneous analytic Finsler submersion ρ:M→B. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds M where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when M is compact and the orbits of C are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then M coincides with the attainable set of each point. In other words, fixed two points of M, one can travel from one point to the other along horizontal broken geodesics.In addition, we show that each orbit O(q) of C associated to a singular Finsler foliation coincides with M, when the flag curvature is positive, i.e., we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case.

Некоторые вопросы, связанные с реализацией множеств притяжения с точностью до наперед заданной окрестности

Questions connected with implementation of attraction sets (AS) in attainability problem with constraints of asymptotic nature (CAN) are considered. It is investigated the possibility of AS implementation accurate to arbitrary neighborhood in class of closures of attainability sets corresponding to concrete sets from the family generating CAN. Moreover, some relations for AS generated by different CAN are considered (disjunction conditions of AS are investigated). General constructions of neighborhood implementation of AS were applied in the case when these AS were considered in the space of ultrafilters of broadly understood measurable space (MS). In particular, the case when CAN are defined by a filter was investigated in detail; for this case, under non-restrictive conditions on the original MS, the set of ultrafilters majorizing the original filter is implemented as AS. In this case (of ultrafilter space) variants of equipment of ultrafilter set with topologies of Stone and Wallman types are investigated separately.

On the Pythagorean orthogonality of Hilbert bounded linear operators

In this work, we characterize the Pythagorean orthogonality of bounded linear operators using the concept of norm attainment set and disjoint support in Hilbert spaces.

On the Attainable Set of Iterative Differential Inclusions

ABSTRACT In this paper, we consider a first-order iterative differential inclusion. We study the existence of solutions and some topological proprieties of the attainable set, where the right hand side is an upper semi-continuous multifunction with convex values. Then, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity.

Some problems of target approach for nonlinear control system at a fixed time moment

The game problem of target approach for nonlinear control system with a target set in a finite-dimensional phase space at a fixed time is studied. The problem is formulated and studied within the framework of concepts and constructions of the theory of positional differential games, created by N.N. Krasovskii and A.I. Subbotin in the second half of the 20th century. One of the central problems of the theory of positional differential games is the problem of calculating positional sets of uptake in convergence game problems. The paper investigates a key stability property in the theory of positional differential games, which is a characteristic of some closed sets in the space of positions of a controlled system that are convenient for the first player to conduct the game. It is important that this property is also characteristic for solvability sets in convergence problems: the inclusion of the concept of stability in research makes it possible to obtain analytical descriptions of solvability sets in some specific convergence problems and to develop algorithms for approximate calculation of the solution in a number of specific problems. Some modifications of the definition of a $u$-stable bridge in the considered convergence problem and a system of sets approximating the attainability set are given. Three specific problems on the convergence of mechanical systems are also presented, computer solutions are simulated, and graphical simulation results are presented.

ATTAINABLE SETS OF INTEGRAL CONSTRAINED SEIR CONTROL SYSTEM WITH NONLINEAR INCIDENCE

In this survey, we consider the dynamics of a contagious disease spread by employing a nonlinear dynamical control system of differential equations. It considers treatment and vaccination as key control parameters to discern their influence on disease control. The study, approximate the attainable sets of a given control system and presents visual results, while also discussing potential biological applications of their findings.

How to tighten the control set?

We examine the problem of reducing the control set in a dynamical system so that the solution set and the attainable sets remain essentially unchanged. We cite some classical results to exhibit the problem and promote a set-valued approach. A necessary condition is formulated by using a concept of set-valued derivative, which can be regarded as an extension of the classical Relaxation theorem.

Safety flight envelope calculation and protection control of UAV based on disturbance observer

In this paper, as for the unmanned air vehicle (UAV) under external disturbance, an attainable-equilibrium-set-based safety fight envelope (SFE) calculation method is proposed, based on which a prescribed performance protection control scheme is presented. Firstly, the existing definition of the SFE based on attainable equilibrium set (AES) is extended to make it consistent and suitable for the UAV system under disturbance. Secondly, a higher-order disturbance observer (HODO) is developed to estimate the disturbances and the disturbance estimation is applied in the computation of the SFE. Thirdly, by using the calculated SFE, a desired safety trajectory based on the time-varying safety margin function and first-order filter is developed to prevent the states of the UAV system from exceeding the SFE. Moreover, an SFE protection controller is proposed by combining the desired safety trajectory, backstepping method, HODO design, and prescribed performance (PP) control technique. In particular, the closed-loop system is established on the basis of disturbance estimation error, filter error, and tracking error. Finally, the stability of the closed-loop system is verified by the Lyapunov stability theory, and the simulations are presented to illustrate the effectiveness of the proposed control scheme.

Control theory approach to continuous-time finite state mean field games

<p style='text-indent:20px;'>In the paper, we study the dependence of solutions of the continuous-time finite state mean field game on initial distribution of players. Our approach relies on the concept of value multifunction of the mean field game that is a mapping assigning to an initial time and an initial distribution a set of expected outcomes of the representative player corresponding to solutions of the mean field game. Using the reformulation of the finite state mean field game as a control problem with mixed constraints, we give the sufficient condition on a given multifunction to be a value multifunction in the terms of the viability theory. The maximal multifunction (i.e., the mapping assigning to an initial time and an initial distribution the whole set of values corresponding to solutions of the mean field game) is characterized via the backward attainability set for the certain control system.</p>

Regional Controllability and Minimum Energy Control of Delayed Caputo Fractional-Order Linear Systems

We study the regional controllability problem for delayed fractional control systems through the use of the standard Caputo derivative. First, we recall several fundamental results and introduce the family of fractional-order systems under consideration. Afterward, we formulate the notion of regional controllability for fractional systems with control delays and give some of their important properties. Our main method consists of defining an attainable set, which allows us to prove exact and weak controllability. Moreover, the main results include not only those of controllability but also a powerful Hilbert uniqueness method, which allows us to solve the minimum energy optimal control problem. More precisely, an explicit control is obtained that drives the system from an initial given state to a desired regional state with minimum energy. Two examples are given to illustrate the obtained theoretical results.

Some remarks on orthogonality of bounded linear operators-II

Let $$\mathbb{X,Y}$$ be normed linear spaces. We continue exploring the validity of the Bhatia–Šemrl (BŠ) Property: “An operator $$T \in \mathbb{L(X,Y)}$$ satisfies Bhatia–Šemrl Property if for any $$A \in \mathbb{L(X,Y)},T \bot_{B} A$$ implies that there exists a unit vector $$x \in \mathbb{X} $$ such that $$\lVert T_{x} \lVert = \lVert T \lVert $$ and $$T_{x}\bot_{B}A_{x}$$ .” A pair of normed linear spaces $$\mathbb{X,Y}$$ is said to be a BŠ pair if for every $$T\in \mathbb{L(X,Y),T}$$ satisfies the BŠ Property if and only if $$M_{T} = D{\cup}(-D)$$ , where D is a non-empty connected subset of $$S_{\mathbb{X}} $$ . We show that $$\ell^{n}_{1},\mathbb{Y}$$ is a BŠ pair for any normed linear space $$\mathbb{Y}$$ , and also obtain some other results in this context. In particular, using the notion of norm attainment set,we characterize the space $$\ell^{3}_{\infty}$$ among all 3-dimensional polyhedral Banach spaces whose unit ball have exactly eight extreme points.

What do people with aphasia want from the Queen Square Intensive Comprehensive Aphasia Programme and do they achieve it? A quantitative and qualitative analysis of their short, medium, long-term and economic goals

ABSTRACT Background The most effective model for achieving therapist-delivered, high-dose SLT for People with Aphasia (PWA) is through Intensive Comprehensive Aphasia Programmes (ICAPs). ICAPs are often assessed using standardised outcome measures; however, as SLT is personalised, it is of interest to examine individualised goal-based outcome measures as well. In the Queen Square ICAP, we use a goal-setting approach (Goal Attainment Setting [GAS]) where the PWA and their therapist negotiate which goals to work on and over what timescales. This process involves recording and scoring the agreed goals, which makes them amenable to formal quantitative and qualitative analysis. Aims The aim of this study was twofold. Firstly, to test the hypothesis that a pre- versus post- ICAP analysis of individual’s goal scores would show statistically significant and clinically meaningful improvements. Secondly, to better understand what PWA wanted to achieve from the ICAP service, we performed a qualitative analysis across all agreed goals. Methods & Procedures Forty-four PWA who varied in aphasia severity from mild to severe took part. PWA jointly set goals with their therapists using the SMART framework (Specific, Measurable, Achievable, Relevant, and Time-Bound). The goals were split into four categories: short (3 weeks), medium (3-6 months), long-term (12 months) and economic, (defined as any outcome that will improve, either directly or indirectly, the economic system that the PWA lives within). Quantitative scores were obtained for each PWA both pre- and post- ICAP and were analysed using paired t-tests, with subsequent ANOVAs to investigate possible confounding factors. The qualitative analysis was carried out by two researchers not involved in delivering the ICAP. Data was collapsed across all goal categories and analysed using thematic analysis. Outcomes & Results Quantitatively, statistically significant gains were made across all four goal categories (ps < 0.001). Unstandardized effect-sizes were clinically significant (ΔGAS ~16). Qualitatively, we identified five main themes: staying connected with the world, understanding aphasia better, raising awareness, the importance of having a work identity and managing personal relationships. Conclusions Quantitative goal-setting for PWA in the context of an ICAP provides robust evidence that PWA can achieve a variety of aspirational goals given high enough doses of specialist input from SLTs and a clinical psychologist. Although the ICAP only spanned 3 weeks, PWA continued to reach medium, long-term and even economic goals up to a year post-recruitment. This is the first time that economic goals have been captured in PWA using GAS. The qualitative analysis describes what the PWA wanted to achieve from participating in our ICAP, while the quantitative analyses demonstrate how much they succeeded in doing so.

Short Takeoff and Landing Strategy for Small-Scale Thrust-Vectoring Vertical/Short Takeoff and Landing Vehicles

This work provides a feasible solution to the shipborne short landing problem for thrust-vectoring V/STOL vehicles. The short takeoff and shipborne rolling vertical landing strategy was designed in this work. First, the strategy design reference was established by flying performance and mission requirements, including the short takeoff and landing performance, deceleration performance, trajectory stability, velocity stability, and conversion corridor, using attainable equilibrium set methods based on the six dimensions of freedom model of the study object. Then, a piecewise short takeoff landing strategy was designed based on the references, together with a nonlinear dynamic inverse-based control designed frame for strategy execution. Finally, the hardware-in-loop Monte-Carlo simulation was implemented for the strategy feasibility verification. The proposed short takeoff and landing strategy satisfies the shipborne short takeoff and landing mission requirements. The short takeoff shortens the taxiing distance by 40% compared to a normal takeoff. With a 20% perturbance on all model parameters, the touchdown speed can be controlled to 14 ± 1 m/s, and the landing point position can be constrained inside a 5 m radius circle with almost zero lateral displacements.

Multirotor Unmanned Aerial Vehicle Configuration Optimization Approach for Development of Actuator Fault-Tolerant Structure

Presently, multirotor unmanned aerial vehicles (UAV) are utilized in numerous applications. Their design governs the system’s controllability and operation performance by influencing the achievable forces and moments produced. However, unexpected causalities, such as actuator failure, adversely affect their controllability, which raises safety concerns about their service. On the other hand, their design flexibility allows further design optimization for various performance requirements, including actuator failure tolerance. Thus, this study proposed an optimization framework that can be employed to design a novel actuator fault-tolerant multirotor UAV configuration. The approach used an attainable moment set (AMS) to evaluate the achievable moment from a multirotor configuration; similarly, standard deviation geometries (SDG) were employed to define performance requirements. Therefore, given a UAV configuration, actuator fault situation, and SDG derived from the designed mission requirement, the suggested optimization framework maximizes the scaling factor of SDG and fits it into the AMS by adjusting the design parameters up to a sufficient margin. The framework is implemented to optimize selected parameters of the Hexacopter-type of parcel delivery multirotor UAV developed by the PNU drone, and a simulation was conducted. The result showed that the optimized configuration of the UAV achieved actuator fault tolerance and operation-performing capability in the presence of a failed actuator.

Estimation of portfolio efficiency via stochastic DEA

Traditional data envelopment analysis (DEA) and diversification DEA are two common data-driven evaluation approaches, which have been widely used in the estimation of portfolio efficiency. The above two DEA approaches usually use the risk and expected return indicators to build the input-output process of portfolios. However, this input-output process derived from the risk and expected return is inconsistent with the actual investment process, since the real input should be the initial wealth, and the output should be the terminal wealth. To address this problem, we propose a novel input-output process based on the initial and terminal wealth of portfolios. We transform the terminal wealth into the rate of return and construct a stochastic attainable set by using portfolio returns. We provide three deterministic estimation approaches to deal with the stochastic attainable set, and then obtain three deterministic attainable sets. We further propose three stochastic DEA models to estimate the portfolio efficiency by using the above three deterministic attainable sets. Finally, we provide an empirical analysis to assess the portfolio efficiency of 50 open-ended funds in China. The results show that there are some differences in the portfolio efficiency and its ranking between the proposed DEA models and the existing DEA models, which further verify the rationality of the proposed DEA models.