Control Localized on Thin Structures for the Linearized Boussinesq System

We study optimal control problems for the linearized Boussinesq system when the control is supported on a submanifold of the boundary of the domain. This type of problem belongs to the class of optimal control problems with measures as controls, which has been studied recently by several authors. We are mainly interested in the optimality conditions for such problems. It is known that the differentiability properties needed to obtain the optimality conditions are more demanding, in terms of regularity of the data, than what is needed to prove the existence of optimal controls. Here we are able to derive the optimality conditions by taking advantage of the particular structure of the controls.     

Instantaneous control for traffic flow

The solution methods for optimal control problems with coupled partial differential equations as constraints are computationally costly and memory intensive; in particular for problems stated on networks, this prevents the methods from being relevant. We present instantaneous control problems for the optimization of traffic flow problems on road networks. We derive the optimality conditions, investigate the relation to the full optimal control problem and prove that certain properties of the optimal control problem carry over to the instantaneous one. We propose a solution algorithm and compare quality of the computed controls and run‐times. Copyright © 2006 John Wiley & Sons, Ltd.     

On the control of time discretized dynamic contact problems

We consider optimal control problems with distributed control that involve a time-stepping formulation of dynamic one body contact problems as constraints. We link the continuous and the time-stepping formulation by a nonconforming finite element discretization and derive existence of optimal solutions and strong stationarity conditions. We use this information for a steepest descent type optimization scheme based on the resulting adjoint scheme and implement its numerical application.     

Optimal control in coefficients of elliptic equations with state constraints

This paper is concerned with state constrained optimal control problems of elliptic equations, the control being a coefficient of the partial differential equation. Existence of an optimal control is proved and optimality conditions are derived. We perform finite-element approximations of optimal control problems and state some convergence results: we prove convergence of optimal controls and states as well as convergence of Lagrange multipliers.This research was partially supported by the Dirección General de Investigación Científica y Técnica (Madrid).     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

Optimality conditions for a class of inverse optimal control problems with partial differential equations

ABSTRACT

We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function. The upper-level problem is used to identify the parameters of the lower-level problem. Our main focus is the derivation of first-order necessary optimality conditions. We prove C-stationarity of local solutions of the inverse optimal control problem and give a counterexample to show that strong stationarity might be violated at a local minimizer.     

Stochastic control problems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay. We state conditions for certain classes of such systems under which the stochastic control problems become finite-dimensional. These conditions are illustrated with three applications. First, we solve some linear quadratic problems with delay. Then we find the optimal consumption rate in a financial market with delay. Finally, we solve explicitly a deterministic fluid problem with delay which arises from admission control in ATM communication networks.     

New optimality conditions for nonsmooth control problems

This work considers nonsmooth optimal control problems and provides two new sufficient conditions of optimality. The first condition involves the Lagrange multipliers while the second does not. We show that under the first new condition all processes satisfying the Pontryagin Maximum Principle (called MP-processes) are optimal. Conversely, we prove that optimal control problems in which every MP-process is optimal necessarily obey our first optimality condition. The second condition is more natural, but it is only applicable to normal problems and the converse holds just for smooth problems. Nevertheless, it is proved that for the class of normal smooth optimal control problems the two conditions are equivalent. Some examples illustrating the features of these sufficient concepts are presented.     

半线性椭圆方程支配系统的最优性条件

本文讨论了可能具有多值解的椭圆型偏微分方程支配系统的最优控制问题,我们通过构造一个抛物方程控制问题的逼近序列,并利用抛物方程控制问题的结果,得到了椭圆系统最优控制的必要条件．     

Mixed Frank–Wolfe Penalty Method with Applications to Nonconvex Optimal Control Problems

We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.     

Relaxation of Minimax Optimal Control Problems with Infinite Horizon

A minimax optimal control problem with infinite horizon is studied. We analyze a relaxation of the controls, which allows us to consider a generalization of the original problem that not only has existence of an optimal control but also enables us to approximate the infinite-horizon problem with a sequence of finite-horizon problems. We give a set of conditions that are sufficient to solve directly, without relaxation, the infinite-horizon problem as the limit of finite-horizon problems.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

Parametric Sensitivity Analysis of Perturbed PDE Optimal Control Problems with State and Control Constraints

We study parametric optimal control problems governed by a system of time-dependent partial differential equations (PDE) and subject to additional control and state constraints. An approach is presented to compute the optimal control functions and the so-called sensitivity differentials of the optimal solution with respect to perturbations. This information plays an important role in the analysis of optimal solutions as well as in real-time optimal control.The method of lines is used to transform the perturbed PDE system into a large system of ordinary differential equations. A subsequent discretization then transcribes parametric ODE optimal control problems into perturbed nonlinear programming problems (NLP), which can be solved efficiently by SQP methods.Second-order sufficient conditions can be checked numerically and we propose to apply an NLP-based approach for the robust computation of the sensitivity differentials of the optimal solutions with respect to the perturbation parameters. The numerical method is illustrated by the optimal control and sensitivity analysis of the Burgers equation.Communicated by H. J. Pesch     

Weakly efficient solutions and pseudoinvexity in multiobjective control problems

In this paper, we introduce new pseudoinvexity conditions on functionals involved in a multiobjective control problem, called W-KT-pseudoinvexity and W-FJ-pseudoinvexity. We prove that all Kuhn-Tucker or Fritz-John points are weakly efficient solutions if and only if these conditions are fulfilled. We relate weakly efficient solutions to optimal solutions of weighting control problems. We generalize recently obtained optimality results of known mathematical programming problems and control problems. We illustrate these results with an example.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

带跳的倒向重随机系统的最大值原理及其应用

本文研究带跳的倒向重随机系统的随机控制问题的最优性条件。在控制域为凸且控制变量进入所有系数条件下,分别以局部形式和全局形式给出必要性最优条件和充分性最优条件。把上述最大值原理应用于重随机线性二次最优控制问题,得到唯一的最优控制,并且给出应用的例子。     

Maximum principle for optimal control problems with intermediate constraints

We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.     

Optimal Control Problems for a Class of Semilinear Multisolution Elliptic Equations

We study the optimal control problem for a class of elliptic problems that may possess multiple solutions. We obtain necessary conditions for optimal control by constructing a related parabolic problem and using known results for the parabolic problem.     

S-convergence in the theory of homogenization of problems of optimal control

We give a definition of homogenized problems of optimal control in terms of variational S-limits and establish sufficient conditions for a family of problems of optimal control described by nonlinear operator relations to be compact with respect to the operation of homogenization.