An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.     

Schauder Estimates for Elliptic and Parabolic Equations

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.     

Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form U_t-Δφ(u)=O, whereφ■C~1(R~1)is a strictly monotone increasing function.Clearly,the above equation has strong degeneracy,i.e.,the set of zero points ofφ′(·)is permitted to have zero measure. This is an answer to an open problem in[13,p.288].     

Distributional Methods for a Class of Functional Equations and Their Stabilities

We consider a class of n-dimensional Pompeiu equations and that of Pexider equations and their Hyers Ulam stability problems in the spaces of Schwartz distributions. First, reducing the given distribution version of functional equations to differential equations we find their solutions. Secondly, using approximate identities we prove the Hyers Ulam stability of the equations.     

Optimal Control Problem Governed by Semilinear Parabolic Equation and its Algorithm

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.     

Hoeder Continuity of Weak Solutions for Parabolic Equations with Nonstandard Growth Conditions

In this paper, we investigate the interior regularity including the local boundedness and the interior HSlder continuity of weak solutions for parabolic equations of the p（x, t）-Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem.     

Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

We consider the optimal control of solutions of first order Hamilton–Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.     

A Primal-Dual Active Set Method for Bilaterally Control Constrained Optimal Control of the Navier–Stokes Equations

Abstract

This paper is devoted to the analysis and numerical solution of distributed optimal control of the Navier–Stokes equations in presence of bilateral pointwise control constraints. The analysis of the problem involves the proof of existence of an optimal solution, as well as the presentation of necessary and sufficient conditions for optimality. For the numerical solution of the problem we apply a primal-dual active set strategy and show global and local convergence properties of the method. Finally, some numerical experiments, which illustrate the performance of the method, are discussed.     

A General Version of the Retract Method for Discrete Equations

We study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u（k ＋ 1） = F（k, u（k））, k ∈ N（a） = {a, a ＋ 1, a ＋ 2 }, a ∈ N,N= {0, 1,... } and F ： N（a） × R^n→R^n. A general principle for the existence of at least one solution with graph staying for every k ∈ N（a） in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result.     

Deterministic Time-inconsistent Optimal Control Problems - an Essentially Cooperative Approach

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N -person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.     

Optimal Control of Magnetohydrodynamic Equations with State Constraint

This work is concerned with the maximum principle for optimal control problem governed by magnetohydrodynamic equations, which describe the motion of a viscous incompressible conducting fluid in a magnetic field and consist of a subtle coupling of the Navier-Stokes equation of viscous incompressible fluid flow and the Maxwell equation of electromagnetic field. An integral type state constraint is considered.     

Optimal controls of Boussinesq equations with state constraints

This work is concerned with the maximum principle for an optimal control problem governed by Boussinesq equations. Some integral type state constraints are considered.     

Optimal control of damped Klein-Gordon equations with state constraints

In this paper, we study the maximum principles for optimal control problems governed by the damped Klein-Gordon equations with state constraints. And we prove the existence of the optimal parameter and deduce the necessary conditions on the optimal parameter.     

Superconvergence for Optimal Control Problem Governed by Nonlinear Elliptic Equations

In this article, we investigate the superconvergence of the finite element approximation for optimal control problem governed by nonlinear elliptic equations. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We give the superconvergence analysis for both the control variable and the state variables. Finally, the numerical experiments show the theoretical results.     

Optimal Control Problems with State Constraint Governed by Magnetohydrodynamic Equations

This work deals with the existence of optimal solution and the maximum principle for optimal control problem governed by MHD equations with state constraint of pointwise type in three-dimension. Strong results in two-dimension also are given.     

Optimal Control of Differential–Algebraic Equations of Higher Index, Part 1: First-Order Approximations

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of these problems which can be transformed to index one control problems. For this class of higher index DAEs, we derive first-order approximations and adjoint equations for the functionals defining the problem. These adjoint equations are then used to state, in the accompanying paper, the necessary optimality conditions in the form of a weak maximum principle. The constructive way used to prove these optimality conditions leads to globally convergent algorithms for control problems with state constraints and defined by higher index DAEs.     

Optimal control of semilinear parabolic systems with state constraint

The aim of this work is to obtain the existence of optimal solution and maximum principle for optimal control problem with pointwise type state constraint governed by semilinear parabolic systems with certain polynomial-like nonlinearity. Application to optimal control problems of the phase transition system is given.     

Optimal Control of Differential–Algebraic Equations of Higher Index,Part 2: Necessary Optimality Conditions

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of problems which can be transformed to index one control problems. For these problems we show in the accompanying paper that, if the solutions to the adjoint equations are well–defined, then the first-order approximations to the functionals defining the problem can be expressed in terms of the adjoint variables. In this paper we show that the solutions to the adjoint equations are essentially bounded measurable functions. Then, based on the first order approximations, we derive the necessary optimality conditions for the considered class of control problems. These conditions do not require the transformation of the DAEs to index-one system; however, higher-index DAEs and their associated adjoint equations have to be solved.     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.     

Asymptotic Analysis of State Constrained Semilinear Optimal Control Problems

The asymptotic behavior of state-constrained semilinear optimal control problems for distributed-parameter systems with variable compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit. We gratefully acknowledge the support of the DAAD. The paper was prepared during the visit of the first author at the Institute of Applied Mathematics II, University Erlangen-Nuremberg in 2003.