scholarly journals OPTIMAL CONTROL FOR A CONTROLLED ILL-POSED WAVE EQUATION WITHOUT REQUIRING THE SLATER HYPOTHESIS

2020 ◽  
Vol 6 (1) ◽  
pp. 84
Author(s):  
Abdelhak Hafdallah
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Missing Data ◽  
Incomplete Data ◽  
Initial Condition ◽  
Control Sequence ◽  
Empty Interior ◽  
Ill Posed ◽  
Well Posed ◽  
Admissible Controls

In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with  ncomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.

scholarly journals Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Pengbin Feng ◽  
Erkinjon T. Karimov
Keyword(s):  
Wave Equation ◽  
Mixed Type ◽  
Orthonormal System ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Initial Time ◽  
Inverse Source Problem ◽  
Ill Posed ◽  
Inverse Source Problems ◽  
Well Posed

AbstractIn the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

No-Regret Optimal Control Characterization for an Ill-Posed Wave Equation

2017 ◽  
Vol 41 (3) ◽  
pp. 283-288 ◽  
Author(s):  
Abdelhak Hafdallah ◽  
Abdelhamid Ayadi ◽  
Chafia Laouar
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Ill Posed

scholarly journals On stochastic optimal control laws

1973 ◽  
Vol 52 ◽  
pp. 1-30 ◽  
Author(s):  
Makiko Nisio
Keyword(s):  
Stochastic Process ◽  
Optimal Control ◽  
Stochastic Optimal Control ◽  
Admissible Control ◽  
Optimal Controls ◽  
Initial Condition ◽  
Control Laws ◽  
The Matrix ◽  
Vector Valued ◽  
Admissible Controls

Let us begin by recalling the existence of optimal controls for a class of stochastic differential equationswith given initial condition X(0) = x, where B is an n-dimensional Brownian motion and the control U is a stochastic process. As admissible controls, let us allow all non-anticipative process U(t) = (U1(t),…Um(t)) ∈ Γ where Γ is a compact subset of Rm. We call Γ a control region. Assume that the matrix valued functional β and the n-vector valued α satisfy a Lipscitz condition in X and some growth condition. Then we have a unique solution Xu for an admissible control U.

scholarly journals Averaged No-Regret Control for an Electromagnetic Wave Equation Depending upon a Parameter with Incomplete Initial Conditions

2021 ◽  
Author(s):  
Abdelhak Hafdallah ◽  
Mouna Abdelli
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Electromagnetic Wave ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Initial Conditions ◽  
Control Sequence ◽  
Potential Term ◽  
Average Control ◽  
Optimality Systems

This chapter concerns the optimal control problem for an electromagnetic wave equation with a potential term depending on a real parameter and with missing initial conditions. By using both the average control notion introduced recently by E. Zuazua to control parameter depending systems and the no-regret method introduced for the optimal control of systems with missing data. The relaxation of averaged no-regret control by the averaged low-regret control sequence transforms the problem into a standard optimal control problem. We prove that the problem of average optimal control admits a unique averaged no-regret control that we characterize by means of optimality systems.

The near-field singularity predicted by the spiral Green’s function in acoustics and electrodynamics

1991 ◽  
Vol 433 (1888) ◽  
pp. 451-459 ◽  
Keyword(s):  
Wave Equation ◽  
Point Source ◽  
Wave Speed ◽  
Near Field ◽  
Radial Component ◽  
Extended Source ◽  
Ill Posed ◽  
Field Singularity ◽  
Theory Of Generalized Functions ◽  
Well Posed

The retarded solution of the wave equation for a point source in circular motion, whose speed exceeds the wave speed, is singular on a spiralling tube-like surface that is at rest in the rest frame of the point source. When solving the wave equation for a corresponding extended source, therefore, we are faced with integrals over the volume of the source which are improper and need to be handled either with the aid of the theory of generalized functions or by Hadamard’s method of finite parts. In this paper, after isolating the finite part of the gradient of the retarded potential due to a rotating extended source, we calculate the asymptotic values of the coefficients in its Fourier representation and show that the radial component of this gradient does not remain finite at those points within the source which move with the wave speed, and so lie on the boundary of the domain of hyperbolicity of the equation of the mixed type to which the wave equation in this case reduces. This latter singularity arises because the problem in question, though well posed physically, is in fact mathematically ill posed.

Convergence and stability analysis of the half thresholding based few-view CT reconstruction

2020 ◽  
Vol 28 (6) ◽  
pp. 829-847
Author(s):  
Hua Huang ◽  
Chengwu Lu ◽  
Lingli Zhang ◽  
Weiwei Wang
Keyword(s):  
Noise Suppression ◽  
Reconstructed Image ◽  
Reference Image ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Reconstruction Algorithms ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Stability ◽  
Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

scholarly journals Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 573
Author(s):  
Marzia Sara Vaccaro ◽  
Francesco Paolo Pinnola ◽  
Francesco Marotti de Sciarra ◽  
Raffaele Barretta
Keyword(s):  
Boundary Conditions ◽  
Structural Engineering ◽  
Beam Deflection ◽  
Soil Reaction ◽  
Differential Problem ◽  
Reaction Field ◽  
Ill Posed ◽  
Elastic Responses ◽  
Benchmark Solutions ◽  
Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

scholarly journals Two Inverse Problems Solution by Feedback Tracking Control

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 137
Author(s):  
Vladimir Turetsky
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Feedback Linearization ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Linear Quadratic Tracking ◽  
Ill Posed ◽  
Quadratic Optimal Control ◽  
Quadratic Tracking

Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates.

scholarly journals PENDUGAAN DATA HILANG DENGAN METODE YATES DAN ALGORITMA EM PADA RANCANGAN LATTICE SEIMBANG

2015 ◽  
Vol 4 (2) ◽  
pp. 74
Author(s):  
MADE SUSILAWATI ◽  
KARTIKA SARI
Keyword(s):  
Missing Data ◽  
Em Algorithm ◽  
Basic Concept ◽  
Incomplete Data ◽  
Likelihood Function ◽  
Animal Husbandry ◽  
Lattice Design ◽  
The Em Algorithm ◽  
Missing Data Estimation ◽  
Data Estimation

Missing data often occur in agriculture and animal husbandry experiment. The missing data in experimental design makes the information that we get less complete. In this research, the missing data was estimated with Yates method and Expectation Maximization (EM) algorithm. The basic concept of the Yates method is to minimize sum square error (JKG), meanwhile the basic concept of the EM algorithm is to maximize the likelihood function. This research applied Balanced Lattice Design with 9 treatments, 4 replications and 3 group of each repetition. Missing data estimation results showed that the Yates method was better used for two of missing data in the position on a treatment, a column and random, meanwhile the EM algorithm was better used to estimate one of missing data and two of missing data in the position of a group and a replication. The comparison of the result JKG of ANOVA showed that JKG of incomplete data larger than JKG of incomplete data that has been added with estimator of data. This suggest  thatwe need to estimate the missing data.

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