Algorithms for solving discrete optimal control problems with infinite time horizon and determining minimal mean cost cycles in a directed graph as decision support tool

Time-discrete systems with a finite set of states are considered. Discrete optimal control problems with infinite time horizon for such systems are formulated. We introduce a certain graph-theoretic structure to model the transitions of the dynamical system. Algorithms for finding the optimal stationary control parameters are presented. Furthermore, we determine the optimal mean cost cycles. This approach can be used as a decision support strategy within such a class of problems; especially so-called multilayered decision problems which occur within environmental emission trading procedures can be modelled by such an approach.     

Mathematical modeling and control of population systems: Applications in biological pest control

The aim of this paper is to apply methods from optimal control theory, and from the theory of dynamic systems to the mathematical modeling of biological pest control. The linear feedback control problem for nonlinear systems has been formulated in order to obtain the optimal pest control strategy only through the introduction of natural enemies. Asymptotic stability of the closed-loop nonlinear Kolmogorov system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. Numerical simulations for three possible scenarios of biological pest control based on the Lotka–Volterra models are provided to show the effectiveness of this method.     

具有跳跃的扩散过程的最优控制及相关的积分-微分型分布参数系统的最优控制问题

§1.引言在[1]、[2]及[3]中考虑了可以用偏微分方程方法处理的一类具有连续轨道的扩散过程的最优控制问题.近来[4]研究了具有跳跃的扩散过程的最优控制问题,证明了 reward函数满足 Bellman 方程.     

Optimal strategies to diminish a pest population via bilinear controls

In this work we model and analyze two control strategies to diminish a pest population using traps. The action of any trap depends on its age. Both problems contain bilinear control rates. The large-time behavior of the model with time-periodic inflow is investigated. The first control strategy deals with a finite horizon problem while the second one is related to a time-periodic problem. We obtain Pontryagin’s principle for both control problems. A special attention is given to the periodic problem. Pontryagin’s principle is used to derive a conceptual gradient-type algorithm to approximate the optimal solution. Numerical tests are given.     

Stochastic design optimization of asynchronous flexible assembly systems

This paper presents the application of the stochastic quasigradient method (SQG) of Ermoliev and Gaivaronski to the performance optimization of asynchronous flexible assembly systems (AFAS). These systems are subject to blocking and starvation effects that make complete analytic performance modeling difficult. A hybrid algorithm is presented in this paper which uses a queueing network model to set the number of pallets in the system and then an SQG algorithm is used to set the buffer spacings to obtain optimal system throughput. Different forms of the SQG algorithm are examined and the specification of optimal buffer sizes and pallet numbers for a variety of assembly systems with ten stations are discussed. The combined Network-SQG method appears to perform well in obtaining a near optimal solution in this discrete optimization example, even though the SQG method was primarily designed for application to differentiable performance functionals. While a number of both theoretical and practical problems remain to be resolved, a heuristic version of the SQG method appears to be a reasonable technique for analyzing optimization problems for certain complex manufacturing systems.     

Asymptotic Behavior of the Value Functions of Discrete-Time Discounted Optimal Control

We study deterministic discounted optimal control problems associated with discrete-time systems. It is shown that, for small discount rates, the controllability properties of the underlying system can guarantee the convergence of the discounted value function to the value function of the average yield. An application in the theory of exponential growth rates of discrete inclusions is presented. This application motivates the analysis of infinite-horizon optimal control problems with running yields that are unbounded from below.     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

Dynamics analysis of a pest management prey–predator model by means of interval state monitoring and control

In this work, a new pest management strategy by means of interval state monitoring is introduced into a prey–predator model, i.e. when the pest density exceeds the slightly harmful level but is below the damage level, the biological control is adopted in case of the predator density below a maintainable level, once the pest density exceeds the damage level, the chemical control is adopted. In order to determine the frequency of the chemical control and yield of releases of the predator, analysis on the existence of order-1 or order-2 periodic orbit is carried out by the construction of Poincaré map. The results could make the pest control strategy to be a periodic one without real-time monitoring the species. In addition, the stability and attractiveness of the periodic orbit are obtained by geometry approach, which ensures a certain robustness of control, i.e., even though the species densities are detected inaccurately or with a deviation, the system will be eventually stable at the periodic orbit under the control action. Furthermore, to obtain the optimum chemical control strength and yield releases of the predator, an optimization problem is constructed. The analytical results presented in the work are validated by numerical simulations for a specific model.     

An approximation procedure for solutions to Mayer problems with cost functional not of classC

We consider here control problems in the Mayer form, with a cost functional which is continuous, but not necessarily of classC ¹. The usual necessary conditions for such problems cannot be applied, since they require that the cost functional beC ¹. We describe a convergence procedure, based upon approximations to the cost functional, that will yield an optimal trajectory to such control systems. Criteria are given which justify that this approximation procedure will yield a trajectory and control, satisfying certain prescribed conditions, which minimize the above cost functional within a specified class of such pairs. Examples are presented.     

Stochastic control problems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay. We state conditions for certain classes of such systems under which the stochastic control problems become finite-dimensional. These conditions are illustrated with three applications. First, we solve some linear quadratic problems with delay. Then we find the optimal consumption rate in a financial market with delay. Finally, we solve explicitly a deterministic fluid problem with delay which arises from admission control in ATM communication networks.     

Multiobjective optimal control of a four-body kinematic chain

Optimal control of mechanical systems is an active area of research. However, so far, most contributions are taking only one single objective into account, whereas for many practical problems, one is interested in optimizing several conflicting objectives at the same time. Applying singleobjective optimization to each of them leads to several trajectories each being optimal for one objective, but ignoring all others. In contrast to that, combining all objectives and using multiobjective optimization leads to a variety of trade off solutions taking all objectives into account simultaneously. We use the direct discretization method DMOCC (Discrete Mechanics and Optimal Control for Constrained systems) to approximate trajectories of the underlying optimal control problems, resulting in restricted optimization problems of high dimension. For the multiobjective part, we apply a reference point technique which successively utilizes an auxiliary distance function to gain the trade off solutions. The presented approach is illustrated by the multiobjective optimal control of a constrained multibody system. A four-body kinematic chain is controlled in a rest to rest maneuver, for which minimal control effort and minimal required maneuver time are the conflicting objectives. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of distributed time delay systems with Dirichlet boundary conditions

Various optimization problems associated with the optimal controlof distributed parameter systems with time lags appearing inthe boundary conditions have been studied recently by Wang (1975),Knowles (1978), Wong (1987), Kowalewski (1987a,b, 1988a,b,c,1990a,b,c,d, 1991, 1993a,b,c,d, 1995) and Kowalewski & Duda(1992). In this paper optimal boundary control problems fordistributed systems described by linear partial differentialequations of parabolic and hyperbolic type in which constanttime delays appear in the state equations are considered. Sufficientconditions for the existence of a unique solutions of such equationswith the Dirichlet boundary conditions are proved. The performancefunctional has the quadratic form. The time horizon T is fixed.Finally, we impose some constraints on the control. Making useof Lions' scheme (Lions 1971) necessary and sufficient conditionsof optimality for the Dirichlet problem with the quadratic performancefunctional and constrained control are derived. The flow chartof the algorithm, which can be used in the numerical solvingof certain optimization problems for distributed parameter systems,is also presented.     

Continuity of optimal control costs and its application to weak KAM theory

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry–Mather problems corresponding to these systems.     

On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems

In this paper we consider a (one-shot) multigrid strategy for solving the discretized optimality system (KKT system) of a PDE-constrained optimization problem. In particular, we discuss the construction of an additive Schwarz-type smoother for a certain class of optimal control problems. A rigorous multigrid convergence analysis is presented. Numerical experiments are shown which confirm the theoretical results. The work was supported by the Austrian Science Fund (FWF) under grant SFB 013/F1309.     

On a class of dynamic programming problems whose optimal controls and states are independent of the future

When applying dynamic programming for optimal decision making one usually needs considerable knowledge about the future. This knowledge, e.g. about future functions and parameters, necessary to determine optimal control policies, however, is often not available and thus precludes the application of dynamic programming.In the present paper it is shown that for a certain class of dynamic programming problems the optimal control policy is independent of the future. To illustrate the results an application in inventory control is given and further applications in the theories of economic growth and corporate finance are listed in the references.     

OPTIMAL STRATEGY FOR ERADICATION OF MULTIPLE POPULATIONS USING THE STERILE INSECT TECHNIQUE

Sterile insect technique (SIT) provides an attractive alternative to chemical pesticides for insect pest eradication. Because mass rearing facilities are expensive both to construct and to maintain, project managers have a limited number of sterile insects to use in any single eradication campaign. The problem of availability of sufficient numbers of sterile insects can be magnified when two or more independent pest infestations occur at the same time. In this paper we present the results of a modeling study in which dynamic programming is used to compute the optimal allocation strategy for release of sterile insects to control two independent infestations of Mediterranean fruit fly, Ceratitis capitata (Wied.) (medfly). We assume that a fixed number of sterile flies are available per week. Both deterministic and stochastic analyses are presented.     

Optimal control of a class of strongly nonlinear parabolic systems

This paper considers some typical optimal control problems for a class of strongly nonlinear parabolic systems. After some necessary preparation, it is shown that the family of admissible trajectories is a weakly closed and weakly sequentially compact subset of a reflexive Banach space and that the set of attainable states at any given time is a weakly compact subset of a Hilbert space. Using these basic results, proofs of existence of optimal controls are presented. A terminal control problem, a special Bolza problem, and a time optimal control problem are solved, and the necessary conditions of optimality for the corresponding control problems are given.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

An applied cell mapping method for optimal control problems

From the application point of view, a series of modifications are proposed for the cell mapping method discussed in Ref. 1 for the optimal control analysis of dynamical systems. The cell order around the target set is rearranged. A set of common discriminate principles is used for the selection of the optimal one among competing control strategies of the same cost. Inequality constraints of the system are taken into account. The number of elements in the set of allowable time intervals is not prescribed, but left open. These modifications seem to make the cell mapping method more efficient for analyzing feedback systems and for obtaining their global optimal control information. The algorithms presented in this paper could broaden the application of the cell mapping approach of Ref. 1 to a wider class of engineering problems.     

Optimal control of the motion of a quasilinear oscillatory system by small forces : PMM vol. 39, n 6, 1975, pp. 995–1005

We solve certain optimal control problems for the motion of a single-frequency oscillatory system which in the unperturbed state consists of an arbitrary number of oscillating elements. The solution is performed in the first approximation with respect to a small parameter . We assume that the frequency depends upon slow time, while the control goes only into the perturbing terms, so that the system is formally weakly controllable [1], But since the time interval over which the process evolves is a quantity ˜1/, all the controlled quantities are able to vary substantially [2, 3], i.e. we investigate the case, interesting in practice, of small but protracted control forces. As mechanical examples we calculate some optimal control problems for the oscillations of systems of the plane oscillator type, etc.