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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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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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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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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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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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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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]

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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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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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

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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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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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]

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).

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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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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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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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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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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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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