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Abstract: Abstract In this paper, we show existence and uniqueness of solutions of the infinite horizon McKean–Vlasov FBSDEs using two different methods, which lead to two different sets of assumptions. We use these results to solve the infinite horizon mean field type control problems and mean field games. PubDate: 2022-11-07

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Abstract: Abstract We study the stochastic Hamilton–Jacobi–Bellman (HJB) equation with jump, which arises from a non-Markovian optimal control problem with a recursive utility cost functional. The solution to the equation is a predictable triplet of random fields. We show that the value function of the control problem, under some regularity assumptions, is the solution to the stochastic HJB equation; and a classical solution to this equation is the value function and characterizes the optimal control. With some additional assumptions on the coefficients, an existence and uniqueness result in the sense of Sobolev space is shown by recasting the stochastic HJB equation as a backward stochastic evolution equation in Hilbert spaces with the Brownian motion and Poisson jump. PubDate: 2022-11-07

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Abstract: Abstract In this work, we study an initial value control problem for a doubly nonlinear PDE perturbed by multiplicative Lévy noise. We first establish wellposedness of a weak martingale solution. Monotonicity arguments have been exploited in the proofs. A path-wise uniqueness of weak martingale solutions is settled via the standard \(L^1\) -method. We formulate the associated control problem, and establish existence of a weak optimal solution of the underlying problem via variational approach. PubDate: 2022-11-07

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Abstract: Abstract In this paper, we consider the Korteweg-de Vries equation posed on the periodic domain \({\mathbb {T}}\) . We show that the Korteweg-de Vries equation is globally approximately controllable by a two dimensional external force. The proof is based on the Agrachev-Sarychev approach in geometric control theory. PubDate: 2022-11-07

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Abstract: Abstract We consider the optimal control of a system governed by the Navier–Stokes equations with stochastic Dirichlet boundary conditions. Control conditions imposed only on the boundary are associated with reduced regularity of the system, as compared to distributed controls. To ensure the well-posedness of the solutions and the efficiency of numerical simulations, the stochastic boundary conditions and controls are required to belong almost surely to the Sobolev space of functions having first order weak derivative along the boundary. To simulate the system, numerical solutions are approximated using the stochastic collocation/finite element approach with sparse grid techniques and Monte Carlo methods which are applied to the boundary random field. An optimality system is derived for a matching-type cost functional. Error estimates are derived for the optimal state, the adjoint state and boundary control variables. Numerical examples for the deterministic cases are provided and compared in which the controls are applied on a part of or on the whole boundary. Simulations for the stochastic cases are also made with sparse grid and Monte Carlo methods to retrieve the statistical information of the optimal solution. PubDate: 2022-11-07

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Abstract: Abstract This article presents a new Carleman inequality for a linear Schrödinger equation which is suitable for both bounded and unbounded domains. We characterize the conditions on the auxiliary function necessary to obtain the global inequality. The novelty of this result is the construction of the auxiliary function on some unbounded domains and for a corresponding valid control region \(\omega \) . As a consequence, we prove some results on the controllability of a linear Schrödinger equation on unbounded domains. PubDate: 2022-11-07

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Abstract: Abstract We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects. PubDate: 2022-11-07

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Abstract: Abstract This article proposes a new discrete framework for approximating solutions to two dimensional shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework. PubDate: 2022-11-07

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Abstract: Abstract In this paper, we study three-dimensional Kirchhoff equations with critical growth and singular nonlinearity. We are concerned with the qualitative analysis of solutions to the following nonlocal problem $$\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+b\displaystyle \int _{\Omega } \nabla u ^2dx\right) \Delta u=\lambda u^{-\gamma }+u^5, &{}\mathrm {in}\ \ \Omega , \\ u>0, &{}\mathrm {in}\ \ \Omega , \\ u=0, &{}\mathrm {on}\ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^3\) is a bounded domain with smooth boundary, \(0<\gamma <1\) , and \(a,\,b,\,\lambda \) are positive constants. By combining variational methods with some delicate decomposition techniques, we obtain the existence of two positive solutions in the case of low perturbations of the singular nonlinearity, namely for small values of the parameter \(\lambda \) . PubDate: 2022-11-07

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Abstract: Abstract The flow of nematic liquid crystals can be described by a highly nonlinear stochastic hydrodynamical model, thus is often influenced by random fluctuations, such as uncertainty in specifying initial conditions and boundary conditions. In this article, we consider a 2-D stochastic nematic liquid crystals with the velocity field perturbed by affine-linear multiplicative white noise, with random initial data and random boundary conditions. Our main objective is to obtain the global well-posedness of the stochastic equations under the sufficient Malliavin regularity of the initial condition. The Malliavin calculus techniques play important roles when we obtain the global existence of the solutions to the stochastic nematic liquid crystal model with random initial and boundary conditions. PubDate: 2022-11-07

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Abstract: Abstract The purpose of this paper is to provide a systematic discussion of a generalized barycenter based on a variant of unbalanced optimal transport (UOT) that defines a distance between general non-negative, finitely supported measures by allowing for mass creation and destruction modeled by some cost parameter. They are denoted as Kantorovich–Rubinstein (KR) barycenter and distance. In particular, we detail the influence of the cost parameter to structural properties of the KR barycenter and the KR distance. For the latter we highlight a closed form solution on ultra-metric trees. The support of such KR barycenters of finitely supported measures turns out to be finite in general and its structure to be explicitly specified by the support of the input measures. Additionally, we prove the existence of sparse KR barycenters and discuss potential computational approaches. The performance of the KR barycenter is compared to the OT barycenter on a multitude of synthetic datasets. We also consider barycenters based on the recently introduced Gaussian Hellinger–Kantorovich and Wasserstein–Fisher–Rao distances. PubDate: 2022-11-07

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Abstract: Abstract This paper deals with the analysis of the internal control of a free-boundary problem for the 1D heat equation with local and nonlocal nonlinearities. We prove a local null controllability result with distributed controls, locally suported in space. The proof is based on Schauder’s fixed point theorem combined with some appropriate specific estimates. PubDate: 2022-11-07

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Abstract: Abstract Vector-borne diseases, in particular arboviruses, represent a major threat to human health. In the fight against these viruses, the endosymbiotic bacterium Wolbachia has become in recent years a promising tool as it has been shown to prevent the transmission of some of these viruses between mosquitoes and humans. In this work, we investigate optimal population replacement strategies, which consists in replacing optimally the wild population by a population carrying the aforementioned bacterium, making less likely the appearance of outbreaks of these diseases. We consider a two species model taking into account both wild and Wolbachia infected mosquitoes. To control the system, we introduce a term representing an artificial introduction of Wolbachia-infected mosquitoes. Assuming a high birth rate, we reduce the model to a simpler one regarding the proportion of infected mosquitoes. We study strategies optimizing a convex combination either of cost and time or cost and final proportion of mosquitoes in the population. We fully analyze each of the introduced problem families, proving a time monotonicity property on the proportion of infected mosquitoes and using a reformulation of the problem based on a suitable change of variable. Our results are useful in considerably more general contexts which we present. PubDate: 2022-11-07

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Abstract: Abstract We investigate the existence and stability of cylindrical symmetric static solutions to the Landau–Lifshitz equation for multidirectional ferromagnets in the three-dimensional (3D) cylindrical surfaces \({\varvec{H_c}} = (x,y,\sqrt{{c^2} - {x^2} - {y^2}} )\) , supplemented with Dirichlet boundary condition. After introducing the cylindrical coordinates transformation, the Landau–Lifshitz equation in Cartesian coordinates, existing a multi-direct effective field, is transformed into that in cylindrical symmetric coordinates, and then the existence of static solutions of the cylindrical symmetric model is proved. The theory of elliptic PDE of second order and the Schauder fixed-point theorem are fundamental ingredients in the existence analysis. Here, the two most notable things are to cope with the multi-direct effective field and the increased terms derived from the transformation. What’s more, we observe that the linearized operator mentioned in this paper can’t be extended to the self-adjoint operator so that the spectral problem on the linearized operator is so complicated, and thus we need to provide more careful estimates to establish the Lyapunov stability of cylindrical symmetric solutions of the time-dependent Landau–Lifshitz equation in the 3D cylindrical surfaces \({\varvec{H_c}}\) . PubDate: 2022-11-07

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Abstract: Abstract In recent work (Lohmann et al. in J Math Pures Appl, 2022, https://doi.org/10.1016/j.matpur.2022.05.021, Theorem 1.3.4) we have shown the equivalence of the widely used nonconvex (generalized) branched transport problem with a shape optimization problem of a street or railroad network, known as (generalized) urban planning problem. The argument was solely based on an explicit construction and characterization of competitors. In the current article we instead analyse the dual perspective associated with both problems. In more detail, the shape optimization problem involves the Wasserstein distance between two measures with respect to a metric depending on the street network. We show a Kantorovich–Rubinstein formula for Wasserstein distances on such street networks under mild assumptions. Further, we provide a Beckmann formulation for such Wasserstein distances under assumptions which generalize our previous result in [16]. As an application we then give an alternative, duality-based proof of the equivalence of both problems under a growth condition on the transportation cost, which reveals that urban planning and branched transport can both be viewed as two bilinearly coupled convex optimization problems. PubDate: 2022-10-14

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Abstract: Abstract We consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin’s maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide numerical examples in which the SHM problem is solved by means of our approach. PubDate: 2022-10-13

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Abstract: Abstract In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochastic process. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. On the one hand, it transforms the infinite-dimensional optimization problem into a finite-dimensional one. On the other hand, it allows us to provide the well-posedness of the problem and approximation. Finally, the results are illustrated with numerical results, where classical model for the growth of a population are considered. PubDate: 2022-10-13

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Abstract: Abstract We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on \(\lambda \) units of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. We study the impact of risk aversion on the interaction between the producer and the consumer as well as on the derivative price. In mathematical terms, we are dealing with a two-player linear-quadratic McKean–Vlasov stochastic differential game. Using methods based on the martingale optimality principle and BSDEs, we find a Nash equilibrium and characterize the corresponding strategies and payoffs in semi-explicit form. Furthermore, we compute the two indifference prices (one for the producer and one for the consumer) induced by that equilibrium and we determine the quantity \(\lambda \) such that the players agree on the price. Finally, we illustrate our results with some numerics. In particular, we focus on how the risk aversions and the volatility control costs of the players affect the derivative price. PubDate: 2022-09-14 DOI: 10.1007/s00245-022-09907-7

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Abstract: Abstract Local exponential stabilization of the three-dimensional Navier–Stokes system to a given reference trajectory via receding horizon control (RHC) is investigated. The RHC enters as the linear combinations of a finite number of actuators. The actuators are spatial functions and can be chosen in particular as indicator functions whose supports cover only a part of the spatial domain. PubDate: 2022-09-14 DOI: 10.1007/s00245-022-09900-0

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Abstract: Abstract In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation. PubDate: 2022-09-14 DOI: 10.1007/s00245-022-09908-6