A dual extremum principle for a population equation

A dual extremum principle for the Verhulst-Pearl population equation is constructed using a complementary variational technique. The dual formulation utilizes a minimum principle recently developed by Leitmann to convert the functional optimization problem into a parameter optimization problem.This research was supported in part by NASA Grant No. NGR-36-010-024. The first author would like to thank Dr. W. Stadler, Department of Mechanical Engineering, University of California, Berkeley, for his valuable suggestions.     

Optimal control of a large turboalternator

The optimal torque and voltage control for a large turbogenerator is found by using the minimum norm formulation. It should be noted that the model used is highly nonlinear. Numerical results are presented.This work was supported in part by the National Research Council of Canada, Grant No. A4146.     

A minimum principle for a population equation

A minimum principle for the Verhulst-Pearl population equation is given in terms of the integral of the inverse of population. This principle is contrasted with one given earlier by Volterra in terms of the integral of population.This research was supported by NSF under Grant No. GP-2405.     

Control of partially observed diffusions

The optimal control of a partially observed diffusion is discussed when the control parameter is present in both the drift and diffusion coefficients. Using a differentiation result of Blagovescenskii and Freidlin, and adapting techniques of Bensoussan, we obtain a stochastic minimum principle.This research was partially supported by NSERC Grant A7964, by the US Air Force Office of Scientific Research Contract AFOSR-86-0332, and by the US Army Research Office Contract DAAL03-87-K-0102.     

Pointwise quasi-Newton method for unconstrained optimal control problems,II

In this paper, the necessary optimality conditions for an unconstrained optimal control problem are used to derive a quasi-Newton method where the update involves only second-order derivative terms. A pointwise update which was presented in a previous paper by the authors is changed to allow for more general second-order sufficiency conditions in the control problem. In particular, pointwise versions of the Broyden, PSB, and SR1 update are considered. A convergence rate theorem is given for the Broyden and PSB versions.This research was supported by NSF Grant No. DMS-89-00410, by NSF Grant No. INT-88-00560, by AFOSR Grant No. AFOSR-89-0044, and by the Deutsche Forschungsgemeinschaft.     

A conjugate-point condition for a class of differential games

A conjugate-point necessary condition is derived for a class of differential games. This is done by considering the conjugate-point condition for the minimum problem and maximum problem associated with a differential game. Two definitions of a conjugate point and two conjugate-point necessary conditions result. These two definitions and necessary conditions are shown to be equivalent and are combined into one definition and one necessary condition.This research was primarily supported by the National Science Foundation under Traineeship No. 1721-80-282 and Grant No. GK-3341. Additional support was obtained from NASA Institutional Grant No. NGR-15-005-021.     

On the optimum design of piston rings

We study a family of control problems that arise in the design of dynamically loaded bearings which have a minimum power loss; in particular, we study the design of piston rings with minimum parasitic power loss.This research was partially supported by NSF Grant No. MCS-77-05624.     

Control of a hybrid conditionally linear Gaussian process

A control problem is considered where the coefficients of the linear dynamics are functions of a noisily observed Markov chain. The approximation introduced is to consider these coefficients as functions of the filtered estimate of the state of the chain; this gives rise to a finite-dimensional conditional Kalman filter. A minimum principle and a new equation for an adjoint process are obtained.This research was partially supported by NSERC under Grant A-7964, by the US Air Force Office of Scientific Research under Contract AFOSR-86-0332, and by the US Army Research Office under Contract DAAL03-87-0102.The authors obtained these results during a visit to UCSD by the first author in January 1990. This author wishes to thank Professor D. D. Sworder and his department for their hospitality.     

Maximum principle of optimal control for functional differential systems

We show that the maximum principle holds for optimal periodic control problems governed by functional differential equations under a Lipschitz condition on the value functional. Generalizations to other boundary conditions are also considered.This research was partially supported by NSF Grant No. DMS-84-01719.The first author was partially supported by the Science Fund of the Chinese Academy of Sciences, Beijing, China.     

Adaptive control of discounted Markov decision chains

In this paper, we consider discounted-reward finite-state Markov decision processes which depend on unknown parameters. An adaptive policy inspired by the nonstationary value iteration scheme of Federgruen and Schweitzer (Ref. 1) is proposed. This policy is briefly compared with the principle of estimation and control recently obtained by Schäl (Ref. 4).This research was supported in part by the Consejo Nacional de Ciencia y Tecnología under Grant No. PCCBBNA-005008, in part by a grant from the IBM Corporation, in part by the Air Force Office of Scientific Research under Grant No. AFOSR-79-0025, in part by the National Science Foundation under Grant No. ECS-0822033, and in part by the Joint Services Electronics Program under Contract No. F49620-77-C-0101.     

An optimal control problem with a random stopping time

This paper deals with a stochastic optimal control problem where the randomness is essentially concentrated in the stopping time terminating the process. If the stopping time is characterized by an intensity depending on the state and control variables, one can reformulate the problem equivalently as an infinite-horizon optimal control problem. Applying dynamic programming and minimum principle techniques to this associated deterministic control problem yields specific optimality conditions for the original stochastic control problem. It is also possible to characterize extremal steady states. The model is illustrated by an example related to the economics of technological innovation.This research has been supported by NSERC-Canada, Grants 36444 and A4952; by FCAR-Québec, Grant 88EQ3528, Actions Structurantes; and by MESS-Québec, Grant 6.1/7.4(28).     

Dynamically similar control systems and a globally minimum gain control technique: IMSC

The concept of dynamically similar control systems is introduced. The necessary and sufficient conditions to minimize a quadratic modal gain measure are given for dynamically similar closed-loop control systems. The globally minimum modal gain is obtained when the independent modal space control (IMSC) is used. Corollaries of the results for the control of infinite-dimensional structural distributed parameter systems (DPS) are given. Based on the results, a modal interaction parameter (MIP) is defined for all control systems. The minimum value of MIP is zero and uniquely corresponds to the IMSC. A nonzero value of MIP corresponds to all other coupled control (CC) designs and implies suboptimality relative to the IMSC design. The relative optimality of the real-space gain matrices of the IMSC and the CC designs depends on the actuator locations for the IMSC. Based on this, a real-space interaction parameter (RIP) is defined. A positive value of RIP renders IMSC optimal in its real-space gain matrix. The MIP and RIP are indications of suboptimality of a particular control technique and can be used to tune-up the control design via actuator locations. Actuator distribution criteria are suggested for both CC and IMSC designs, based on the values of MIP and RIP, respectively.This work was supported by the National Science Foundation, Grant No. MEA-82-04920.     

Sufficient conditions for the optimal control of a class of systems with continuous lags

Sufficient conditions in the form of a maximum principle are obtained for the optimal control of a system described by integro-differential equations and subject to some specified path constraints. The conditions are relaxed to allow for jumps in the adjoint variables at the junction points, provided a certain convexity hypothesis is satisfied for the constraint set at these points.This research was partially supported at Stanford University by the Office of Naval Research, Contract No. N-00014-67-A-0112-0011, by the National Science Foundation, Grant No. GP-31393, and by the US Atomic Energy Commission, Contract No. AT(04-3)-326-PA-18. It was also supported by the Department of Economics at Rice University.     

Minimum-fuel control of high-order systems

The minimum-fuel control problem is of special interest in various space systems. To date, solutions of minimum-fuel control problems have been carried out for relatively low-order systems. Space structures, however, are generally characterized by a large number of degrees of freedom, so that minimum-fuel control of such systems requires a new approach. In the independent modal-space control (IMSC) method, the control laws are designed in the modal space for each mode independently. The minimum-fuel problem reduces to that of a set of independent second-order systems, so that minimum-fuel control is possible. This paper shows how the IMSC method can be used to control a space structure with a minimum amount of fuel. A numerical example is presented.This research was supported by NASA Research Grant No. NAG-1-225, sponsored by the Spacecraft Control Branch, Langley Research Center.     

Dynamics and optimization of a distributed sales-advertising model

A general sales-advertising model is developed in which the state of the system represents a population distribution over a parameter space. With appropriate interpretations, this can include income, family size, geographic distributions, etc. Effects of information diffusion, interaction, and population migration are included. Under fairly general conditions, it is shown that such model are well posed and that there exists an optimal control.This work was supported by NSERC Grant No. A-4619 and by Grant No. AFOSR-82-0271. Thanks are due to G. Haines and T. Mitchell.     

Minimal trajectories of nonconvex differential inclusions

We consider an optimization problem with endpoint constraints associated with a nonconvex differential inclusion. We give a necessary condition of the maximum principle type for a solution of the problem. Following the approach from Ref. 1, the condition is stated in terms of single-valued selections of the convexified right-hand side of the inclusion.This work was supported in part by the National Science Foundation, Grant No. DMS-86-01774.     

Penalty approximation of a Sobolev optimal control problem with state constraints

In this paper, we study the approximation by the penalty method of a control problem governed by a pseudo-parabolic equation with a noncoercive control functional and with control and state constraints. The existence of solutions to the penalized problems is established. In addition, the convergence of the penalized problems to the solution, the Lagrange multipliers, and the minimum value of the original problem is studied. The results apply to Sobolev and parabolic equations as well.This work was partially supported by the National Science Foundation, Grant No. MCS-79-02037. The author would like to thank Professor A. B. Schwarzkopf for his helpful comments on this paper.     

A note on minimum distance estimates

It is shown that minimum distance estimates enjoy the invariance property of maximum likelihood estimates. Supported under Grant No. 18-2773 of the American University of Beirut.     

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).     

Feedback approximation of the minimum energy control for linear infinite-dimensional systems with zero terminal state

The paper studies the minimum energy control problem for linear infinite-dimensional systems with an unbounded input operator and zero terminal state. This problem is approximated by the minimum energy control problem with a small terminal state for which the solution is derived in feedback form. The operators which comprise the feedback are described in terms of differential relations which, depending on circumstances, involve Liapunov or Riccati differential equations. A detailed example illustrates how the general results apply to the wave equation with control in Dirichlet boundary condition.This work was supported by the Polish Ministry of National Education under Grant DNS-T/02/097/90-2.