Optimal control systems with state variable jump discontinuities

Consideration is given to continuous-time, parameter-dependent optimal control problems with state-variable jump discontinuities atN variable interior times. A maximum principle involving known costate jump conditions is stated and is proved by transforming the problem into a standard Mayer control problem. An illustrative example for fisheries management is included.This work was partially supported by a grant from Control Data. The authors are grateful to Professor T. L. Vincent for drawing their attention to Refs. 4–6 listed below.     

Optimal control problems with state constraints for semilinear distributed-parameter systems

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.This research was partly supported by the National Science Foundation under Grant DMS-92-21819.     

Optimal control of time-delay systems with distributed parameters

An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.The author wishes to express his deep gratitude to Professors J. M. Sloss and S. Adali for the valuable guidance and constant encouragement during the preparation of this paper.     

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.     

Optimal control of self-adjoint nonlinear operator equations in Hilbert spaces

In this paper, we formulate and study a general optimal control problem governed by nonlinear operator equations described by unbounded self-adjoint operators in Hilbert spaces. This problem extends various particular control models studied in the literature, while it has not been considered before in such a generality. We develop an efficient way to construct a finite-dimensional subspace extension of the given self-adjoint operator that allows us to design the corresponding adjoint system and finally derive an appropriate counterpart of the Pontryagin Maximum Principle for the constrained optimal control problem under consideration by using the obtained increment formula for the cost functional and needle type variations of optimal controls.     

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.     

A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information

The paper is concerned with a stochastic optimal control problem in which the controlled system is described by a fully coupled nonlinear forward-backward stochastic differential equation driven by a Brownian motion. It is required that all admissible control processes are adapted to a given subfiltration of the filtration generated by the underlying Brownian motion. For this type of partial information control, one sufficient (a verification theorem) and one necessary conditions of optimality are proved. The control domain need to be convex and the forward diffusion coefficient of the system can contain the control variable. This work was partially supported by Basic Research Program of China (Grant No. 2007CB814904), National Natural Science Foundation of China (Grant No. 10325101) and Natural Science Foundation of Zhejiang Province (Grant No. Y605478, Y606667)     

Optimal control of heterogeneous systems: Basic theory

A general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side-conditions. The heterogeneity is represented by a parameter taking values in an abstract measurable space, so that both continuous and discrete heterogeneity, as well as probabilistic heterogeneity without density, are included. A distributed and a lumped controls are involved, the latter appearing also in the side conditions. An existence theorem is proved for the uncontrolled system, and the sensitivity of the solution with respect to the control variables is estimated. The main result is an optimality condition in the form of the Pontryagin local maximum principle. A global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces. A number of possible applications are outlined: age-structured systems, size-structured systems, (nonlocal) advection-reaction equations, static parametric heterogeneity in epidemiology, and two-stage control systems with uncertain switching time.     

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.     

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).     

具有年龄结构的种群线性动力系统的最优收获控制问题

§ 1　Introduction and setting of the problemThe optimal control of age-dependent population dynamics has been intensivelystudied in the last two decades and there is now a vast stock of literature on the topic ofoptimal control problems ofage-structured population dynamics.(see [1 -9] ) .To the bestof our knowledge,the works of Brokate[3,4] are the firstto deal with this topic.Since then,many authors devote to the optimal harvesting problem.In this aspect,we refere to thefundamental papers o…     

Optimal control of a one product recovery system with backlogging

In this paper a product recovery system for one product is investigated.The system contains one inventory for returned and recoverableitems and one for serviceable items. Demands are satisfied fromserviceable inventory where backlogging of demands is allowed.In addition, there is the possibility of disposal for the returnedproducts. We assume deterministic but dynamic return and demandrates and a linear cost structure. The Pontryagin maximum principleis used to determine the optimal production, remanufacturingand disposal policy.     

Optimal control of fluid dynamic systems with state constraint of pointwise type

The aim of this work is to obtain the maximum principle by spike perturbation for the optimal control problem with pointwise type state constraint governed by 3-dimensional fluid dynamic systems.     

Maximum principle for the optimal control of systems with continuous leads

A sufficiency theorem is provided for the optimal control of systems with continuous leads wherein the motion of today's state is governed by the future trajectory of the control and the state. An application to the economics of dynamic limit pricing is given.This research had its origin in the doctoral dissertation of the author. The comments of H. Ryder are gratefully acknowledged.     

Optimal skill mix: An application of the maximum principle for systems with retarded controls

In this paper, we analyze the optimal skill mix in a model with two kinds of imperfectly substitutable labor, skilled and unskilled. The population is characterized by a distribution of innate abilities, and individuals are trained according to optimal rules or market rules (with imperfect expectations); the length of each individual's training period depends upon his innate ability. The market and optimal rules are characterized and compared, and corrective policies are investigated. This model represents a major advance over earlier models, which are based on the following assumptions: (a) either unskilled and skilled labor are perfectly substitutable or training is a necessary condition for employment; (b) individuals are innately identical; (c) in most cases, training occurs either instantaneously or with fixed lag.Earlier versions of this paper were delivered at the IFAC/IFORS International Conference on Dynamic Modelling of National Economies held in Coventry, England. A Guest Lecture was delivered by the first author to the 14th Biennial Seminar of the Canadian Mathematical Congress on Optimal Control Theory and its Applications held in London, Ontario, 1973. The authors acknowledge with appreciation support from a National Science Foundation Grant in Economics to Carnegie-Mellon University; in addition, the second author was supported by a Ford Foundation Faculty Research Fellowship during the academic year 1970–71. The authors are indebted to D. Cass, R. E. Lucas, and G. L. Thompson for valuable comments on an earlier draft, although all remaining errors of commission and omission are ours.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

Optimal control of age-structured population dynamics for spread of universally fatal diseases

This article is concerned with the optimal control problem of age-structured population dynamics for the spread of universally fatal diseases. The existence and uniqueness of solution of the system, which consists of a group of partial differential equations with nonlocal boundary conditions, is proved. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of the Pontryagin maximum principle of the system. The necessary condition is presented for the optimal control problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made.     

Optimal control of a class of systems with continuous lags: Dynamic programming approach and economic interpretations

This paper derives a maximum principle for dynamic systems with continuous lags, i.e., systems governed by integrodifferential equations, using dynamic programming. As a result, the adjoint variables can be provided with useful economic interpretations.This research was supported by NSERC Grant No. A4619.