Optimality conditions and adjoint state for a perturbed boundary optimal control system

In this paper, we are concerned with finding optimal controls for a class of linear boundary optimal control systems associated to a Laplace operator on a regular bounded domain in the n-dimensional Euclidean space. For these systems, in previous works (see [1,2]), we proved existence of the (perturbed) states and optimal controls, and studied their behaviour. The purpose of this paper is to establish the system of optimality conditions, investigate the adjoint states, and prove their strong convergence in some Sobolev spaces.     

Optimal Control of a Class of Phase Change Processes with Terminal State Observation

A class of optimal control problems for nonlinear evolutionary processes governed by two-phase Stefan problems is analyzed. The processes with terminal state observation are considered in the case of one space dimension. Approximate optimal solutions (controls, as well as the corresponding states and adjoint states), referring to the problems with time-averaged state observation are shown to converge to the appropriate solutions for the original problem.     

A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable $y$ and its flux $σ$ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables $z$ and $ω$, and also a variational inequality for the control variable $u$ is derived. As we can see the two resulting systems for the unknown state variable $y$ and its flux $σ$ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states $z$ and $ω$ are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable $y$ and its flux $σ$. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.     

Hölder Continuity of Adjoint States and Optimal Controls for State Constrained Problems

We investigate Hölder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Hölder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Hölder continuous, while they have the two sided lower Hölder continuity property for less regular constraints. Finally, we provide sufficient conditions for Hölder type regularity of optimal controls.     

Optimal control and optimal trajectories of regional macroeconomic dynamics based on the Pontryagin maximum principle

The Pontryagin maximum principle is used to prove a theorem concerning optimal control in regional macroeconomics. A boundary value problem for optimal trajectories of the state and adjoint variables is formulated, and optimal curves are analyzed. An algorithm is proposed for solving the boundary value problem of optimal control. The performance of the algorithm is demonstrated by computing an optimal control and the corresponding optimal trajectories.     

Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints

In this paper we consider a model elliptic optimal control problem with finitely many state constraints in two and three dimensions. Such problems are challenging due to low regularity of the adjoint variable. For the discretization of the problem we consider continuous linear elements on quasi-uniform and graded meshes separately. Our main result establishes optimal a priori error estimates for the state, adjoint, and the Lagrange multiplier on the two types of meshes. In particular, in three dimensions the optimal second order convergence rate for all three variables is possible only on properly refined meshes. Numerical examples at the end of the paper support our theoretical results.     

On relations of the adjoint state to the value function for optimal control problems with state constraints

We consider an optimal control problem under state constraints and show that to every optimal solution corresponds an adjoint state satisfying the first order necessary optimality conditions in the form of a maximum principle and sensitivity relations involving the value function. Such sensitivity relations were recently investigated by P. Bettiol and R.B. Vinter for state constraints with smooth boundary. In the difference with their work, our setting concerns differential inclusions and nonsmooth state constraints. To obtain our result we derive neighboring feasible trajectory estimates using a novel generalization of the so-called inward pointing condition.     

Optimal Control of Point Processes with Noisy Observations: The Maximum Principle

This paper studies the optimal control problem for point processes with Gaussian white-noised observations. A general maximum principle is proved for the partially observed optimal control of point processes, without using the associated filtering equation . Adjoint flows—the adjoint processes of the stochastic flows of the optimal system—are introduced, and their relations are established. Adjoint vector fields , which are observation-predictable, are introduced as the solutions of associated backward stochastic integral-partial differential equtions driven by the observation process. In a heuristic way, their relations are explained, and the adjoint processes are expressed in terms of the adjoint vector fields, their gradients and Hessians, along the optimal state process. In this way the adjoint processes are naturally connected to the adjoint equation of the associated filtering equation . This shows that the conditional expectation in the maximum condition is computable through filtering the optimal state, as usually expected. Some variants of the partially observed stochastic maximum principle are derived, and the corresponding maximum conditions are quite different from the counterpart for the diffusion case. Finally, as an example, a quadratic optimal control problem with a free Poisson process and a Gaussian white-noised observation is explicitly solved using the partially observed maximum principle. Accepted 8 August 2001. Online publication 17 December, 2001.     

Classical differential geometry solution of the brachistochrone tunnel problem

The Frenet-Serret equations of classical differential geometry are used to describe the quickest descent tunneling path problem. The optimal tunnel is shown to have a constant turn rate with zero torsion and is equivalent to Edelbaum's hypocycloid solution. The solutions are obtained using the maximum principle and singular arc conditions. The optimal curvature is a first-order singular arc and the optimal torsion is a second-order singular arc. Our treatment includes both the normal and the abnormal optimal control problems. Our problem is abnormal for the case where the final speed is zero. Analytical solutions for the optimal time histories are derived for all states and all adjoint states. One of Leitmann's sufficiency field theorems is used to establish optimality of the solutions.     

The turnpike property in finite-dimensional nonlinear optimal control

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory.     

一类具有三种状态的可修排队系统研究

研究一类具有三种状态的可修排队模型主算子的豫解集.通过研究该主算子的共轭算子的豫解集得到此主算子的豫解集.     

Optimal Control Problems for Evolution Equations of Parabolic Type with Nonlinear Perturbations

In this paper, we study the optimal control problems governed by the semilinear parabolic type equation in Hilbert spaces. Under Lipschitz continuity condition of the nonlinear term, we can obtain the optimal conditions and maximal principles for a given equation, which are described by the adjoint state corresponding to the given equation without the rigorous conditions for the nonlinear term.     

基于隐式离散极大值原理的聚合物驱最优注入策略

为了获得聚合物驱油的最大利润,建立了确定最佳聚合物注入浓度的最优控制模型.利用全隐式差分格式将连续模型离散化得到离散系统的状态方程.通过隐含离散系统的极大值原理获得了该最优控制问题的必要条件.给出了基于梯度的数值求解方法,在求解状态方程的过程中直接构造了伴随问题的系数矩阵.通过一个三维聚合物驱模型的计算实例表明了所提出方法的可行性和有效性.     

A Joint Tikhonov Regularization and Augmented Lagrange Approach for Ill-Posed State Constrained Control Problems with Sparse Controls

Abstract

We provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem without any additional L² regularization terms. The sparsity is guaranteed by an additional L¹ term. Here, the modification of the classical augmented Lagrange method guarantees us uniform boundedness of the multiplier that corresponds to the state constraints. We present a coupling between the regularization parameter introduced by the Tikhonov regularization and the penalty parameter from the augmented Lagrange method, which allows us to prove strong convergence of the controls and their corresponding states. Moreover, convergence results proving the weak convergence of the adjoint state and weak*-convergence of the multiplier are provided. Finally, we demonstrate our method in several numerical examples.     

Shooting method for the numerical solution of optimal control problems with bounded state variables

Abtract Various methods have been proposed for the numerical solution of optimal control problems with bounded state variables. In this paper, a new method is put forward and compared with two other methods, one of which makes use of adjoint variables whereas the other does not. Some conclusions are drawn on the usefulness of the three methods involved.     

Sharp error estimate of Grünwald-Letnikov scheme for a multi-term time fractional diffusion equation

Mixed and hybrid finite element discretizations for distributed optimal control problems governed by an elliptic equation are analyzed. A cost functional keeping track of both the state and its gradient is studied. A priori error estimates and super-convergence properties for the continuous and discrete optimal states, adjoint states, and controls will be given. The approximating finite-dimensional systems will be solved by adding penalization terms for the state and the associated Lagrange multipliers. In general, performing optimization, discretization, hybridization, and penalization in any order lead to the same optimality system. Numerical examples based on the Raviart–Thomas finite elements will be presented.

     

Junction times in singular optimal control

Finding junction times on partially singular optimal control trajectories is in general difficult analytically and especially so when the optimal control is explicitly dependent on both the state and adjoint variables. This paper presents a new approach for linear problems whereby variations in the final state, measured from a reference optimal solution, can be used to find corresponding junction times. The approach is shown to be effective when applied to a problem which has an optimal solution involving a single bang/singular junction for certain choices of end state.     

The solution and sensitivity of a general optimal control problem

A solution method for the general optimal control problem is presented. This can be used to solve optimal control problems for which the system dynamics are not necessarily described by differential or difference equations. Having obtained the solution it is of theoretical and practical interest to investigate the sensitivity of the optimal trajectory to perturbations. Indicators of this sensitivity are the adjoint variables derived in the maximum principle. A method of deriving the adjoint variables from the solution is described. To illustrate the solution method and the determination of the adjoint variables a problem in the urban housing market is used.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons

We formulate an S-I-R (Susceptible, Infected, Immune) spatiotemporal epidemic model as a system of coupled parabolic partial differential equations with no-flux boundary conditions. Immunity is gained through vaccination with the vaccine distribution considered a control variable. The objective is to characterize an optimal control, a vaccine program which minimizes the number of infected individuals and the costs associated with vaccination over a finite space and time domain. We prove existence of solutions to the state system and existence of an optimal control, as well as derive corresponding sensitivity and adjoint equations. Techniques of optimal control theory are then employed to obtain the optimal control characterization in terms of state and adjoint functions. To illustrate solutions, parameter values are chosen to model the spread of rabies in raccoons. Optimal distributions of oral rabies vaccine baits for homogeneous and heterogeneous spatial domains are compared. Numerical results reveal that natural land features affecting raccoon movement and the relocation of raccoons by humans can considerably alter the design of a cost-effective vaccination regime. We show that the use of optimal control theory in mathematical models can yield immediate insight as to when, where, and what degree control measures should be implemented.