Optimal control of a heat transfer problem with convective boundary condition

We consider the problem of controlling the solution of the heat equation with the convective boundary condition taking the heat transfer coefficient as the control. We take as our cost functional the sum of theL ²-norms of the control and the difference between the temperature attained and the desired temperature. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We show how the solution depends in a differentiable way on the control using appropriate a priori estimates. We establish existence and uniqueness of the solution of the optimality system, and thus determine the unique optimal control in terms of the solution of the optimality system.This research was sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under Contract DE-AC05-84OR21400 with the Martin Marietta Energy Systems. The authors thank David R. Adams for his assistance in clarifying the proof of Proposition 2.1 and appreciate the comments of the referees for needed revisions.     

Optimal control of the viscous weakly dispersive Degasperis-Procesi equation

In this paper, we study the optimal control problem for the viscous weakly dispersive Degasperis-Procesi equation. We deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Then, according to optimal control theories and distributed parameter system control theories, the optimal control of the viscous weakly dispersive Degasperis-Procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive Degasperis-Procesi equation is proved.     

Optimal dynamic pricing of inventories with stochastic demand and discounted criterion

We consider a continuous time dynamic pricing problem for selling a given number of items over a finite or infinite time horizon. The demand is price sensitive and follows a non-homogeneous Poisson process. We formulate this problem as to maximize the expected discounted revenue and obtain the structural properties of the optimal revenue function and optimal price policy by the Hamilton-Jacobi-Bellman (HJB) equation. Moreover, we study the impact of the discount rate on the optimal revenue function and the optimal price. Further, we extend the problem to the case with discounting and time-varying demand, the infinite time horizon problem. Numerical examples are used to illustrate our analytical results.     

Optimal control of parameter distributed systems with impulses

This paper concerns optimal control problems with impulses. The optimal magnitude of impulses and the spatial position of impulses are studied. We obtain maximum principles for these problems.     

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.     

Group analysis of KdV equation with time dependent coefficients

We study the generalized KdV equation having time dependent variable coefficients of the damping and dispersion from the Lie group-theoretic point of view. Lie group classification with respect to the time dependent coefficients is performed. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are obtained. These subalgebras are then used to construct a number of similarity reductions and exact group-invariant solutions, including soliton solutions, for some special forms of the equations.     

Optimal maintenance of systems with Markovian mission and deterioration

We consider the maintenance of a mission-based system that is designed to perform missions consisting of a random sequence of phases or stages with random durations. A finite state Markov process describes the mission process. The age or deterioration process of the system is described by another finite state Markov process whose generator depends on the phases of the mission. We discuss optimal repair and optimal replacement problems, and characterize the optimal policies under some monotonicity assumptions. We also provide numerical illustrations to demonstrate the structure of the optimal policies.     

Optimal harvesting policy for general stochastic Logistic population model

We consider some optimal harvesting policies for a general stochastic Logistic population model. For two management objectives, that are maximum sustainable yield and the maximum retained profits, the optimal harvesting policies are obtained. Meanwhile, the optimal harvest effort, the maximum of expectation of sustainable yield (or retained profits) and the corresponding variance are given.     

Optimal retirement strategy with a negative wealth constraint

This paper investigates an optimal consumption, portfolio, and retirement time choice problem of an individual with a negative wealth constraint. We obtain analytical results of the optimal consumption, investment, and retirement behaviors and discuss the effect of the negative wealth constraint on the optimal behaviors. We find that, as an individual can borrow more with better credit, she is more likely to retire at a higher wealth level, to consume more, and to invest more in risky assets.     

Optimal harvesting policy for stochastic Logistic population model

In this paper, we consider some optimal harvesting policies for single population models, in which the harvest effort and the intrinsic growth rate are disturbed by environment noises. We choose the maximum sustainable yield and the maximum retained profits as two management objectives, and obtain the optimal harvesting policies, respectively. For the two objectives, we give the optimal harvest effort that maximizes the sustainable yield (or retained profits), the maximum of expectation of sustainable yield (or retained profits) and the corresponding variance. Their explicit expressions are determined by the coefficients of equation and the disturbance intensity.     

Optimal control of a competitive system with age-structure

We investigate optimal control of a first order partial differential equation (PDE) system representing a competitive population model with age structure. The controls are the proportions of the populations to be harvested, and the objective functional represents the profit from harvesting. The existence and unique characterization of the optimal control pair are established.     

Optimal recovery of solutions of the generalized heat equation in the unit ball from inaccurate data

We consider the problem of optimal recovery of solutions of the generalized heat equation in the unit ball. Information is given at two time instances, but inaccurate. The solution is to be constructed at some intermediate time. We provide the optimal error and present an algorithm which achieves this error level.     

一类非线性奇异最优控制问题的离散解法

本文研究非线性奇异最优控制问题的离散解法。利用在每个“小”时间区间上的积分形式来刻画奇异最优控制的特征,并构造了求解问题的差分方程,同时建立预估、校正格式给出了最优控制近似解的计算方法。     

Optimal control problem for viscous Cahn-Hilliard equation

This paper is concerned with the viscous Cahn-Hilliard equation, which arises in the dynamics of viscous first order phase transitions in cooling binary solutions. The optimal control under boundary condition is given and the existence of optimal solution to the equation is proved.     

Backward stochastic differential equations with two barriers and generalized reflection

We prove the existence and uniqueness of solutions of backward stochastic differential equations (BSDEs) with generalized reflection at time dependent càdlàg barriers. The reflection model we consider includes, as special cases, the standard reflection as well as the mirror reflection studied earlier in the theory of forward stochastic differential equations. We also show that the solution of BSDEs with generalized reflection corresponds to the value of an optimal stopping problem.     

Optimal financing and dividend control of the insurance company with proportional reinsurance policy

We consider the optimal control problem of the insurance company with proportional reinsurance policy. The management of the company controls the reinsurance rate, dividends payout as well as the equity issuance processes to maximize the expected present value of the dividends minus the equity issuance until the time of bankruptcy. This is the first time that the financing process in an insurance model has been considered, which is more realistic. To find the solution of the mixed singular-regular control problem, we firstly construct two categories of suboptimal models, one is the classical model without equity issuance, the other never goes bankrupt by equity issuance. Then we identify the value functions and the optimal strategies corresponding to the suboptimal models depending on the relationships between the coefficients.     

Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs

We consider an optimal impulse control problem on reinsurance, dividend and reinvestment of an insurance company. To close reality, we add fixed and proportional transaction costs to this problem. The value of the company is associated with expected present value of net dividends pay out minus the net reinvestment capitals until ruin time. We focus on non-cheap proportional reinsurance. We prove that the value function is a unique solution to associated Hamilton–Jacobi–Bellman equation, and establish the regularity property of the viscosity solution under a weak assumption. We solve the non-uniformly elliptic equation associated with the impulse control problem. Finally, we derive the value function and the optimal strategy of the control problem.     

Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy

In this paper, we consider an optimal financing and dividend control problem of an insurance company. The management of the insurance company controls the dividends payout, equity issuance and the excess-of-loss reinsurance policy. In our model, the dividends are assumed to be paid out continuously, which is of interest from the perspective of financial modeling. The objective is to find the strategy which maximizes the expected present values of the dividends payout minus the equity issuance up to the time of ruin. We solve the optimal control problem and identify the optimal strategy by constructing two categories of suboptimal control problems.     

Optimal times for constrained nonlinear control problems without local controllability

We study optimal times to reach a given closed target for controlled systems with a state constraint. Our goal is to characterize these optimal time functions in such a way that it is possible to compute them numerically and we do not need to compute trajectories of the controlled system. In this paper we provide new results using viability theory. This allows us to study optimal time functions free from the controllability assumptions classically made in the partial differential equations approach.     

Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior

We consider an M/M/1 queueing system in which the queue length may or may not be observable by a customer upon entering the system. The “observable” and “unobservable” models are compared with respect to system properties and performance measures under two different types of optimal customer behavior, which we refer to as “selfishly optimal” and “socially optimal”. We consider average customer throughput rates and show that, under both types of optimal customer behavior, the equality of effective queue-joining rates between the observable and unobservable systems results in differences with respect to other performance measures such as mean busy periods and waiting times. We also show that the equality of selfishly optimal queue-joining rates between the two types of system precludes the equality of socially optimal joining rates, and vice versa.