Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems

We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known second-order sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabolic one with boundary control and the elliptic one with boundary or distributed control, are discretized by a finite difference method. The discrete SSC are stated and numerically verified in all cases providing an indication of optimality where only necessary conditions had been studied before.     

Sequence-based necessary second-order optimality conditions for semilinear elliptic optimal control problems with nonsmooth data

Positivity - The main goal of this note is to formulate sequence-based necessary second-order optimality conditions for a semilinear elliptic optimal control problem, with a pointwise pure state...     

Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem. Accepted 6 May 1997     

Second-order necessary optimality conditions for semilinear elliptic optimal control problems

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized.     

Existence and large-time asymptotics for solutions of semilinear parabolic systems with measure data

We study the Cauchy–Dirichlet problem for monotone semilinear uniformly elliptic second-order parabolic systems in divergence form with measure data. We show that under mild integrability conditions on the data, there exists a unique probabilistic solution of the system. We also show that if the operator and the data do not depend on time, then the solution of the parabolic system converges as t → ∞ to the solution of the Dirichlet problem for an associated elliptic system. In fact, we prove some results on the rate of the convergence.     

Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regularization parameter are derived.     

Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems

This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation, namely, a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second-order optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second-order sufficient condition holds, we give a formula for the second-order expansion of the value of the problem as well as the directional derivative of the optimal control, when the cost function is perturbed. Then we extend the theory of second-order optimality conditions to the case of vector-valued controls when the feasible set is defined by local and smooth convex constraints. When the space dimension n is greater than 3, the results are based on a two norms approach, involving spaces L ² and L ^s , with s>n/2 . Accepted 27 January 1997     

OPTIMAL HARVESTING OF A SPATIALLY EXPLICIT FISHERY MODEL

Abstract We consider an optimal fishery harvesting problem using a spatially explicit model with a semilinear elliptic PDE, Dirichlet boundary conditions, and logistic population growth. We consider two objective functionals: maximizing the yield and minimizing the cost or the variation in the fishing effort (control). Existence, necessary conditions, and uniqueness for the optimal harvesting control for both cases are established. Results for maximizing the yield with Neumann (no‐flux) boundary conditions are also given. The optimal control when minimizing the variation is characterized by a variational inequality instead of the usual algebraic characterization, which involves the solutions of an optimality system of nonlinear elliptic partial differential equations. Numerical examples are given to illustrate the results.     

Error estimates of the lumped mass finite element method for semilinear elliptic problems

We are concerned with the semilinear elliptic problems. We first investigate the L²-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory.     

Optimal Control of Semilinear Elliptic Variational Bilateral Problem

This paper is concerned with an optimal control problem for semilinear elliptic variational inequalities associated with bilateral constraints. Existence and optimality conditions of the optimal pair are established. Received August 17, 1998, Accepted December 28, 1998     

Regularity of solutions for an optimal control problem with mixed control-state constraints

A class of optimal control problems for a semilinear elliptic partial differential equation with mixed control-state constraints is considered. Existence results of an optimal control and necessary optimality conditions are stated. Moreover, a projection formula is derived that is equivalent to the necessary optimality conditions. As main result, the Lipschitz continuity of the optimal control is obtained.     

On the semicontinuity of the solution map to a parametric boundary control problem

This paper studies solution stability of a parametric boundary control problem governed by semilinear elliptic equation and nonconvex cost function with mixed state control constraints. Using the direct method and the first-order necessary optimality conditions, we obtain the upper semicontinuity and continuity of the solution map with respect to parameters.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

Sensitivity analysis and the adjoint update strategy for an optimal control problem with mixed control-state constraints

In this article, an optimal control problem subject to a semilinear elliptic equation and mixed control-state constraints is investigated. The problem data depends on certain parameters. Under an assumption of separation of the active sets and a second-order sufficient optimality condition, Bouligand-differentiability (B-differentiability) of the solutions with respect to the parameter is established. Furthermore, an adjoint update strategy is proposed which yields a better approximation of the optimal controls and multipliers than the classical Taylor expansion, with remainder terms vanishing in L ^∞.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several 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 and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

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.     

Indirect Obstacle Minimax Control for Elliptic Variational Inequalities

A minimax control problem for a coupled system of a semilinear elliptic equation and an obstacle variational inequality is considered. The major novelty of such problem lies in the simultaneous presence of a nonsmooth state equation (variational inequality) and a nonsmooth cost function (sup norm). In this paper, the existence of optimal controls and the optimality conditions are established.     

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.     

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.     

A Transformation Approach in Shape Optimization: Existence and Regularity Results

In this article, we consider a model shape optimization problem. The state variable solves an elliptic equation on a star-shaped domain, where the radius is given via a control function. First, we reformulate the problem on a fixed reference domain, where we focus on the regularity needed to ensure the existence of an optimal solution. Second, we introduce the Lagrangian and use it to show that the optimal solution possesses a higher regularity, which allows for the explicit computation of the derivative of the reduced cost functional as a boundary integral. We finish the article with some second-order optimality conditions.