Publications

2021

• Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems
  
  • Bonazzoli Marcella
  • Claeys Xavier
  • Nataf Frédéric
  • Tournier Pierre-Henri
  Journal of Scientific Computing, Springer Verlag, 2021, 89. We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though). (10.1007/s10915-021-01631-8)
  DOI : 10.1007/s10915-021-01631-8
• EXISTENCE, UNIQUENESS AND REGULARITY FOR THE STOCHASTIC ERICKSEN-LESLIE EQUATION
  
  • de Bouard Anne
  • Hocquet Antoine
  • Prohl Andreas
  Nonlinearity, IOP Publishing, 2021. We investigate existence and uniqueness for the liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in L p-based spaces, for every p > 2. Thanks to a bootstrap principle together with a Gyöngy-Krylov-type compactness argument, this will ultimately lead us to prove the existence of a particular class of global solutions which are partially regular, strong in the probabilistic sense, and taking values in the "critical space" L 2 × H 1 . (10.1088/1361-6544/ac022e)
  DOI : 10.1088/1361-6544/ac022e
• Outliers Detection in Networks with Missing Links
  
  • Gaucher Solenne
  • Klopp Olga
  • Robin Geneviève
  Computational Statistics and Data Analysis, Elsevier, 2021, 164, pp.107308. Outliers arise in networks due to different reasons such as fraudulent behavior of malicious users or default in measurement instruments and can significantly impair network analyses. In addition, real-life networks are likely to be incompletely observed, with missing links due to individual non-response or machine failures. Identifying outliers in the presence of missing links is therefore a crucial problem in network analysis. In this work, we introduce a new algorithm to detect outliers in a network that simultaneously predicts the missing links. The proposed method is statistically sound: we prove that, under fairly general assumptions, our algorithm exactly detects the outliers, and achieves the best known error for the prediction of missing links with polynomial computation cost. It is also computationally efficient: we prove sub-linear convergence of our algorithm. We provide a simulation study which demonstrates the good behavior of the algorithm in terms of outliers detection and prediction of the missing links. We also illustrate the method with an application in epidemiology, and with the analysis of a political Twitter network. The method is freely available as an R package on the Comprehensive R Archive Network. (10.1016/j.csda.2021.107308)
  DOI : 10.1016/j.csda.2021.107308
• Some EM-type algorithms for incomplete data model building
  
  • Lavielle Marc
  , 2021. We propose an extension of the EM algorithm and its stochastic versions for the construction of incomplete data models when the selected model minimizes a penalized likelihood criterion. This optimization problem is particularly challenging in the context of incomplete data, even when the model is relatively simple. However, by completing the data, the E-step of the algorithm allows us to simplify this problem of complete model selection into a classical problem of complete model selection that does not pose any major difficulties. We then show that the criterion to be minimized decreases with each iteration of the algorithm. Examples of the use of these algorithms are presented for the identification of regression mixture models and the construction of nonlinear mixed-effects models.
• Regenerative properties of the linear Hawkes process with unbounded memory
  
  • Graham Carl
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2021, 31 (6), pp.2844-2863. We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function, which may have unbounded support. These results are applicable to sliding window statistical estimators. We exploit independence in the Poisson cluster point process decomposition, and the regeneration times are not stopping times for the Hawkes process. The regeneration time is interpreted as the renewal time at zero of a M/G/infinity queue, which yields a formula for its Laplace transform. When the transfer function admits some exponential moments, we stochastically dominate the cluster length by exponential random variables with parameters expressed in terms of these moments. This yields explicit bounds on the Laplace transform of the regeneration time in terms of simple integrals or special functions yielding an explicit negative upper-bound on its abscissa of convergence. These regenerative results allow, e.g., to systematically derive long-time asymptotic results in view of statistical applications. This is illustrated on a concentration inequality previously obtained with coauthors. (10.1214/21-AAP1664)
  DOI : 10.1214/21-AAP1664
• Numerical simulation of rigid particles in Stokes flow: lubrication correction for general shapes of particles
  
  • Lefebvre-Lepot Aline
  • Nabet Flore
  Mathematical Modelling of Natural Phenomena, EDP Sciences, 2021, 16, pp.45. We address the problem of numerical simulation of suspensions of rigid particles in a Stokes flow. We focus on the inclusion of the singular short range interaction effects (lubrication effects) in the simulations when the particles come close one to another. The problem is solved without introducing new hypothesis nor model. As in Lefebvre-Lepot et al. [J. Fluid Mech. 769 (2015) 369–386], the key idea is to decompose the velocity and pressure flows in a sum of a singular and a regular part. In this article, the singular part is computed using an explicit asymptotic expansion of the solution when the distance goes to zero. This expansion is similar to the asymptotic expansion proposed in Hillairet and Kelai [Asymptotic Anal. 95 (2015) 187–241] but is more appropriate for numerical simulations of suspensions. It can be computed for any locally convex (that is the particles have to be convex close to the contact point) and regular shape of particles. Using Hillairet and Kelai [Asymptotic Anal. 95 (2015) 187–241] as an intermediate result, we prove that the remaining part is regular in the sense that it is bounded independently of the distance. As a consequence, only a small number of degrees of freedom are necessary to obtain accurate results. The method is tested in dimension 2 for clusters of two or three aligned particles with general rigid velocities. We show that, as expected, the convergence is independent of the distance. (10.1051/mmnp/2021037)
  DOI : 10.1051/mmnp/2021037
• A Non-Nested Infilling Strategy for Multi-Fidelity based Efficient Global Optimization
  
  • Sacher Matthieu
  • Le Maitre Olivier
  • Duvigneau Régis
  • Hauville Frédéric
  • Durand Mathieu
  • Lothode C.
  International Journal for Uncertainty Quantification, Begell House Publishers, 2021, 11 (1), pp.1-30. Efficient Global Optimization (EGO) has become a standard approach for the global optimization of complex systems with high computational costs. EGO uses a training set of objective function values computed at selected input points to construct a statistical surrogate model, with low evaluation cost, on which the optimization procedure is applied. The training set is sequentially enriched, selecting new points, according to a prescribed infilling strategy, in order to converge to the optimum of the original costly model. Multi-fidelity approaches combining evaluations of the quantity of interest at different fidelity levels have been recently introduced to reduce the computational cost of building a global surrogate model. However, the use of multi-fidelity approaches in the context of EGO is still a research topic. In this work, we propose a new effective infilling strategy for multi-fidelity EGO. Our infilling strategy has the particularity of relying on non-nested training sets, a characteristic that comes with several computational benefits. For the enrichment of the multi-fidelity training set, the strategy selects the next input point together with the fidelity level of the objective function evaluation. This characteristic is in contrast with previous nested approaches, which require estimation all lower fidelity levels and are more demanding to update the surrogate. The resulting EGO procedure achieves a significantly reduced computational cost, avoiding computations at useless fidelity levels whenever possible, but it is also more robust to low correlations between levels and noisy estimations. Analytical problems are used to test and illustrate the efficiency of the method. It is finally applied to the optimization of a fully nonlinear fluid-structure interaction system to demonstrate its feasibility on real large-scale problems, with fidelity levels mixing physical approximations in the constitutive models and discretization refinements. (10.1615/Int.J.UncertaintyQuantification.2020032982)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2020032982
• Coupling techniques for nonlinear hyperbolic equations. II. Resonant interfaces with internal structure
  
  • Boutin Benjamin
  • Coquel Frédéric
  • Lefloch Philippe G.
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2021, 16 (2), pp.283-315. In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions. (10.3934/nhm.2021007)
  DOI : 10.3934/nhm.2021007
• A weak solution theory for stochastic Volterra equations of convolution type
  
  • Jaber Eduardo Abi
  • Cuchiero Christa
  • Larsson Martin
  • Pulido Sergio
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2021, 31 (6), pp.2924-2952. We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in $L^p$ spaces. The main tools are new a priori estimates on Sobolev--Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.
• Log-Sobolev Inequality for the Continuum Sine-Gordon Model
  
  • Bauerschmidt Roland
  • Bodineau Thierry
  Commun.Pure Appl.Math., 2021, 74 (10), pp.2064-2113. We derive a multiscale generalisation of the Bakry-Émery criterion for a measure to satisfy a log-Sobolev inequality. Our criterion relies on the control of an associated PDE well-known in renormalisation theory: the Polchinski equation. It implies the usual Bakry-Émery criterion, but we show that it remains effective for measures that are far from log-concave. Indeed, using our criterion, we prove that the massive continuum sine-Gordon model with β < 6π satisfies asymptotically optimal log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. (10.1002/cpa.21926)
  DOI : 10.1002/cpa.21926
• Transmission eigenvalues for multipoint scatterers
  
  • Grinevich Piotr
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2021, 9 (4), pp.17–25. We study the transmission eigenvalues for the multipoint scatterers of the Bethe-Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions d = 2 and d = 3. We show that for these scatterers: 1) each positive energy E is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex E is an interior transmission eigenvalue of infinite multiplicity. The case of dimension d = 1 is also discussed. (10.32523/2306-6172-2021-9-4-17-25)
  DOI : 10.32523/2306-6172-2021-9-4-17-25
• Flag-approximability of convex bodies and volume growth of Hilbert geometries
  
  • Vernicos Constantin
  • Walsh Cormac
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2021, 54, pp.1297-1315. We show that the volume entropy of a Hilbert geometry on a convex body is exactly twice the flag-approximability of the body. We then show that both of these quantities are maximized in the case of the Euclidean ball. We also compute explicitly the asymptotic volume of a convex polytope, which allows us to prove that simplices have the least asymptotic volume, as was conjectured by the first author. (10.24033/asens.2482)
  DOI : 10.24033/asens.2482
• A Bayesian Approach for Quantile Optimization Problems with High-Dimensional Uncertainty Sources
  
  • Sabater Christian
  • Le Maitre Olivier
  • Congedo Pietro Marco
  • Görtz Stefan
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2021, 376, pp.113632. Robust optimization strategies typically aim at minimizing some statistics of the uncertain objective function and can be expensive to solve when the statistic is costly to estimate at each design point. Surrogate models of the uncertain objective function can be used to reduce this computational cost. However, such surrogate approaches classically require a low-dimensional parametrization of the uncertainties, limiting their applicability. This work concentrates on the minimization of the quantile and the direct construction of a quantile regression model over the design space, from a limited number of training samples. A Bayesian quantile regression procedure is employed to construct the full posterior distribution of the quantile model. Sampling this distribution, we can assess the estimation error and adjust the complexity of the regression model to the available data. The Bayesian regression is embedded in a Bayesian optimization procedure, which generates sequentially new samples to improve the determination of the minimum of the quantile. Specifically, the sample infill strategy uses optimal points of a sample set of the quantile estimator. The optimization method is tested on simple analytical functions to demonstrate its convergence to the global optimum. The robust design of an airfoil’s shock control bump under high-dimensional geometrical and operational uncertainties serves to demonstrate the capability of the method to handle problems with industrial relevance. Finally, we provide recommendations for future developments and improvements of the method. (10.1016/j.cma.2020.113632)
  DOI : 10.1016/j.cma.2020.113632
• Concurrent shape optimization of the part and scanning path for additive manufacturing
  
  • Boissier Mathilde
  • Allaire Grégoire
  • Tournier Christophe
  , 2021.
• Reciprocal association between participation to a national election and the epidemic spread of COVID-19 in France: nationwide observational and dynamic modeling study.
  
  • Zeitoun Jean-David
  • Faron Matthieu
  • Manternach Sylvain
  • Fourquet Jerome
  • Lavielle Marc
  • Lefevre Jeremie
  European Journal of Public Health, Oxford University Press (OUP): Policy B - Oxford Open Option D, 2021. Objective: To investigate possible reciprocal associations between the intensity of the COVID-19 epidemic in France and the level of participation at national elections. Design: Observational study and dynamic modelling using a sigmoidal mixed effects model. Setting: All hospitals where patients were admitted for COVID-19. Participants: All admitted patients from March 18, 2020 to April 17, 2020. Main outcome measures: Abstention and admission rate for COVID-19. Results: Mean abstention rate in 2020 among departments was 52.5%+/-6.4 and had increased by a mean of 18.8% as compared with the 2014 election. There was a high degree of similarity of abstention between the two elections among the departments (p<0.001). Among departments with a high outbreak intensity before the election, those with a higher participation were not affected by a significantly higher number of COVID-19 admissions after the elections. The sigmoidal model fitted the data from the different departments with a high degree of consistency. The covariate analysis showed that a significant association between participation and number of admitted patients was observed for both elections (2020: B=-5.36, p<1e-9 and 2014: B=-3.15, p<1e-6) contradicting a direct specific causation of the 2020 election. Participation was not associated with the position of the inflexion point suggesting no effect in the speed of spread. Conclusions: Our results suggest that the surrounding intensity of the COVID-19 epidemic in France did not have any local impact on citizens participation to a national election. The level of participation to the 2020 election had no impact on the spread of the pandemic. (10.1101/2020.05.14.20090100)
  DOI : 10.1101/2020.05.14.20090100
• Phase-field approximation for a class of cohesive fracture energies with an activation threshold
  
  • Chambolle Antonin
  • Crismale Vito
  Advances in Calculus of Variation, Walter de Gruyter GmbH, 2021, 14 (4), pp.475-497. We study the Γ-limit of Ambrosio-Tortorelli-type functionals Dε(u, v), whose dependence on the symmetrised gradient e(u) is different in Au and in e(u) − Au, for a C-elliptic symmetric operator A, in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano in [43]. In particular we prove that G(S)BD functions with bounded A-variation are (S)BD. (10.1515/acv-2019-0018)
  DOI : 10.1515/acv-2019-0018
• Detecting seed bank influence on plant metapopulation dynamics
  
  • Louvet Apolline
  • Machon Nathalie
  • Mihoub Jean‐baptiste
  • Robert Alexandre
  Methods in Ecology and Evolution, 2021, 12 (4), pp.655-664. Seed banks are known to play a key role in plant metapopulations. However, detecting seed banks remains challenging and requires intense monitoring efforts. Assessing the genuine effect of seed banks on plant metapopulation dynamics (rather than their presence) may offer a much easier while still biologically relevant way to overcome this issue. In this study, we developed a new metric: the seed bank characteristic event (SBCE) probability. Instead of detecting seed bank directly, the SBCE probability measures seed bank contribution to the observed metapopulation dynamics. Exploring seed bank parameters (colonization, germination and seed bank death probabilities, initial proportion of patches containing a seed bank), a wide range of monitoring durations (from 3 to 10 years) and number of patches in the metapopulation (from 10 to 1,000 patches), we examined the conditions under which the SBCE probability is correctly estimated. To test the robustness of our approach, we further introduced false negatives, false positives or parameter heterogeneity between patches. Finally, we applied the SBCE probability method to the monitoring of tree bases plant species in Paris, France, to assess the applicability of the method to real‐world datasets and increase the understanding of plant metapopulation dynamics within an urban environment. Our results indicate that the SBCE probability is well‐estimated when enough monitoring years or number of patches are considered, and for probabilities of false negatives or false positives of up to 0.1. However, the SBCE probability estimation is not robust to colonization probability heterogeneity between patches. When we applied the SBCE probability method to the real monitoring dataset, we found a contrasted contribution of the seed bank to the observed metapopulation dynamics from one street and one species to another. The study suggests that the measurement of seed bank contribution is less data‐demanding than assessment of seed bank presence. Applying the estimation method to the monitoring of tree bases plant species highlights a significant contribution of the seed bank to plant metapopulation dynamics in an urban environment, and illustrates how the method can be applied on real‐world datasets. (10.1111/2041-210x.13547)
  DOI : 10.1111/2041-210x.13547
• VARIATIONAL APPROXIMATION OF INTERFACE ENERGIES AND APPLICATIONS
  
  • Amstutz Samuel
  • Gourion Daniel
  • Zabiba Mohammed
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2021, 23 (23), pp.59-102. Minimal partition problems consist in finding a partition of a domain into a given number of components in order to minimize a geometric criterion. In applicative fields such as image processing or continuum mechanics, it is standard to incorporate in this objective an interface energy that accounts for the lengths of the interfaces between components. The present work is focused on the theoretical and numerical treatment of minimal partition problems with such interface energies. The considered approach is based on a Γ-convergence approximation combined with convex analysis techniques. (10.4171/IFB/450)
  DOI : 10.4171/IFB/450
• Fast Incremental Expectation Maximization for finite-sum optimization: nonasymptotic convergence
  
  • Fort Gersende
  • Gach Pierre
  • Moulines Eric
  Statistics and Computing, Springer Verlag (Germany), 2021, 31 (48). Fast Incremental Expectation Maximization (FIEM) is a version of the EM framework for large datasets. In this paper, we first recast FIEM and other incremental EM type algorithms in the {\em Stochastic Approximation within EM} framework. Then, we provide nonasymptotic bounds for the convergence in expectation as a function of the number of examples $n$ and of the maximal number of iterations $\kmax$. We propose two strategies for achieving an $\epsilon$-approximate stationary point, respectively with $\kmax = O(n^{2/3}/\epsilon)$ and $\kmax = O(\sqrt{n}/\epsilon^{3/2})$, both strategies relying on a random termination rule before $\kmax$ and on a constant step size in the Stochastic Approximation step. Our bounds provide some improvements on the literature. First, they allow $\kmax$ to scale as $\sqrt{n}$ which is better than $n^{2/3}$ which was the best rate obtained so far; it is at the cost of a larger dependence upon the tolerance $\epsilon$, thus making this control relevant for small to medium accuracy with respect to the number of examples $n$. Second, for the $n^{2/3}$-rate, the numerical illustrations show that thanks to an optimized choice of the step size and of the bounds in terms of quantities characterizing the optimization problem at hand, our results desig a less conservative choice of the step size and provide a better control of the convergence in expectation. (10.1007/s11222-021-10023-9)
  DOI : 10.1007/s11222-021-10023-9
• Qualitative indicator functions for imaging crack networks using acoustic waves
  
  • Audibert Lorenzo
  • Chesnel Lucas
  • Haddar Houssem
  • Napal Kevish
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2021. We consider the problem of imaging a crack network embedded in some homogeneous background from measured multi-static far field data generated by acoustic plane waves. We propose two novel approaches that can be seen as extensions of linear sampling-type methods and that provide indicator functions which are sensitive to local cracks densities. The first approach uses multiple frequencies data to compute spectral signatures associated with artificially embedded localized obstacles. The second approach also exploits the idea of incorporating an artificial background but uses data for a single frequency. The indicator function is built using a similar concept as for differential sampling methods: compare the solution of the interior transmission problem for healthy inclusion with the one with embedded cracks. The performance of the methods is tested and discussed on synthetic examples and the numerical results are compared with the ones obtained using the classical factorization method. (10.1137/20M134650X)
  DOI : 10.1137/20M134650X
• Approximating the Total Variation with Finite Differences or Finite Elements
  
  • Chambolle Antonin
  • Pock Thomas
  , 2021, 22, pp.383--417. We present and compare various types of discretizations which have been proposed to approximate the total variation (mostly, of a grey-level image in two dimensions). We discuss the properties of finite differences and finite elements based approach and compare their merits, in particular in terms of error estimates and quality of the reconstruction. (10.1016/bs.hna.2020.10.005)
  DOI : 10.1016/bs.hna.2020.10.005
• A semi-supervised method for the characterization of degradation of nuclear power plants steam generators
  
  • Pinciroli Luca
  • Baraldi Piero
  • Shokry Ahmed
  • Zio Enrico
  • Seraoui Redouane
  • Mai Carole
  Progress in Nuclear Energy, Elsevier, 2021, 131, pp.103580. (10.1016/j.pnucene.2020.103580)
  DOI : 10.1016/j.pnucene.2020.103580
• Size matters for OTC market makers: General results and dimensionality reduction techniques
  
  • Bergault Philippe
  • Guéant Olivier
  Mathematical Finance, Wiley, 2021, 31 (1), pp.279-322. (10.1111/mafi.12286)
  DOI : 10.1111/mafi.12286
• ELASTOPLASTIC TOPOLOGY OPTIMIZATION AND CYCLICALLY LOADED STRUCTURES VIA DIRECT METHODS FOR SHAKEDOWN
  
  • Boissier Mathilde
  • Deaton Joshua
  • Beran Philip
  • Vermaak Natasha
  Structural and Multidisciplinary Optimization, Springer Verlag, 2021, 64 (1), pp.189-217. For the first time, the lower bound shakedown theorem is integrated into a level set based topol-ogy optimization framework to identify lightweight elastoplastic designs. Shakedown is a cyclic elastoplastic behavior in which, upon cycling beyond the elastic limit, the accumulation of plastic strain arrests and purely elastic behavior is recovered. In contrast to most elastoplastic toplogy optimization, the use of a lower bound shakedown limit allows elastoplastic shakedown limits to be rigorously estimated using only the elastic solution. Under small deformations assumptions, this amounts to solving one simple partial differential equation, avoiding the non-linearity associated with plasticity, and thus simplifying the resolution process. Numerical results are provided for several benchmark examples. The results highlight the design performance enhancements attributed to allowing elastoplastic shakedown to occur instead of designing to first yield. In particular, up to 10% reduction in weight is found for the simple structures considered. (10.1007/s00158-021-02875-6)
  DOI : 10.1007/s00158-021-02875-6
• State-constrained control-affine parabolic problems II: second order sufficient optimality conditions
  
  • Aronna Maria Soledad
  • Frédéric Bonnans Joseph
  • Kröner Axel
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2021. In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform. (10.1137/19M1286906)
  DOI : 10.1137/19M1286906