Constructive methods in the nonlinear problems of control theory

We consider nonlinear control problems of transformation of a substance from one state into another given some boundary data. Our goal is to recover the second order differential operators with three time-independent coefficients. Some constructive analytical methods of study are proposed that use, in particular, the Burman-Lagrange inversion formulas for analytic functions. Some formulas are given for solving the control problems in a few particular cases of the initialboundary data. Alongside the theoretical study, we carry out, on the basis of a logical system of symbolic computation, the practical implementation of the algorithms and programs for automatic generation of the formulas that give the solutions of control problems.     

On boundary state constrained control problems

The aim of this paper is to present a simple qualification condition for control elliptic problems involving constraints both on the control and the state, when there are both a distributed and a boundary control. We establish a general maximum principle and give some examples. Then we study the particular case when the control is only a boundary one and we present a Lagrangian Algorithm with a few numerical results.     

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.     

Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations

We study the numerical approximation of Neumann boundary optimal control problems governed by a class of quasilinear elliptic equations. The coefficients of the main part of the operator depend on the state function, as a consequence the state equation is not monotone. We prove that strict local minima of the control problem can be approximated uniformly by local minima of discrete control problems and we also get an estimate of the rate of this convergence. One of the main issues in this study is the error analysis of the discretization of the state and adjoint state equations. Some difficulties arise due to the lack of uniqueness of solution of the discrete equations. The theoretical results are illustrated by numerical tests.     

Boundary Layers in Periodic Homogenization of Neumann Problems

This paper is concerned with a family of second‐order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first‐order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L² in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher‐order convergence rate for Neumann problems with nonoscillating data. © 2018 Wiley Periodicals, Inc.     

Homogenization of low-cost control problems on perforated domains

The aim of this paper is to study the asymptotic behaviour of some low-cost control problems in periodically perforated domains with Neumann condition on the boundary of the holes. The optimal control problems considered here are governed by a second order elliptic boundary value problem with oscillating coefficients. It is assumed that the cost of the control is of the same order as that describing the oscillations of the coefficients. The asymptotic analysis of small cost problem is more delicate and need the H-convergence result for weak data. In this connection, an H-convergence result for weak data under some hypotheses is also proved.     

Regularity of solutions of boundary value problems for parabolic equations of arbitrary order in weighted Hölder spaces

We consider initial-boundary value problems for a uniformly parabolic equation of arbitrary order 2m in a noncylindrical domain whose lateral boundary is nonsmooth with respect to t. We assume that the lower-order coefficients and the right-hand side of the equation, generally speaking, grow to infinity no more rapidly than some power function when approaching the parabolic boundary of the domain, all coefficients of the equation are locally Hölder, and their Hölder constants can grow near that boundary. We construct a smoothness scale of solutions of such problems in weighted Hölder classes of functions whose higher derivatives may grow when approaching the parabolic boundary of the domain.     

Finite element methods for elliptic optimal control problems with boundary observations

We study in this paper the finite element approximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domains. Some numerical experiments are carried out at the end of the paper to support our theoretical findings.     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.     

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝ^N. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.     

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.     

Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems—Strong convergence of optimal controls

A class of optimal control problems for a parabolic equation with nonlinear boundary condition and constraints on the control and the state is considered. Associated approximate problems are established, where the equation of state is defined by a semidiscrete Ritz-Galerkin method. Moreover, we are able to allow for the discretization of admissible controls. We show the convergence of the approximate controls to the solution of the exact control problem, as the discretization parameter tends toward zero. This result holds true under the assumption of a certain sufficient second-order optimality condition.Dedicated to the 60th birthday of Lothar von Wolfersdorf     

Optimality necessary conditions in singular stochastic control problems with nonsmooth data

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau-Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients.     

On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.     

The Dirichlet and Regularity Problems for Some Second Order Linear Elliptic Systems on Bounded Lipschitz Domains

In this paper, we investigate divergence-form linear elliptic systems on bounded Lipschitz domains in \(\mathbb {R}^{d+1}, d \ge 2\), with L² boundary data. The coefficients are assumed to be real, bounded, and measurable. We show that when the coefficients are small, in Carleson norm, compared to one that is continuous on the boundary, we obtain solvability for both the Dirichlet and regularity boundary value problems given that the coefficients satisfy a certain “pseudo-symmetry” condition.     

Solution of the control problem for a linear descriptor system with rectangular coefficient matrix

We pose a control problem for linear stationary algebro-differential system whose coefficients are rectangular matrices. We obtain a controllability criterion for the partially given output function and the boundary values of the state function. We construct a control function and a state function under minimal smoothness requirements on the output function.     

Series solutions for Laplace''s equation with nonhomogeneous mixed boundary conditions and irregular boundaries

Series solutions to Laplace's equation provide accurate analytical results for problems with regular boundary geometries. This paper develops the series approach, providing a solution methodology for problems with nonhomogeneous, mixed boundary conditions, defined on irregular boundaries. The series coefficients are obtained by first transforming to Poisson's equation with some homogeneous boundary conditions, and then modifying the eigenfunction expansion theory to cater for irregular boundary geometry. The computational details for evaluating the series coefficients are given, and the method demonstrated by a practical example.