Normalized solution to Schrödinger system with Bopp–Podolsky-type self-attraction

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

There is much current interest in general equality constrained quadratic programming problems, both for their own sake and for their applicability to active set methods for nonlinear programming. In the former case, typically, the issues are existence of solutions and their determination. In the latter instance, nonexistence of solutions gives rise to directions of infinite descent. Such directions may subsequently be used to determine a more desirable active set.The generalised Cholesky decomposition of relevant matrices enables us to answer the question of existence and to determine directions of infinite descent (when applicable) in an efficient and stable manner.The methods examined are related to implementations that are suitable for null-space, range-space and Lagrangian methods.

In dynamical systems with a large number of agents, competitive, and cooperative phenomenaoccur in a broad range of designed and natural settings. Such as communications, environmental,biological, transportation, trading, and energy systems, and they underlie mucheconomic and financial behavior. Analysis of such systems is intractable using the classicalfinite N-players game theoretic methods is often intractable. The mean-field games (MFG)framework was developed to study these large systems, modeling them as a continuum ofrational agents that interact in a non-cooperative way.In this thesis, we address some theoretical aspects and propose a definition of relaxedsolution for MFG that allows establishing uniqueness under minimal regularity hypothesis.We also propose a price impact model, that is a modification of the Merton’s portfolio problemwhere we consider that assets’ transactions influence their prices.We also study numerical methods for continuous time finite-state MFG that satisfy amonotonicity condition, and for time-dependent first-order nonlocal MFG. MFG is determinedby a system of differential equations with initial and terminal boundary conditions. Thesenon-standard conditions make the numerical approximation of MFG difficult. Using themonotonicity condition, we build a flow that is a contraction and whose fixed points solveboth for stationary and time-dependent MFG.We also develop Fourier approximation methods for the solutions of first-order nonlocalmean-field games (MFG) systems. Using Fourier expansion techniques, we approximate agiven MFG system by a simpler one that is equivalent to a convex optimization problem overa finite-dimensional subspace of continuous curves. We solve this problem using a variant ofa primal-dual hybrid gradient method.Finally, we introduce a price-formation model where a large number of small players canstore and trade electricity. Our model is a constrained MFG where the price is a Lagrangemultiplier for the supply versus demand balance condition. We establish the existence of aunique solution using a fixed-point argument. Then, we study linear-quadratic models thathold specific solutions, and we find that the dynamic price depends linearly on the instantaggregated consumption.

Uniform array synthesis is treated as a problem of approximation in the mean square sense and in the Sobolev sense are considered. Following Tichonov's regularization technique for solving inverse problems , the array excitation current is subjected to a certain constraint in order to make the design less-sensitive to construction errors. A variational formulation of the problem using lagrange multiplier method is presented. The equation determining the excitation current is obtained in a general operator form and is then put in a form amenable to numerical computation. Existence and uniqueness of solution are proved. Some example problems are presented.

We prove the existence and uniqueness of weak solutions of the stationary magneto-micropolar equations with mixed boundary conditions for velocity, including Navier slip condition. We study an optimal boundary control problem associated to weak solutions of these equations. By using the Lagrange multipliers method, we obtain first-order necessary conditions from which we derive an optimality system.

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Asymptotic behaviors of normalized solutions for a class of Choquard equations

In this paper, the existence and uniqueness of solutions to a coupled formally symmetric system of fractional differential equations with nonlinear p-Laplacian operator and nonlinear fractional differential-integral boundary conditions are obtained by using the matrix eigenvalue method. The Hyers–Ulam stability of the coupled formally symmetric system is also presented with certain growth conditions. By using the topological degree theory and nonlinear functional analysis methods, some sufficient conditions for the existence and uniqueness of solutions to this coupled formally symmetric system are established. Examples are provided to verify our results.

Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations

In this paper we propose a well-balanced finite volume/augmented Lagrangian method for compressible visco-plastic models focusing on a compressible Bingham type system with applications to dense avalanches. For the sake of completeness we also present a method showing that such a system may be derived for a shallow flow of a rigid-viscoplastic incompressible fluid, namely for incompressible Bingham type fluid with free surface. When the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics, see for instance [N.J. Balmforth et al., J. Fluid Mech (2002)]. When the motion is a little bit quicker, shallow water theory for non-Newtonian flows may be applied, for instance assuming a Navier type boundary condition at the bottom. We start from the variational inequality for an incompressible Bingham fluid and derive a shallow water type system. In the case where Bingham number and viscosity are set to zero we obtain the classical Shallow Water or Saint-Venant equations obtained for instance in [J.F. Gerbeau, B. Perthame, DCDS (2001)]. For numerical purposes, we focus on the one-dimensional in space model: We study associated static solutions with sufficient conditions that relate the slope of the bottom with the Bingham number and domain dimensions. We also propose a well-balanced finite volume/augmented Lagrangian method. It combines well-balanced finite volume schemes for spatial discretization with the augmented Lagrangian method to treat the associated optimization problem. Finally, we present various numerical tests.

Viscosity solutions of nonlinear integro-differential equations

We investigate a class of fractional Schrödinger–Poisson systems with critical exponents and steep potential well. By using the constraint variational method and a quantitative deformation lemma, the existence of positive ground state solutions and sign-changing ground state solutions is obtained. It is shown that the energy of the sign-changing ground state solution is strictly larger than twice the energy of the positive ground state solution. Moreover, we also study the asymptotic behavior of the sign-changing ground state

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution for all λ ≠ 0 and sufficiently small ε > 0. Moreover, for each λ ≠ 0 we show that uε converges to the positive ground state solution of the associated limit problem and concentrates to a maximum point of b(x) in certain sense as ε → 0. Furthermore, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.