Inverse Theorem Research Articles

Local controllability of reaction-diffusion systems around nonnegative stationary states

We consider an×nnonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝN. This system models reversible chemical reactions. We act on the system throughmcontrols (1 ≤m<n), localized in some arbitrary nonempty open subsetωof the domainΩ. We prove the local exact controllability to nonnegative (constant) stationary states in any timeT> 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole spaceL∞(Ω)n. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in aL∞-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system.

Internal null controllability of the generalized Hirota-Satsuma system

The generalized Hirota-Satsuma system consists of three coupled nonlinear Korteweg-de Vries (KdV) equations. By using two distributed controls it is proven in this paper that the local null controllability property holds when the system is posed on a bounded interval. First, the system is linearized around the origin obtaining two decoupled subsystems of third order dispersive equations. This linear system is controlled with two inputs, which is optimal. This is done with a duality approach and some appropriate Carleman estimates. Then, by means of an inverse function theorem, the local null controllability of the nonlinear system is proven.

Adiabatic Invariants of Herglotz Type for Perturbed Nonconservative Lagrangian Systems

Both conservative and nonconservative systems can be studied simultaneously using the differential variational principle of Herglotz type. For a perturbed system, in which parameters change with time, it is useful to find adiabatic invariants. Based on the Herglotz differential variational principle, we study the perturbation of infinitesimal transformations and adiabatic invariants for perturbed nonconservative Lagrangian systems. From the generalized Euler-Lagrange equation and the invariance condition for the Hamilton-Herglotz action under the group of infinitesimal transformations, we obtain an exact invariant of Herglotz type for a holonomic nonconservative system. We propose a definition of higher-order adiabatic invariants of Herglotz type and obtain such adiabatic invariants for nonconservative Lagrangian systems with small perturbations. We prove the corresponding inverse theorem of adiabatic invariants. As examples of using the obtained results, we consider an oscillator with square damping and a system with two degrees of freedom.

Approximation by trigonometric polynomials in weighted Morrey spaces

In this paper we investigate the best approximation by trigonometric polynomials in weighted Morrey spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$, where the weight function $w$ is in the Muckenhoupt class $A_{p}(I_{0})$ with $1 < p < \infty$ and $I_{0}=[0, 2\pi]$. We prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces $\mathcal{\widetilde{M}}_{p,\lambda}(I_{0},w)$ the closure of $C^{\infty}(I_{0})$ in $\mathcal{M}_{p,\lambda}(I_{0},w)$. We give the characterization of $K-$functionals in terms of the modulus of smoothness and obtain the Bernstein type inequality for trigonometric polynomials in the spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$.

A Time-dependent Interaction Problem Between an Electromagnetic Field and an Elastic Body

This paper considers the scattering of a time-dependent electromagnetic plane wave by a bounded elastic body. An appropriate decomposition of the coupling interface conditions is proposed according to the Voigt’s model between the electromagnetic and elastic medium. The original unbounded scattering problem is equivalently reduced into an initial-boundary value problem in a bounded domain by introducing an exact transparent boundary condition (TBC) on a sufficiently large sphere. Making use of the Lax-Milgram lemma, the abstract inversion theorem of Laplace transform and the energy method, we verify the well-posedness and stability for the reduced problem. Moreover, a priori estimates are established for the electromagnetic field and elastic displacement by taking special test functions directly in the time domain variational formulation.

A new type of adiabatic invariants for disturbed Birkhoffian system of Herglotz type

The generalized variational principle of Herglotz type provides an effective way to study the problems of conservative and non-conservative systems in a unified way. According to the differential variational principle of Herglotz type, we study the adiabatic invariants for a disturbed Birkhoffian system in this paper. Firstly, the differential equations of motion of the Birkhoffian system based upon this variational principle are given, and the exact invariant of Herglotz type of the system is introduced. Secondly, a new type of adiabatic invariants for the system under the action of small perturbation is obtained. Thirdly, the inverse theorem of adiabatic invariant for the disturbed Birkhoffian system of Herglotz type is obtained. Finally, an example is given.

Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M

AbstractIn the Orlicz type spaces 𝓢M, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and PeetreK-functionals in the spaces 𝓢M.

Local null controllability of coupled degenerate systems with nonlocal terms and one control force

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanuvilov, is obtained from the global null controllability of the linearized system provided by Lyusternik's Inverse Mapping Theorem. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.

Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

Some Theorems of Approximation Theory in Weighted Smirnov Classes with Variable Exponent

Let $${ G\subset {\mathbb {C}} }$$ be a Jordan domain with rectifiable Dini smooth boundary $$\varGamma $$. In this work, we investigate approximation properties of matrix transforms constructed via Faber series in weighted Smirnov classes with variable exponent. Moreover, direct and inverse theorems of approximation theory in weighted Smirnov classes with variable exponent are proved and some results related to constructive characterization in generalized Lipschitz classes are obtained.

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

This paper studies the internal control of the Korteweg–de Vries–Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear.

A new type of adiabatic invariants for disturbed non-conservative nonholonomic system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212, 11272227, and 10972151) and the Innovation Program for Postgraduade in Higher Education Institutions of Jiangsu Province, China (Grant No. KYCX18 2548).

According to the Herglotz variational principle and differential variational principle of Herglotz type, we study the adiabatic invariants for a non-conservative nonholonomic system. Firstly, the differential equations of motion of the non-conservative nonholonomic system based upon the generalized variational principle of Herglotz type are given, and the exact invariant for the non-conservative nonholonomic system is introduced. Secondly, a new type of adiabatic invariant for the system under the action of a small perturbation is obtained. Thirdly, the inverse theorem of the adiabatic invariant is given. Finally, an example is given.

Multidimensional fractional Fourier transform and generalized fractional convolution

ABSTRACTIn this paper, we prove inversion theorems and Parseval identity for the multidimensional fractional Fourier transform. Analogous to the existing fractional convolutions on functions of single variable, we also introduce a generalized fractional convolution on functions of several variables and we derive their properties including convolution theorem and product theorem for the multidimensional fractional Fourier transform.

De Finetti Theorems for Braided Parafermions

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).

Existence and uniqueness of the Karcher mean on unital C⁎-algebras

The Karcher mean on the cone Ω of invertible positive elements of the C⁎-algebra B(E) of bounded operators on a Hilbert space E has recently been extended to a contractive barycentric map on the space of L1-probability measures on Ω. In this paper we first show that the barycenter satisfies the Karcher equation and then establish the uniqueness of the solution. Next we establish that the Karcher mean is real analytic in each of its coordinates, and use this fact to show that both the Karcher mean and Karcher barycenter map exist and are unique on any unital C⁎-algebra. The proof depends crucially on a recent result of the author giving a converse of the inverse function theorem.

Temperature Fields Generated by a Circular Heat Source: Solution of a Composite Solid of Two Different Isotropic Semi-Infinite Media

Abstract The problem of a three-dimensional (3D) heat flow from a circular heat source (CHS) embedded inside a composite solid of two isotropic but different semi-infinite media is solved for the first time in this paper. This CHS asymmetrical measurement setup is useful when two identical samples are not available for measurement. Two different time-dependent temperature fields are derived for the composite semi-infinite media, as well as their corresponding heat fluxes. The derivation of the 3D solution uses first principles with basic assumptions and employs the Hankel and Laplace transforms. The Laplace inversion theorem is used to find the inverse Laplace transform of the temperature functions, since no tabulated inverse transform functions are available for this case. The solution is exact with no approximations and is given in an integral form, which can easily be evaluated numerically. This solution is a generic one and can be applied to more complex asymmetrical setups, such as the case involving thermal contact resistances.

Approximation theorems for multivariate Taylor-Abel-Poisson means

We obtain direct and inverse approximation theorems of functions of several variables by Taylor-Abel-Poisson means in the integral metrics. We also show that norms of multipliers in the spaces $L_{p,Y}(\mathbb T^d)$ are equivalent for all positive integers $d.$

Fourier Transform Inversion Using an Elementary Differential Equation and a Contour Integral

Let f be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that f is absolutely continuous such that f and are Lebesgue integrable. A function g is defined by . This differential equation has a well-known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires straightforward manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little machinery. With only minor changes the method is also used to prove the inversion theorem for functions of several variables and to prove Riemann’s localization theorem.

Simplifying coefficients in differential equations for generating function of Catalan numbers

In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients in two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers.

A coupling rational finite element-boundary element method for a laterally loaded pile in transversely isotropic poroelastic soils

A coupling rational finite element-boundary element method for the interaction between a laterally loaded pile and multilayered transversely isotropic poroelastic soils is developed in this study. Based on the Timoshenko beam theory, the pile is modeled by the rational finite element method (RFEM), which can avoid shear locking. The extended precise integration method is adopted to obtain the consolidation solution of transversely isotropic poroelastic soils, which is used as the kernel function of the boundary element method (BEM). By coupling of BEM and RFEM, the interaction equations between the pile and soils are established and solved in the Laplace-Hankel transform domain, and the actual solution is obtained by the numerical inversion theorem. Several examples are tested to verify the theory and to investigate the key parameters of the laterally loaded pile in transversely isotropic poroelastic soils.