The Experts below are selected from a list of 1494 Experts worldwide ranked by ideXlab platform
Karl Kunisch - One of the best experts on this subject based on the ideXlab platform.
-
optimal control of the linear wave equation by time depending bv controls a semi smooth newton approach
Mathematical Control and Related Fields, 2020Co-Authors: Sebastian Engel, Karl KunischAbstract:An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by \begin{document}$ H^1 $\end{document} functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the \begin{document}$ H^1- $\end{document} regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the Distributional Derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.
-
optimal control of the linear wave equation by time depending bv controls a semi smooth newton approach
arXiv: Optimization and Control, 2018Co-Authors: Sebastian Engel, Karl KunischAbstract:An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by H1 functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the H1 regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the Distributional Derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.
Virginia De Cicco - One of the best experts on this subject based on the ideXlab platform.
-
Nonautonomous Chain Rules in BV with Lipschitz Dependence
Milan Journal of Mathematics, 2016Co-Authors: Virginia De CiccoAbstract:The aim of this paper is to state a nonautonomous chain rule in BV with Lipschitz dependence, i.e., a formula for the Distributional Derivative of the composite function \({v(x) = B(x, u(x))}\), where \({u : \mathbb{R}^N \rightarrow \mathbb{R}}\) is a scalar function of bounded variation, \({B(\cdot, t)}\) has bounded variation and \({B(x, \cdot)}\) is only a Lipschitz continuous function. We present a survey of recent developments on the nonautonomous chain rules in BV. Formulas of this type are an useful tool especially in view to applications to lower semicontinuity for integral functional (see [12, 14, 15, 16]) and to the conservation laws with discontinuous flux (see [8, 10, 11]).
-
a chain rule formula in the space bv and applications to conservation laws
Siam Journal on Mathematical Analysis, 2011Co-Authors: Graziano Crasta, Virginia De CiccoAbstract:In this paper we prove a new chain rule formula for the Distributional Derivative of the composite function $v(x)=B(x,u(x))$, where $u:\mathopen]a,b\mathclose[\to\mathbb{R}^d$ has bounded variation, $B(x,\cdot)$ is continuously differentiable, and $B(\cdot,u)$ has bounded variation. We propose an application of this formula in order to deal in an intrinsic way with the discontinuous flux appearing in conservation laws in one space variable.
-
A chain rule formula in BV and applications to conservation laws
SIAM Journal on Mathematical Analysis, 2011Co-Authors: Graziano Crasta, Virginia De CiccoAbstract:In this paper we prove a new chain rule formula for the Distributional Derivative of the composite function v(x) = B(x,u(x)), where u :)a,b(! R d has bounded variation, B(x,·) is continuously differentiable and B(·,u) has bounded variation. We propose an application of this formula in order to deal in an intrinsic way with the discontinuous flux appearing in conservation laws in one space variable.
Sebastian Engel - One of the best experts on this subject based on the ideXlab platform.
-
optimal control of the linear wave equation by time depending bv controls a semi smooth newton approach
Mathematical Control and Related Fields, 2020Co-Authors: Sebastian Engel, Karl KunischAbstract:An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by \begin{document}$ H^1 $\end{document} functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the \begin{document}$ H^1- $\end{document} regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the Distributional Derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.
-
optimal control of the linear wave equation by time depending bv controls a semi smooth newton approach
arXiv: Optimization and Control, 2018Co-Authors: Sebastian Engel, Karl KunischAbstract:An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by H1 functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the H1 regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the Distributional Derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.
Erik Talvila - One of the best experts on this subject based on the ideXlab platform.
-
The one-dimensional heat equation in the Alexiewicz norm
Advances in Pure and Applied Mathematics, 2015Co-Authors: Erik TalvilaAbstract:AbstractA distribution on the real line has a continuous primitive integral if it is the Distributional Derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock–Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Θ
-
The one-dimensional heat equation in the Alexiewicz norm
arXiv: Analysis of PDEs, 2015Co-Authors: Erik TalvilaAbstract:A distribution on the real line has a continuous primitive integral if it is the Distributional Derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock--Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $\Theta_t(x)=\exp(-x^2/(4t))/\sqrt{4\pi t}$ be the heat kernel. With initial data $f$ that is the Distributional Derivative of a continuous function, it is shown that $u_t(x):=u(x,t):=f\ast\Theta_t(x)$ is a classical solution of the heat equation $u_{11}=u_2$. The estimate $\|f\ast\Theta_t\|_\infty\leq\|f\|/\sqrt{\pi t}$ holds. The Alexiewicz norm is $\|f\|=\sup_I|\int_If|$, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, $\|u_t-f\|\to 0$ as $t\to 0^+$. The solution of the heat equation is unique under the assumptions that $\|u_t\|$ is bounded and $u_t\to f$ in the Alexiewicz norm for some integrable $f$. The heat equation is also considered with initial data that is the $n$th Derivative of a continuous function and in weighted spaces such that $\int_{-\infty}^\infty f(x)\exp(-ax^2)\,dx$ exists for some $a>0$. Similar results are obtained.
-
The L p primitive integral
Mathematica Slovaca, 2014Co-Authors: Erik TalvilaAbstract:For each 1 ≤ p < ∞, a space of integrable Schwartz distributions L′ p , is defined by taking the Distributional Derivative of all functions in L p . Here, L p is with respect to Lebesgue measure on the real line. If f ∈ L′ p such that f is the Distributional Derivative of F ∈ L p , then the integral is defined as \(\int\limits_{ - \infty }^\infty {fG} = - \int\limits_{ - \infty }^\infty {F(x)g(x)dx} \), where g ∈ L q , \(G(x) = \int\limits_0^x {g(t)dt} \) and 1/p + 1/q =1. A norm is ‖f‖ p ′ = ‖F‖ p . The spaces L′ p and L p are isometrically isomorphic. Distributions in L′ p share many properties with functions in L p . Hence, L′ p is reflexive, its dual space is identified with L q , there is a type of Holder inequality, continuity in norm, convergence theorems, Gateaux Derivative. It is a Banach lattice and abstract L-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes L′1 into a Banach algebra isometrically isomorphic to the convolution algebra on L 1. Spaces of higher order Derivatives of L p functions are defined. These are also Banach spaces isometrically isomorphic to L p .
-
Fourier Series with the Continuous Primitive Integral
Journal of Fourier Analysis and Applications, 2012Co-Authors: Erik TalvilaAbstract:Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the Distributional Derivative of a continuous function. This space of distributions is denoted ${\mathcal{A}}_{c}(\mathbb{T})$ and is a Banach space under the Alexiewicz norm, $\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|$ , the supremum being taken over intervals of length not exceeding 2 π . It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L ^1 Fourier series continue to hold for this larger space, with the L ^1 norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form $\hat{f}(n)=o(n)$ as | n |→∞. The convolution is defined for $f\in{\mathcal{A}}_{c}(\mathbb{T})$ and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate $\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}$ . For $g\in L^{1}(\mathbb{T})$ , $\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}$ . As well, $\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n)$ . There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in ${\mathcal{A}}_{c}(\mathbb{T})$ . The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D _ n be the Dirichlet kernel and let $f\in L^{1}(\mathbb{T})$ . Then $\|D_{n}\ast f-f\|_{\mathbb{T}}\to0$ as n →∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.
Gianni Dal Maso - One of the best experts on this subject based on the ideXlab platform.
-
Fine Properties of Functions with Bounded Deformation
Archive for Rational Mechanics and Analysis, 1997Co-Authors: Luigi Ambrosio, Alessandra Coscia, Gianni Dal MasoAbstract:The paper is concerned with the fine properties of functions in , the space of functions with bounded deformation. We analyse the set of Lebesgue points and the set where these functions have one-sided approximate limits. Moreover, following the analogy with , we decompose the symmetric Distributional Derivative into an absolutely continuous part , a jump part , and a Cantor part . The main result of the paper is a structure theorem for functions, showing that these parts of the Derivative can be recovered from the corresponding ones of the one-dimensional sections. Moreover, we prove that functions are approximately differentiable in almost every point of their domain.