Global principle for free-endpoint problems in optimal control and differential games

It is shown that the maximum principle in optimal control and the min-max principle in differential games are consequences of a certain global principle which is stated here.     

A control problem for clamped extensible beams

An optimal control problem for a nonlinear partial integro-differential equation is presented. Existence of an optimal control is obtained, and a minimum principle is proved     

Maximum principle for optimal control of distributed-parameter systems with singular mass matrix

A maximum principle for the open-loop optimal control of a vibrating system relative to a given convex index of performance is investigated. Though maximum principles have been studied by many people (see, e.g., Refs. 1–5), the principle derived in this paper is of particular use for control problems involving mechanical structures. The state variable satisfies general initial conditions as well as a self-adjoint system of partial differential equations together with a homogeneous system of boundary conditions. The mass matrix is diagonal, constant, and singular, and the viscous damping matrix is diagonal. The maximum principle relates the optimal control with the solution of the homogeneous adjoint equation in which terminal conditions are prescribed in terms of the terminal values of the optimal state variable. An application of this theory to a structural vibrating system is given in a companion paper (Ref. 6).     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 1: Theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.     

Optimality necessary conditions in singular stochastic control problems with nonsmooth data

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau-Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients.     

带非局部低阶项非线性抛物型方程的时间最优控制

考虑一个带非局部低阶项非线性抛物型方程的时间最优控制问题．首先利用Schauder不动点定理证明了系统的适定性，然后利用Carleman不等式和Kakutani不动点定理证明了容许控制和最优控制的存在性，并且建立了时间最优控制的最大值原理．     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 2: Application

The optimal open-loop control of a beam subject to initial disturbances is studied by means of a maximum principle developed for hyperbolic partial differential equations in one space dimension. The cost functional representing the dynamic response of the beam is taken as quadratic in the displacement and its space and time derivatives. The objective of the control is to minimize a performance index consisting of the cost functional and a penalty term involving the control function. Application of the maximum principle leads to boundary-value problems for hyperbolic partial differential equations subject to initial and terminal conditions. The explicit solution of this system is obtained yielding the expressions for the state and optimal control functions. The behavior of the controlled and uncontrolled beam is studied numerically, and the effectiveness of the proposed control is illustrated.     

Optimal birth control of age-dependent competitive species III. Overtaking problem

A study is made of an overtaking optimal problem for a population system consisting of two competing species, which is controlled by fertilities. The existence of optimal policy is proved and a maximum principle is carefully derived under less restrictive conditions. Weak and strong turnpike properties of optimal trajectories are established.     

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.     

Maximum Principle for Risk-Sensitive Stochastic Optimal Control Problem and Applications to Finance

This article is concerned with a risk-sensitive stochastic optimal control problem motivated by a kind of optimal portfolio choice problem in the financial market. The maximum principle for this kind of problem is obtained, which is similar in form to its risk-neutral counterpart. But the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter. This result is used to solve a kind of optimal portfolio choice problem and the optimal portfolio choice strategy is obtained. Computational results and figures explicitly illustrate the optimal solution and the sensitivity to the volatility rate parameter.     

Optimal control for a class of nonlinear age-distributed population systems

This paper deals with an optimal control problem for a kind of age-dependent biological population systems. The well-posedness of the state system is treated by means of characteristics line and fixed-point principle. Necessary optimality conditions are obtained via tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal controller are established by the use of Ekeland’s principle.     

Necessary extremum conditions in the optimal control problem with state constraints

The optimal control problem with state constraints is examined. An alternative to the available approaches to the study of this problem is proposed. The maximum principle and second-order necessary conditions are proved.     

TO LEARN OR TO CHANGE: OPTIMAL R&D INVESTMENTS UNDER UNCERTAINTY IN THE CASE OF CLIMATE CHANGE

When studying R&D investments in technologies that address potential damage from climate change (termed as “research to change” or RTC), current literature overlooks the effects of purchased learning (i.e., learn through scientific research, termed as “research to learn” or RTL) about climate change. We investigate interactions between optimal R&D investments in RTC and RTL under uncertainty in climate change and research outcomes, while accounting for the positive impact that successful RTL may have on RTC outcome. We find that simultaneously investing in both RTL and RTC may be optimal when the probability that climate change imposes a specific level of damage is either moderate or very high and when RTL cost is relatively low. Whenever RTL and RTC are conducted simultaneously, then they substitute. However, when it is not optimal to conduct RTC and RTL simultaneously, then an increase in RTC cost decreases, at least weakly, RTL investment (i.e., RTL and RTC complement). When the probability that climate change imposes damage increases, then the optimal RTL investment may first decrease and then increase. Moreover, we identify conditions under which either the precautionary principle or the learn‐then‐act principle should be followed regarding R&D investments.     

A proof of a general maximum principle for optimal controls via a multiplier rule on metric space

A unified proof is given of the maximum principle for optimal control with various kinds of constraints by using a multiplier rule on metric spaces.     

Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.The author thanks the referees for careful reading and helpful suggestions and comments.     

Discrepancy based model selection in statistical inverse problems

The reconstruction of solutions in statistical inverse problems in Hilbert spaces requires regularization, which is often based on a parametrized family of proposal estimators. The choice of an appropriate parameter in this family is crucial. We propose a modification of the classical discrepancy principle as an adaptive parameter selection. This varying discrepancy principle evaluates the misfit in some weighted norm, and it also has an incorporated emergency stop. These ingredients allow the order optimal reconstruction when the solution owns nice spectral resolution. Theoretical analysis is accompanied with numerical simulations, which highlight the features of the proposed varying discrepancy principle.     

Optimal control problems with applications to the elimination of substrate diffusing and convecting through immobilized enzyme

Some optimization problems concerning a substrate in a fluid are considered. The concentration of the substrate is affected by diffusion, convection, and elimination by enzymes, and the problem is to find the optimal distribution of enzymes. In this paper, the rate of elimination and the transmission coefficient are optimized. Mathematically, these problems are optimal control problems, and they are analyzed by means of Pontryagin's maximum principle.     

On the maximum principle for a class of discrete dynamical systems with lags

A maximum principle for a class of discrete systems with lags is established constructively within the framework of discrete optimal control theory. An application of the maximum principle to a problem in advertising concludes the paper.     

模糊择近原则在多目标容差设计中的应用

利用模糊数学中的择近原则,以容差-成本模型为基础,通过引入模糊数学中的隶属度函数和贴近度原则,结合实验并设计方法提出了一种模糊容差稳健优化设计方法,较好地解决了多目标容差设计的全局最优问题.最后通过一个实例验证了该方法的合理、有效性.     

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.