Boundary optimal feedback controller for time-periodic Stokes–Oseen flows

This work considered the Dirichlet boundary optimal control of time-periodic Stokes–Oseen equations. The existence of optimal solution and maximum principle are obtained without assuming that the normal component of the control is equal to zero. Moreover, we get the regularity result of the optimal solution via the Euler–Lagrange system. The existence of solution to the HJB equation is proved. The feedback form of the optimal controller is given, and with this feedback controller, we can get for the solution to the periodic Navier–Stokes equations the property of continuous dependence on the outer force term.     

Second-order nonlinear impulsive evolution equations with time-varying generating operators and optimal controls

In this article, the second-order nonlinear impulsive evolution differential equations with time-varying generating operators is considered. Constructing evolution systems generated by time-varying operator matrix, we introduce suitable mild solution of the second-order nonlinear impulsive evolution differential equations. The existence and uniqueness of the mild solutions and the continuous dependence on initial value are proved. The existence of the optimal controls for a Lagrange problem of the systems governed by the second-order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.     

An Evolutionary Boundary Value Problem

We consider a nonlinear initial boundary value problem in a two-dimensional rectangle. We derive variational formulation of the problem which is in the form of an evolutionary variational inequality in a product Hilbert space. Then, we establish the existence of a unique weak solution to the problem and prove the continuous dependence of the solution with respect to some parameters. Finally, we consider a second variational formulation of the problem, the so-called dual variational formulation, which is in a form of a history-dependent inequality associated with a time-dependent convex set. We study the link between the two variational formulations and establish existence, uniqueness, and equivalence results.

     

Existence of finitely optimal solutions for infinite-horizon optimal control problems

In this paper, we investigate the existence of finitely optimal solutions for the Lagrange problem of optimal control defined on [0, ) under weaker convexity and seminormality hypotheses than those of previous authors. The notion of finite optimality has been introduced into the literature as the weakest of a hierarchy of types of optimality that have been defined to permit the study of Lagrange problems, arising in mathematical economics, whose cost functions either diverge or are not bounded below. Our method of proof requires us to analyze the continuous dependence of finite-interval Lagrange problems with respect to a prescribed terminal condition. Once this is done, we show that a finitely optimal solution can be obtained as the limit of a sequence of solutions to a sequence of corresponding finite-horizon optimal control problems. Our results utilize the convexity and seminormality hypotheses which are now classical in the existence theory of optimal control.This research forms part of the author's doctoral dissertation written at the University of Delaware, Newark, Delaware under the supervision of Professor Thomas S. Angell.     

扩散方程的随机Dirichlet问题

令D。表示d+1维欧氏空间R。d的有界子集.旨在用概率方法利用时空布朗运动探讨D。上如下扩散方程的随机D irich let问题:12Δu(x。(t))+q(x。(t))u(x。(t))=tu(x。(t)),x。(t)∈D。(*)其中q是给定的定义在D。上的有界Ho。lder连续函数.本文解决了上述扩散方程(*)的随机Dirichlet问题的解在S3内存在性及唯一性问题.     

The Willmore boundary problem

We consider the Euler–Lagrange equation of the Willmore functional coupled with Dirichlet and Neumann boundary conditions on a given curve. We prove existence of a branched solution, and for Willmore energy < 8π, we prove the existence of a smooth, embedded solution.     

On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints

A family of convex optimal control problems that depend on a real parameterh is considered. The optimal control problems are subject to state space constraints.It is shown that under some regularity conditions on data the solutions of these problems as well as the associated Lagrange multipliers are directionally-differentiable functions of the parameter.The respective right-derivatives are given as the solution and respective Lagrange multipliers for an auxiliary quadratic optimal control problem subject to linear state space constraints.If a condition of strict complementarity type holds, then directional derivatives become continuous ones.     

A New Pareto Optimal Solution in a Lagrange Decomposable Multi-Objective Optimization Problem

In this paper a committee decision-making process of a convex Lagrange decomposable multi-objective optimization problem, which has been decomposed into various subproblems, is studied. Each member of the committee controls only one subproblem and attempts to select the optimal solution of this subproblem most desirable to him, under the assumption that all the constraints of the total problem are satisfied. This procedure leads to a new solution concept of a Lagrange decomposable multi-objective optimization problem, called a preferred equilibrium set. A preferred equilibrium point of a problem, for a committee, may or may not be a Pareto optimal point of this problem. In some cases, a non-Pareto optimal preferred equilibrium point of a problem, for a committee, can be considered as a special type of Pareto optimal point of this problem. This fact leads to a generalization of the Pareto optimality concept in a problem.     

Differential stability of solutions to convex,control constrained optimal control problems

A family of convex, control constrained optimal control problems that depend on a real parameter is considered. It is shown that under some regularity conditions on data the solutions of these problems, as well as the associated Lagrange multipliers are directionally differentiable with respect to parameter. The respective right-derivatives are given as the solution and the associated Lagrange multipliers for some quadratic optimal control problem. If a condition of strict complementarity type hold, then directional derivatives become continuous ones.     

Cahn-Hilliard方程的拟谱逼近

该文讨论用Ｌｅｇｅｎｄｒｅ拟谱方法数值求解非线性Ｃａｈｎ Ｈｉｌｌｉａｒｄ方程的Ｄｉｒｉｃｈｌｅｔ问题．建立了其半离散和全离散逼近格式，它们保持原问题能量耗散的性质．证明了离散解的存在唯一性，并给出了最佳误差估计．数值实验也证实了我们的结果．     

An Optimal Control Problem for Flows with Discontinuities

In this paper, we study a design problem for a duct flow with a shock. The presence of the shock causes numerical difficulties. Good shock-capturing schemes with low continuity properties often cannot be combined successfully with efficient optimization methods requiring smooth functions. A remedy studied in this paper is to introduce the shock location as an explicit variable. This allows one to fit the shock and yields a problem with sufficiently smooth functions. We prove the existence of optimal solutions, Fréchet differentiability, and the existence of Lagrange multipliers. In the second part, we introduce and investigate the discrete problem and study the relations between the optimality conditions for the infinite-dimensional problem and the discretized one. This reveals important information for the numerical solution of the problem. Numerical examples are given to demonstrate the theoretical findings.     

The Existence of Optimal Solution for a Shape Optimization Problem on Starlike Domain

In this paper, a shape optimization problem over a multi-dimensional starlike domain with boundary payoff is considered. The function, which characterizes the boundary of the domain with respect to some ball contained inside domain, is shown to be Lipschitz continuous. The existence of an optimal solution is proved.     

一类非线性方程的Dirichlet问题

设D为R~d(d≥3)中非空规则开子集.本文利用超布朗运动给出了一类非线性方程Dirichlet问题非负有界解的通式.在一定条件下证明了满足在无穷远处有规定极限非负有界解的存在唯一性,并给出了解的概率表达式,推广了Dynkin中的结果.     

Superlinear Dirichlet problems

We investigate the existence of solutions of the Dirichlet problem for the superlinear partial differential equation, which is considered as the Euler–Lagrange equation for a certain integral functional. We present duality and variational principles for this problem.     

On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs

We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes with weights.     

Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.     

Approximation par éléments finis de problèmes elliptiques d'optimisation de forme

We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H¹ with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H¹ whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On a grid-method solution of the Laplace equation in an infinite rectangular cylinder under periodic boundary conditions

We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h ² ln h ⁻¹), where h is the step of a cubic grid.     

Stability for the Dirichlet Problem under Continuous Steiner Symmetrization

In this paper we prove the existence of a deformation transforming an arbitrary open set into the ball, which has the following properties: it keeps constant the measure, the kth eigenvalue of Laplace–Dirichlet operator is continuous from the left and the first eigenvalue is decreasing. The deformation is given by a sequence of continuous Steiner symmetrizations, and the behavior of the eigenvalues is related to the stability of the Dirichlet problem.     

含有广义p-Laplace算子的非线性边值问题解的存在性的研究

该文研究了两类含有广义p-Laplace算子的非线性边值问题. 首先, 利用变分不等式解的存在性的结果, 证明了含有广义p-Laplace算子的非线性Dirichlet边值问题解的存在性. 然后, 提出了一类含有广义p-Laplace算子的非线性Neumann边值问题. 通过深入挖掘这两类非线性边值问题间的关系, 借助于极大单调算子值域的一个扰动结果, 证明了含有广义p-Laplace算子的非线性Neumann边值问题解的存在性. 文中采用了一些新的证明技巧,推广和补充了作者以往的一些研究工作.