Controllability and Optimization Problems

UEFISCDI research project PN-II-RU-TE-2014-4-1109

Publications

Published articles

N. Cîndea, S. Micu, I. Roventa. Boundary controllability for finite-differences semi-discretizations of a clamped beam equation, SIAM J. Cont. Optim., 55(2017),785-817. pdf

This article deals with the boundary observability properties of a space finite-differences semi-discretization of the clamped beam equation. We make a detailed spectral analysis of the system and, by combining numerical estimates with asymptotic expansions, we localize all the eigenvalues of the corresponding discrete operator depending on the mesh size $h$. Then, an Ingham's type inequality and a discrete multiplier method allow us to deduce that the uniform (with respect to $h$) observability property holds if and only if the eigenfrequencies are filtered out in the range ${\cal O}\left(1/h^4\right)$.

N. Cîndea, A. Münch. Simultaneous reconstruction of the solution and the source of hyperbolic equations from boundary measurements: a robust numerical approach, Inverse Problems, Vol. 32, Number 11, 36 pp, (2016). pdf

We introduce a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations posed in $\Omega\times (0,T)$ - $\Omega$ a bounded subset of $\mathbb{R}^N$. We consider the simultaneous reconstruction of both the state and the source term from a partial boundary observation. We employ a least-squares technique and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discuss several examples in the one and two dimensional case.

D. Yang Gao, P. Neff, I. Roventa, C. Thiel. On the Convexity of Nonlinear Elastic Energies in the Right Cauchy-Green Tensor, Journal of Elasticity 127 (2) (2017), 303-308. pdf

We present a sufficient condition under which a weak solution of the Euler-Lagrange equations in nonlinear elasticity is already a global minimizer of the corresponding elastic energy functional. This criterion is applicable to energies $W(F) = \widehat{W}(F^T F) = \widehat{W}(C)$ which are convex with respect to the right Cauchy-Green tensor $C = F^T F$, where $F$ denotes the gradient of deformation. Examples of such energies exhibiting a blow up for $\det F \to 0$ are given.

I. Roventa. Hardy–Littlewood–Polya’s Inequality and a New Concept of Weak Majorization, Mediterranean Journal of Mathematics 13 (2016), 573–583. pdf

In this paper, we study some weak majorization properties with applications for the trees. A strongly notion of majorization is introduced and Hardy–Littlewood–Polya’s inequality is generalized.

C. P. Niculescu, I. Roventa. Hardy-Littlewood-Polya theorem of majorization in the framework of generalized convexity, Carpathian Journal of Mathematics 33, no. 1, (2017), 87-95. pdf

Based on a new concept of generalized relative convexity, a largee xtension of Hardy-Littlewood-Pólya theorem of majorization is obtained. Several applications escaping the classical framework of convexity are included.

A. Madeo, P. Neff , I.-D. Ghiba, G. Rosi. Reflection and transmission of elastic waves in non-local band-gap metamaterials: A comprehensive study via the relaxed micromorphic model, Journal of the Mechanics and Physics of Solids 95 (2016), 441–479. pdf

In this paper we propose to study wave propagation, transmission and reflection in band-gap mechanical metamaterials via the relaxed micromorphic model. To do so, guided by a suitable variational procedure, we start deriving the jump duality conditions to be imposed at surfaces of discontinuity of the material properties in non-dissipative, linear-elastic, isotropic, relaxed micromorphic media. Jump conditions to be imposed at surfaces of discontinuity embedded in Cauchy and Mindlin continua are also presented as a result of the application of a similar variational procedure. The introduced theoretical framework subsequently allows the trans- parent set-up of different types of micro-macro connections granting the description of both (i) internal connexions at material discontinuity surfaces embedded in the considered continua and, as a particular case, (ii) possible connections between different (Cauchy, Mindlin or relaxed micromorphic) continua. The established theoretical framework is general enough to be used for the description of a wealth of different physical situations and can be used as reference for further studies involving the need of suitably connecting different continua in view of (meta-) structural design. In the second part of the paper, we focus our attention on the case of an interface between a classical Cauchy continuum on one side and a relaxed micromorphic one on the other side in order to perform explicit numerical simulations of wave reflection and transmission. This particular choice is descriptive of a specific physical situation in which a classical material is connected to a phononic crystal. The reflective properties of this particular interface are numerically investigated for different types of possible micro-macro connections, so explicitly showing the effect of different boundary conditions on the phenomena of reflec- tion and transmission. Finally, the case of the connection between a Cauchy continuum and a Mindlin one is presented as a numerical study, so showing that band-gap description is not possible for such continua, in strong contrast with the relaxed micromorphic case.

P. Neff, I. Munch, I.D. Ghiba, A. Madeo., On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush. International Journal of Solid and Structures 81, pp. 122-128, 2016 pdf

In a series of papers which are either published [Hadjesfandiari, A., Dargush, G. F., 2011a. Couple stress theory for solids. Int. J. Solids Struct. 48 (18), 2496–2510; Hadjesfandiari, A., Dargush, G. F., 2013. Fundamental solutions for isotropic size-dependent couple stress elasticity. Int. J. Solids Struct. 50 (9), 1253–1265.] or available as preprints [Hadjesfandiari, A., Dargush, G. F., 2010. Polar continuum mechanics. Preprint arXiv:1009.3252; Hadjesfandiari, A. R., Dargush, G. F., 2011b. Couple stress theory for solids. Int. J. Solids Struct. 48, 2496–2510; Hadjesfandiari, A. R., 2013. On the skew-symmetric character of the couple-stress tensor. Preprint arXiv:1303.3569; Hadjesfandiari, A. R., Dargush, G. F., 2015a. Evolution of generalized couple-stress continuum theories: a critical analysis. Preprint arXiv:1501.03112; Hadjesfandiari, A. R., Dargush, G. F., 2015b. Foundations of consistent couple stress theory. Preprint arXiv:1509.06299] Hadjesfandiari and Dargush have reconsidered the linear indeterminate couple stress model. They are postulating a certain physically plausible split in the virtual work principle. Based on this postulate they claim that the second-order couple stress tensor must always be skew-symmetric. Since they do not consider that the set of boundary conditions intervening in the virtual work principle is not unique, their statement is not tenable and leads to some misunderstandings in the indeterminate couple stress model. This is shown by specifying their development to the isotropic case. However, their choice of constitutive parameters is mathematically possible and we show that it still yields a well-posed boundary value problem.

P. Neff, I.D. Ghiba, Loss of ellipticity in additive logarithmic finite strain plasticity. International Journal of Non-Linear Mechanics 81, pp. 122-128, 2016 pdf

In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy $F \mapsto W ( F ) = \widehat W( \log U )$ defined in terms of logarithmic strain $\log U$, where $U = \sqrt{F^T F}$, happens to be everywhere rank-one convex as a function of $F$, the new function $F \mapsto \widetilde W( F ) = W^ (\log U − \log U_p)$ need not remain rank-one convex at some given plastic stretch $U_p$ (viz. $E_p^{\log} := \log U_p$). This is in complete contrast to multiplicative plasticity (and infinitesimal plasticity) in which $F \mapsto W ( F F_p^{−1} )$ remains rank-one convex at every plastic distortion $F_p$ if $F \mapsto W ( F )$ is rank-one convex ( $\nabla u \mapsto \| \text{sym } \nabla u − \varepsilon_p\|^2$ remains convex). We show this disturbing feature of the additive logarithmic plasticity model with the help of a recently introduced family of exponentiated Hencky energies.

I.-L. Stăncuț, I. D. Stîrcu, Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces. Opuscula Mathematica vol. 36, no. 1 (2016), 103-122 pdf

A compact Riemann surface $X$ of genus $g>1$ which has a conformal automorphism $\rho$ of prime order $p$ such that the orbit space $X/\langle \rho \rangle$ is the Riemann sphere is called cyclic $p$-gonal. Exceptional points in the moduli space $M_g$ of compact Riemann surfaces of genus $g$ are unique surface classes whose full group of conformal automorphisms acts with a triangular signature. We study symmetries of exceptional points in the cyclic $p$-gonal locus in $M_g$ for which $\text{Aut}(X)/\langle \rho \rangle$ is a dihedral group $D_n$.

N. Cîndea, S. Micu, I. Roventa. Uniform Observability for a Finite Differences Discretization of a Clamped Beam Equation, IFAC-PapersOnLine 49-8 (2016), 315–320. pdf

The aim of this paper is to prove a uniform observability inequality for a finite differences semi-discretization of a clamped beam equation. A discrete multiplier method is employed in order to obtain the uniform observability of the eigenvectors of the matrix driving the semi-discrete system, corresponding to eigenfrequencies smaller than a precise filtering threshold. This result can be generalized to the uniform observability of every filtered solution. Numerical simulations, concerning the dual controllability problem, illustrate the theoretical results.

I.-L. Stancut. On the existence of infinitely many solutions of a nonlinear Neumann problem involving the m-Laplace operator, Annals of the University of Craiova, Mathematics and Computer Science Series 42, no. 2, (2015), 402-426. pdf

This paper surveys the existence of infinitely many solutions of a nonlinear Neumann problem involving the m-Laplace operator, where the constant m satisfies certain alternative inequalities, and some functions $f(x,u)$ and $g(x,u)$ continuous on $\overline{\Omega}\times\mathbb{R}$ and on $\partial\Omega\times\mathbb{R}$, respectively, and odd with respect to u. We work on a domain $\Omega$ bounded in $\mathbb{R}^{N}$ with smooth boundary. More specifically, we demonstrate the existence of a sequence of solutions which diverge to infinity provided that the nonlinear term is locally superlinear and the existence of a sequence of solutions which converge to zero provided that the nonlinear term is locally sublinear.

Julien Aniort, Laurent Chupin, Nicolae Cindea, Mathematical model of calcium exchange during hemodialysis using a citrate containing dialysate, acepted in Mathematical Medicine and Biology a journal of IMA. pdf

In this paper we propose a mathematical model for the calcium exchange during hemodialysis. This model combines a first part describing the flows of two fuids, blood and dialysate, in a dialyser fiber to a second part which tackle the chemical reactions between several chemical species presented in these fluids. The model governing the fluid flows is obtained by asymptotic analysis and takes into account the anisotropy of fibers of a dialyzer. Several rheologies for blood are proposed to highlight the diferences in flow. The fluid velocity field drives the convective part in the reaction- diffusion system, modelling the exchange of five chemical species present in blood and dialysate. Finally, several numerical experiments illustrate this model emphasizing the calcium balance for a citrate containing dialysate.

I. Munch, P. Neff, A. Madeo, I.-D. Ghiba, The modified couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless, Zeitschrift für Angewandte Mathematik und Mechanik. pdf

We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical hypothesis, implying that the couple stress tensor is traceless and may be symmetric anyway. To this aim, the origin of couple stress is discussed on the basis of certain properties of the total stress itself. In contrast to classical continuum mechanics, the balance of linear momentum and the balance of angular momentum are formulated at an infinitesimal cube considering the total stress as linear and quadratic approximation of a spatial Taylor series expansion.

Submitted works

M. Malin, I. Roventa. Equilibrium problems in the context of relative convexity. pdf

In this paper we solve a generalized equilibrium problem and a Nash equilibrium existence theorem is obtained. The key point is given by an extension of the classical Ky Fan inequality into the framework of rela- tive convexity. More precisely, we prove that Ky Fan type mini-max inequal- ity holds even outside the framework of convexity conditions, namely relative convexity assumptions. The strategy used appeals to a Knaster-Kuratowsky- Mazurkievich argument.

S. Micu, I. Roventa, L. E. Temereanca: Approximation of the controls for the wave equation with a potential. pdf

This article deals with the approximation of the boundary controls of of a 1-D linear wave equation with a potential by using a finite difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh-size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal $L^2$-norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.

Works in progress

Pierre Lissy, Ionel Roventa: Optimal filtration for the approximation of boundary controls for a problem involving fractional Laplacian.

In general, the high frequency numerical spurious oscillations of the solutions of a system lead to a loss of the uniform (with respect to the mesh-size) controllability property of the semi-discrete model. For a very general problem involving fractional Laplacian, by filtering the high frequencies of the initial data in an optimal range, we try to restore the uniform controllability property. The key point is to use a similar multiplier with the one introduced in Pierre Lissy, Ionel Roventa, Optimal filtration for the approximation of boundary controls for the one-dimensional wave equation, https://hal.archives-ouvertes.fr/hal-01338619/, where we have considered a finite-diferences semi-discrete scheme for the approximation of boundary controls in the case of the one-dimensional wave equation. We obtain a relation between the range of filtration and the minimal time of control needed to ensure the uniform controllability, recovering in many cases the usual minimal time to control the (continuous) wave equation. Our aim is to obtain similar results for a more general problem involving fractional Laplacian, covering the cases of wave, heat and beam equations.

Maria Malin, Ionel Roventa, Mihai Adrian Tudor, Iterate polygons in global NPC spaces.

Constantin Niculescu, Ionel Roventa, Weak majorization in global NPC spaces.

Ionel Roventa, Laurentiu Temereanca, Schur convexity properties of the even degree complete homogeneous symmetric polynomials.