Minimum time problems

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A semi-Lagrangian scheme for Lp-penalized minimum time problems

A semi-Lagrangian scheme for Lp-penalized minimum time problems

A semi-Lagrangian scheme for L p -penalized minimum time problems Maurizio Falcone 1 and Dante Kalise 2 and Axel Kr¨oner 3 Abstract— In this paper we consider a semi-Lagrangian scheme for minimum time problems with L p -penalization. The minimum time function of the penalized control problem can be characterized as the solution of a Hamilton-Jacobi Bellman (HJB) equation. Furthermore, the minimum time converges with respect to the penalization parameter to the minimum time of the non-penalized problem. To solve the control problem we formulate the discrete dynamic programming principle and set up a semi-Lagrangian scheme. Various numerical examples are presented studying the effects of different choices of the penalization parameters.
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The Mayer and Minimum Time Problems with Stratified State Constraints *

The Mayer and Minimum Time Problems with Stratified State Constraints *

In this paper, we follow the same line of research undertaken in [20] and derive a new charac- terization of the value function, for Mayer and minimum time problems, in term of unique Bilateral viscosity solution to an adequate Hamilton-Jacobi equation. The main feature of this paper is to assume that the state constraint set K is endowed with a stratified structure while the dynamic data F satisfies standard assumptions of differential inclusion theory along with a structural tangential condition. In particular, the constraint set K may have empty interior or have subdomains of different dimension, conditions precluded by any of the pointed conditions. We will also require that on each stratified subdomain, either the admissible velocities throughout that subdomain form a Lipschitz multifunction, or there are no admissible velocities at all throughout the subdomain. Moreover, our proof techniques require a local controllability assumption in order to treat obstreperous feasible arcs that exhibit a chattering or Zeno-like behavior.
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Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming.

Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming.

To set this paper in perspective, we relate our approach to the previous works [2], [17], [19], [16], which also consider optimal feedback control problems for infinite-dimensional dynamics, using either proper orthogonal decomposition or spectral elements to obtain a low dimensional semi- discrete system. A numerical solution for feedback control problems for nonlinear parabolic equations is considered in [7]. Numerical implementations and approximation results for feedback problems of (second-order) hyperbolic equa- tions using Riccati equations can be found in [14], [15], and references therein. Similar approaches, based on model predictive control, which can be interpreted as a relaxed version of dynamic programming, have been presented in [8], [4]. Minimum time problems were first considered in [6]. For time optimal control problems of parabolic equations see [12], [18], [20], [23], [22], [27]. The novelty of this article resides in the combination of HJB techniques for the computation of time-optimal feedback controllers with model reduction and state observation algorithms providing thus a consistent approach for closed-loop control of infinite- dimensional systems.
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On the minimum time optimal control problem of an aircraft in its climbing phase

On the minimum time optimal control problem of an aircraft in its climbing phase

Abstract In this article, the minimum time optimal control problem of an aircraft in its climbing phase is studied. First, a reduction of the initial dynamics into a three dimensional single-input system with a linear de- pendence with respect to the control is performed. This reduced system is then studied using geometric control techniques. In particular, the maximum principle leads to describe a multi-point boundary value problem which is solved by indirect methods. These methods are the implementation of the maximum principle and are initialized by direct methods. We check that the extremal solution of the boundary value problem satisfies necessary and sufficient conditions of optimality. From this reference case and consider- ing small-time optimal trajectories, we give a local classification with respect to the initial mass and final velocity of BC-extremals for the climbing phase.
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Arc Routing Problems with Time Duration Constraints and Uncertainty

Arc Routing Problems with Time Duration Constraints and Uncertainty

clearer sectoring result with better shape and compactness of each sector can be produced with the TSA, at the expense of cost. After partitioning a large-sized network into small-sized sectors, the robust arc routing problem with time duration is addressed. The deterministic mathematical formulation for the ARPTD was proposed and a general polyhedral uncertainty set of service and deadheading times was defined. After that, the robust counterpart of the deterministic formulation was developed and then solved. Computational experiments showed that the RARPTD can be solved to optimality quickly for small-sized networks. The sensitivity analysis was conducted with respect to the level of uncertainty and the number of vehicles used, which revealed that a higher level of robustness of the optimal solution usually incurs higher costs.
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Algorithms for routing problems in stochastic time-dependent networks

Algorithms for routing problems in stochastic time-dependent networks

The same levels of the number of nodes, of the number of discrete departure times, and of the number of link travel time realizations as in the all-to-one mini[r]

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Cube versus Torus Models for Combinatorial Optimization Problems and the Euclidean Minimum Spanning Tree Constant

Cube versus Torus Models for Combinatorial Optimization Problems and the Euclidean Minimum Spanning Tree Constant

For a sample of points drawn uniformly from either the d-dimensional torus or the d-cube, d > 2, we define a class of random processes with the property of being asymptotic[r]

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Minimum time control problems for non autonomous differential equations

Minimum time control problems for non autonomous differential equations

The main goal of a minimum time problem is to determine the minimal time needed by a controlled sys- tem to reach a given target. When the dynamics of the system does not have an explicit time dependence, the problem has been widely studied with different approaches. Here, we focus on the Hamilton-Jacobi- Bellman (HJB) method. Let us recall that this approach is based on Dynamic Programming Principle (DPP) studied by R. Bellman [8]. It leads to a characterization of the minimal time function as a solution of an HJB equation, which appears to be well-posed in the framework of viscosity solutions introduced by Crandall and Lions [13, 12]. These tools also allow to perform the numerical analysis of the approxi- mation schemes. We refer to [2, Chapter IV], [3] for theoretical studies. Various numerical methods have been also investigated, such as those based on finite difference schemes [14], ENO and WENO schemes [19, 18], discontinuous Galerkin schemes [17], semi-Lagrangian methods [2, Appendix A] and [15].
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Minimum time control of the rocket attitude reorientation associated with orbit dynamics

Minimum time control of the rocket attitude reorientation associated with orbit dynamics

In this paper, we consider the time minimum control of the attitude reorientation coupled with the orbit dynamics of a rocket, denoted in short (MTCP). The chattering phenomenon that may occur according to the terminal conditions under consideration, makes in particular the problem quite difficult. Chattering means that the control switches an infinite number of times over a compact time interval. Such a phenomenon typically occurs when trying to connect bang arcs with a higher-order singular arc (see, e.g., [ 15 , 26 , 37 , 38 ]). In [ 38 ], we studied the planar version of (MTCP), where the system consists of a single-input control-affine system, and we established as well the occurrence of a chattering phenomenon and that the chattering extremals are locally optimal in C 0 topology. ∗
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The Bird Core for Minimum Cost Spanning Tree Problems Revisited: Monotonicity and Additivity Aspects

The Bird Core for Minimum Cost Spanning Tree Problems Revisited: Monotonicity and Additivity Aspects

JEL classification: C71. 1 Introduction One of the classical problems in Operations Research is the problem of find- ing a minimum cost spanning tree (mcst) in a connected network. For al- gorithms solving this problem see Kruskal (1956) and Prim (1957). Claus and Kleitman (1973) discuss the problem of allocating costs among users in

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Time-dependent sources identification for transmission lines problems

Time-dependent sources identification for transmission lines problems

IV. NUMERICAL VALIDATION As a first experiment, we apply the LCCF method described in the previous section to the unshielded coaxial transmission line network shown on Figure 2. The left end of line 1 is matched and all the other load impedances of the network are open loads. Three- dimensional coupling between the branches of the network by radiation are neglected. For all the numerical simulations, we use a home-made Finite-Difference Time Domain (FDTD) code to solve equations (1) for the voltages and the currents. However, since the proposed method is non-intrusive, any commercial software such as CST or HFSS could equally be used.
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Time Reversed Absorbing Condition: Application to inverse problems

Time Reversed Absorbing Condition: Application to inverse problems

1. the reduction of the size of the computational domain by redefining the reference surface on which the receivers appear to be located 2. the location of an unknown inclusion from boundary measurements The first application is reminiscent of the redatuming method introduced in [Ber79]. In our case, we use the wave equation and not a paraxial or parabolic approximation to it. This extends the domain of validity of the redatuming approach. Concerning the second application there is a huge literature that deals with this inverse problem. We mention the MUSIC algorithm [The92] and its application to imaging [LD03], the sampling methods first introduced in [CK96], see the review paper [CCM00] and references therein, and the DORT method [PMSF96]. Mathematical analysis of this kind of approach can be found in [CK98]. These methods were developed in the time-harmonic domain for impenetrable inclusions. The TRAC method is designed in both the time-dependent and harmonic domains and does not rely on any a priori knowledge of the physical properties of the inclusion. It works both for impenetrable and penetrable inclusions.
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Lower and Upper Bounds for Minimum Energy Broadcast and Sensing Problems in Sensor Networks

Lower and Upper Bounds for Minimum Energy Broadcast and Sensing Problems in Sensor Networks

Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue J[r]

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Minimum-maximum

Minimum-maximum

Document n° 42, créée le 31/5/2003 - Mis à jour le 13/7/2007 1. Aire minimum d'une lunule On considère la figure suivante : (C) est un cercle de centre O et de rayon 1, [AB] est un diamètre. À partir d'un point M de [AB], tracer deux demi-cercles de diamètre [AM] et [MB] (voir figure ci- dessous).

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On the minimum time optimal control problem of an aircraft in its climbing phase

On the minimum time optimal control problem of an aircraft in its climbing phase

Abstract In this article, the minimum time optimal control problem of an aircraft in its climbing phase is studied. First, a reduction of the initial dynamics into a three dimensional single-input system with a linear de- pendence with respect to the control is performed. This reduced system is then studied using geometric control techniques. In particular, the maximum principle leads to describe a multi-point boundary value problem which is solved by indirect methods. These methods are the implementation of the maximum principle and are initialized by direct methods. We check that the extremal solution of the boundary value problem satisfies necessary and sufficient conditions of optimality. From this reference case and consider- ing small-time optimal trajectories, we give a local classification with respect to the initial mass and final velocity of BC-extremals for the climbing phase.
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Solving Time Domain Audio Inverse Problems using Nonnegative Tensor Factorization

Solving Time Domain Audio Inverse Problems using Nonnegative Tensor Factorization

V. C ONCLUSIONS In this paper, we have presented a novel approach for time domain signal estimation in the maximum likelihood manner. It relies on the low rank NTF modeling of the power spectrum of the signal and can be applied to many types of problems that were not previously solved using the NMF/NTF model. The proposed algorithm is demonstrated to be very effective for several audio inverse problems while providing multiple advantages compared to other existing methods. For the audio declipping problem, clipped sections of music and speech sig- nals are restored using the proposed approach as well as state of the art methods, and the proposed algorithm is shown to be highly competitive while providing complementary advantages such as naturally handling noise and quantization artefacts and easily incorporating various types of constraints. For audio source separation and mixture declipping, the proposed algorithm is shown to be capable of jointly solving these two separate problems which was not possible with any other method in the literature. Joint handling of these problems is also demonstrated to be more effective than sequentially approaching each problem in case of severe distortions. The proposed algorithm is also shown to be highly effective for the reconstruction of randomly subsampled signals such as in the case of compressive sampling approaches. This advantage of our algorithm is further utilised for the problem of informed source separation, to create a compression scheme which uses the principles of compressive sampling and distributed coding. For this application, the proposed algorithm is not only shown to achieve compression performance equivalent to that of the state of the art, but also shown to have unique advantages, specifically having a very simple encoder as well as the decoding stage being independent of the encoding stage.
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Solving Time Domain Audio Inverse Problems using Nonnegative Tensor Factorization

Solving Time Domain Audio Inverse Problems using Nonnegative Tensor Factorization

will often lead to the loss of a huge amount of available information. Indeed, for example in the case of a clipped audio every STFT frame may be clipped, thus this naive solution would lead to considering the whole signal to be missing, even though there is perhaps only 20 % of the signal that is clipped in the time domain. Another problem of NMF/NTF- based audio inpainting methods [10]–[12] which consider fully-missing STFT coefficients is that NMF/NTF models are phase-invariant and thus they only allow estimating the magnitudes of the missing coefficients. As a result, the phase information, which is very important for audio perceptual quality, still needs to be reconstructed somehow. A popular approach by Griffin and Lim [14] is usually used for the phase reconstruction, but it performs quite poorly in many situations. As an alternative, a so-called high resolution NMF (HR-NMF) approach was proposed [15], [16]. This approach extends the NMF to model temporal dependencies between time-frequency bins, which yields better phase estimates. However, for the moment this approach is quite computationally expensive and it is limited to harmonic sounds. At the same time, when some samples are missing in the time domain and one manages to estimate properly the phase-invariant NMF model and the missing samples from these observations, the resulting phase estimates should be better than those obtained via Griffin and Lim’s approach [14], since missing samples in time domain does not mean completely discarding the phase information in the STFT domain.
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Complexity of ten decision problems in continuous time dynamical systems

Complexity of ten decision problems in continuous time dynamical systems

Questions of complexity related to qualitative properties of differential equations have long been of theoretical interest. A natural question one can ask is if the stability of a system of polynomial differential equations can be decided by a Turing machine in finite time. In [4], Arnold made a well- known conjecture that the contrary is true; i.e. the question is undecidable. To the authors’ knowledge, even though some variants of the question have been studied and answered [12], [16], the question in its original form is so far unresolved. Although the results in this paper do not resolve Arnold’s question, they provide lower bounds on the computational complexity of deciding local asymptotic stability and several similar and related problems.
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Short-time regularity for dynamic evolution problems in perfect plasticity

Short-time regularity for dynamic evolution problems in perfect plasticity

The paper is organized as follows: in Section 2, we recall some useful notations and results about the dynamical elasto-plastic problem. Section 3 is devoted to prove our main theorem. The proof of our main theorem follows the lines of the regularity result of [3]. Although the comparison principle and its proof are very similar to Kato’s inequality in [3], the rest of our arguments needs to be adapted to the general case of perfect plasticity. Indeed, due to the fact that we are dealing with any closed and convex elasticity sets K, the dissipation functional H is now a general positively 1-homogeneous and convex function of the plastic strain measure (in contrast with [3] where the specific choice of the elasticity set leads to a dissipation functional which is just the total variation of the plastic strain measure). Thus, a particular care and measure theoretic arguments (due to the use of convex functions of a measure) are needed to prove our short-time regularity result. Finally, a better choice of test functions in the comparison principle allows us to slightly improve the time regularity of the solution.
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Non overlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Non overlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Methods for Time Harmonic Wave Problems Xavier Claeys, Francis Collino, Patrick Joly and Emile Parolin Abstract . The domain decomposition method (DDM) initially designed, with the celebrated paper of Schwarz in 1870 [22] as a theoretical tool for partial differential equations (PDEs) has become, since the advent of the computer and parallel com- puting techniques, a major tool for the numerical solution of such PDEs, especially for large scale problems. Time harmonic wave problems offer a large spectrum of applications in various domains (acoustics, electromagnetics, geophysics, ...) and occupy a place of their own, that shines for instance through the existence of a natu- ral (possibly small) length scale for the solutions: the wavelength. Numerical DDMs were first invented for elliptic type equations (e.g. the Laplace equation), and even though the governing equations of wave problems (e.g. the Helmholtz equation) look similar, standard approaches do not work in general.
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