Frechet Differentiability Research Articles

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

The solvability of the optimal control problem for a nonlinear Schrödinger equation

In this paper, we analyze the solvability of the optimal control problem for a nonlinear Schr\"{o}dinger equation. A Lions-type functional is considered as the objective functional. First, it is shown that the optimal control problem has at least one solution. Later, the Frechet differentiability of the objective functional is proved and a formula is obtained for its gradient. Finally, a necessary optimality condition is derived.

A Family of Global Attractors for a Class of Generalized Kirchhoff-Beam Equations

The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are proved by prior estimation and Galerkin finite element method. Then the bounded absorption set is obtained by prior estimation, and the family of global attractors for the high-order Kirchhoff-Beam equation is obtained. The Frechet differentiability of the solution semigroup is proved after the linearization of the equation, and the decay of the volume element of the linearization problem is further proved. Finally, the Hausdorff dimension and Fractal dimension of the family of global attractors are proved to be finite.

ОБРАТНАЯ ЗАДАЧА ТИПА УПРАВЛЕНИЯ ОБ ОПРЕДЕЛЕНИИ СТАРШЕГО КОЭФФИЦИЕНТА ОДНОМЕРНОГО ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ

In this paper, we consider one inverse control-type problem of determining the leading coefficient of a one-dimensional parabolic equation. The problem under consideration is a variational statement of a coefficient inverse problem for a parabolic equation. The sought for coefficient of the parabolic equation depends on the spatial variable. An integral boundary condition is set for the parabolic equation. The desired highest coefficient of the parabolic equation plays role of the control function, which is an element of the Sobolev space. The set of admissible control functions belong to the Sobolev space. The objective functional for the control problem is compiled based on the integral overdetermination condition set in the inverse problem. This condition may be interpreted as tasks of a weighted mean of the solution of the equation under consideration as per time variable. The solution of the boundary value problem for a parabolic equation, for each given control function, is defined as a generalized solution from the Sobolev space. The existence of a solution of the considered inverse control-type problem is proved. An adjoint boundary value problem for the control problem under study is introduced. The Frechet differentiability of the objective functional in a set of admissible control functions is proved. In addition, an auxiliary boundary value problem is introduced; and using the solution of this problem, an expression for the gradient of the objective functional is found. The necessary optimality condition for the admissible control function is obtained.

Optimal control of anisotropic Allen-Cahn equations

This paper aims at solving an optimal control problem governed by an anisotropic Allen-Cahn equation numerically. Therefor we first prove the Frechet differentiability of an in time discretized parabolic control problem under certain assumptions on the involved quasilinearity and formulate the first order necessary conditions. As a next step, since the anisotropies are in general not smooth enough, the convergence behavior of the optimal controls is studied for a sequence of (smooth) approximations of the former quasilinear term. In addition the simultaneous limit in the approximation and the time step size is considered. For a class covering a large variety of anisotropies we introduce a certain regularization and show the previously formulated requirements. Finally, a trust region Newton solver is applied to various anisotropies and configurations, and numerical evidence for mesh independent behavior and convergence with respect to regularization is presented.

Shape optimization in elliptic problem with nonlinear boundary conditions

Shape optimization problems have a long history of mathematical study and a wide range of applications. In recent decades there has been an interest in solving these problems with partial differential equation (PDE) constraints. In practice we often meet problems, leading to the problem of finding an “optimal” shape of systems, the behaviour of which is described by elliptic equations. Problems of this type have been studied by many authors from mathematical as well as from computation point of view. When surfaces of a specimen which have been damaged by a corrosion aggressive attack are not accessible to direct inspection, one is forced to rely on over-determined measurements performed on the accessible part of the boundary. In this study, we consider such a non-destructive inspection technique modelled as a shape optimization problem, which consists of determining an unknown part of the boundary of a simply-connected bounded domain. The solution of Laplace equation $u$ satisfies nonlinear Robin condition on an unknown part of boundaty. We have Dirichlet and Neumann boundary conditions on the rest of boundary. We look for a curve, that minimizes the cost functional represented as a sum of regularizing term and $L_2$-norm of the difference of the state variable $u$ and some additional measurement on the part of admissible boundary. The existence of at least one curve is proved for an appropriate choice of the class of admissible curves. This will lead us to study properties of continuity of J and compactness of the set of controls. We shall characterize the shape derivative of the cost functional with respect to perturbations of the domain defined by a sufficiently smooth function. In order obtain necessary optimality condition we use the mapping method. This method transforms the unknown domain to a fixed one. The problem under consideration is essentially nonlinear. We prove Frechet differentiability of the solution operator.

Regularization in Banach Spaces with Respect to the Bregman Distance

The Moreau envelope, also known as Moreau–Yosida regularization, and the associated proximal mapping have been widely used in Hilbert and Banach spaces. They have been objects of great interest for optimizers since their conception more than half a century ago. They were generalized by the notion of the D-Moreau envelope and D-proximal mapping by replacing the usual square of the Euclidean distance with the conception of Bregman distance for a convex function. Recently, the D-Moreau envelope has been developed in a very general setting. In this article, we present a regularizing and smoothing technique for convex functions defined in Banach spaces. We also investigate several properties of the D-Moreau envelope function and its related D-proximal mapping in Banach spaces. For technical reasons, we restrict our attention to the Lipschitz continuity property of the D-proximal mapping and differentiability properties of the D-Moreau envelope function. In particular, we prove the Frechet differentiability property of the envelope and the Lipschitz continuity property of its derivative.

Some Notes on the Differentiability of the Support Function

Let X be a Banach space, C be a nonempty closed convex set and $$\sigma _C$$ be the support function of the set C. In this work, we give some necessary and sufficient conditions for the set C to satisfy $$\mathrm {int}(\mathrm {dom}\sigma _C)\ne \emptyset $$ , which enable us to study the Frechet and Gateaux differentiability of support function on the set C.

The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping

The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.

Identification of the initial temperature from the given temperature data at the left end of a rod

Abstract This study aims to investigate the problem of determining the unknown initial temperature in a variable coefficient heat equation. We obtain the existence and uniqueness of the solution of the optimal control problem considered under some conditions. Using the adjoint problem approach, we get the Frechet differential of the cost functional. We construct a minimizing sequence and give the convergence rate of this sequence. Also, we test the theoretical results in a numerical example by using the MAPLE® program.

Strong Existence and Higher Order Fréchet Differentiability of Stochastic Flows of Fractional Brownian Motion Driven SDEs with Singular Drift

In this paper we present a new method for the construction of strong solutions of SDE’s with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter $$H<\frac{1}{2}$$ . Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE’s in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a “local time variational calculus”. We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

Dependence of solutions and eigenvalues of third order linear measure differential equations on measures

This paper deals with a complex third order linear measure differential equation $${\rm{id}}{(y')^ \bullet} + 2{\rm{i}}q(x)y'{\rm{d}}x + y({\rm{id}}q(x) + {\rm{d}}p(x)) = \lambda y{\rm{d}}x$$ on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. The dependence of eigenvalues on the coefficients p and q is investigated. We prove that the n-th eigenvalue is continuous in p and q when the norm topology of total variation and the weak* topology are considered. Moreover, the Frechet differentiability of the n-th eigenvalue in p and q with the norm topology of total variation is also considered. To deduce these conclusions, we investigate the dependence of solutions of the above equation on the coefficients p and q with different topologies and establish the counting lemma of eigenvalues according to the estimates of solutions.

Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models

Motivated by topics and issues critical to human health, the problem studied in this work derives from the modeling and stabilizing control of electrical cardiac activity in order to maximize the efficiency and safety of treatment for cardiac disease. In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission. The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays. The concept of our minimax control approach consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and disturbances play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain data to maintain the system in a desired state. First, the mathematical model is introduced and its well-posedness is studied. Second, the minimax control problem is formulated. Afterwards the Frechet differentiability of nonlinear solution map from the couple control-disturbance input to the solution of state system is assessed as well as stability of the derived sensitive system. The existence of an optimal solution is proved and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus.

Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” by the same authors, general well-posedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous Cahn–Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A, B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in the quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Frechet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.

Optimal nonlinearity control of Schrödinger equation

We study the optimal nonlinearity control problem for the nonlinear Schrodinger equation \begin{document} $iu_{t} = -\triangle u+V(x)u+h(t)|u|^α u$ \end{document} , which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well-posedness of the equation for \begin{document} $0 , we show the existence of the optimal control. The Frechet differentiability of the objective functional is proved, and the first order optimality system for \begin{document} $N≤ 3$ \end{document} is presented.

Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations, II

The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreĭn trace formula derived. The aim of the present paper is to prove the Frechet differentiability of operator Bernstein functions and the Livschits-Kreĭn trace formula in the multidimensional setting.

A functional calculus of closed operators on Banach space. III. certain topics of perturbation theory

The present paper is a continuation of research of A. A. Atvinovskii and of the author in the area of functional calculus of closed operators on Banach spaces based on Markov and related functions as symbols. The following topics in the perturbation theory are considered: Estimates of bounded perturbations of operator functions with respect to general operator ideal norms, Lipschitz property, moment inequality, Frechet differentiability, analyticity of operator functions under consideration with respect to the perturbation parameter, spectral shift function, and Lifshits–Krein trace formula.

Fréchet differentiability of molecular distribution functions I. $$L^\infty $$ L ∞ analysis

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space, we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the corresponding molecular distribution functions. For certain admissible perturbations of the pair potential and sufficiently small activity, we rigorously establish Frechet differentiability with respect to the supremum norm in the image space—both for bounded domains and in the thermodynamical limit.

Fréchet differentiability of molecular distribution functions II: the Ursell function

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the thermodynamical limit of the Ursell or pair correlation function. For certain admissible perturbations of the pair potential and sufficiently small activity we rigorously establish Frechet differentiability of the Ursell function in the $L^1$ norm. Furthermore, concerning the thermodynamical limit of the pair distribution function we explicitly compute its Fr\'echet derivative as a sum of a multiplication operator and an integral operator.

Frechet differentiability in Besov spaces in the optimal control of parabolic free boundary problems

Abstract We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.