Study of an optimal control problem related to the Solow control model

We study the Solow model of optimal economic growth. The statement of the optimal control problem under consideration contains mixed constraints on the control. A special change of the control is used to reduce the problem to the study of a new controlled object with a usual geometric constraint. The reachable set of the original control system is found, an existence theorem for the optimal control is proved, and sufficient conditions for the absence of singular regimes are obtained.     

Fluid analysis of an input control problem

A two-station network with controllable inputs and sequencing control, proposed by Wein (Oper. Res. 38:1065–1078, 1990), is analyzed. A control is sought to minimize holding cost subject to a throughput constraint. In a Lagrangian formulation, input vanishes in the fluid limit. Several alternative fluid models, including workload formulations, are analyzed to develop a heuristic policy for the stochastic network. Both the fluid heuristic and Wein’s diffusion solution are compared with the optimal policy by solving the dynamic program. Examples with up to six customer classes, using Poisson arrival and service processes, are presented. The fluid heuristic does well at sequencing control but the diffusion gives additional, and better, information on input control. The fluid analysis, in particular whether the fluid priorities are greedy, aids in determining whether the fluid heuristic contains useful information.     

Control of an extending nonlinear elastic cable with an active vibration control strategy

An active control strategy based on the fuzzy sliding mode control (FSMC) is developed in this research for controlling the large-amplitude vibrations of an extending nonlinear elastic cable. The geometric nonlinearity of the cable and the fixed–fixed boundary of the cable are considered. For effectively and accurately control the motion of the cable with the active control strategy developed, the governing equation of the elastic cable is established and transformed into a multi-dimensional dynamic system with the 3rd order Galerkin method. The active control strategy is developed on the basis of the dynamic system, and the control strategy is applicable to multi-dimensional dynamic systems. In the numerical simulation, large-amplitude vibrations of the cable are effectively controlled with the control strategy. The results of the research demonstrate significances for controlling the cable vibrations of an elevator in practice.     

Model-free control of Lorenz chaos using an approximate optimal control strategy

In this paper, we are concerned with model-free control of the Lorenz chaotic system, where only the online input and output are available while the mathematic model of the system is unknown. The problem is formulated from an optimal control perspective and solved using an iterative method. The convergence of the iteration and the stability of the control law are proven in theory. Simulations validate theoretical conclusions and demonstrate the effectiveness of the proposed method.     

Optimal control of an elastic string with viscous damping

We study bilinear optimal control of a wave equation with one spatial dimension. The problem describes oscillations of an elastic string with viscous damping, and the damping coefficient is taken as the control. The objective functional involves driving the state solution close to a desired profile and incurring a cost on the control. The optimal control is characrerized in terms of an optimality system.     

Aspects of an inventory control study

A case study for the design of a spare parts inventory control system is described. The main emphasis is given to the problems of applying and tailoring forecasting and inventory control theory to suit the needs of a particular problem. In the main, well established methods were used and these are not described in detail. New developments arising in the course of the study have been described elsewhere.     

Optimal control problem with an integral equation as the control object

We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.     

Delayed feedback-based bifurcation control in an Internet congestion model

In this paper, the control of Hopf bifurcation in an Internet congestion model with a single link accessed by a single source is presented. By choosing the gain parameter as a bifurcation parameter, it is found that the system without control cannot guarantee a stationary sending rate. Furthermore, Hopf bifurcation occurs when the positive gain parameter of the system exceeds a critical value. For Internet congestion model, a control model based on delayed feedback is proposed and analyzed for delaying the onset of undesirable Hopf bifurcation. Numerical simulations are given to justify the validity of delayed feedback controller in bifurcation control.     

Analysis of an HIV model with post-treatment control

Recent investigation indicated that latent reservoir and immune impairment are responsible for the post-treatment control of HIV infection. In this paper, we simplify the disease model with latent reservoir and immune impairment and perform a series of mathematical analysis. We obtain the basic infection reproductive number $R_{0}$ to characterize the viral dynamics. We prove that when $R_{0}<1$, the uninfected equilibrium of the proposed model is globally asymptotically stable. When $R_{0}>1$, we obtain two thresholds, the post-treatment immune control threshold and the elite control threshold. The model has bistable behaviors in the interval between the two thresholds. If the proliferation rate of CTLs is less than the post-treatment immune control threshold, the model does not have positive equilibria. In this case, the immune free equilibrium is stable and the system will have virus rebound. On the other hand, when the proliferation rate of CTLs is greater than the elite control threshold, the system has stable positive immune equilibrium and unstable immune free equilibrium. Thus, the system is under elite control.     

Quasistrategies in an abstract guidance control problem

We consider an abstract guidance control problem with elements of uncertainty, where the control procedures are identified with set-valued quasistrategies. The goal of the control consists in the guidance to the objective set under phase constraints.We construct the solvability set by the method of program iterations which is well known in the theory of differential games. We prove that the limit of the iterative procedure has the sense of the set of positional absorption (the stable bridge) introduced by N. N. Krasovskii. This limit coincides with the solvability set in the class of quasistrategies.     

Steplike contrast structure in an elementary optimal control problem

For an elementary nonlinear optimal control problem with a scalar differential constraint and with a small parameter multiplying the derivative but without any constraints on the control, the possibility of emerging fast internal phase transitions in the optimal trajectory is shown as based on results concerning contrast structures in the theory of singularly perturbed boundary value problems.     

Local convergence of an algorithm for solving optimal control problems

In this paper, a method is proposed for the numerical solution of optimal control problems with terminal equality constraints. The multiplier method is employed to deal with the terminal equality constraints. It is shown that a sequence of control functions, which converges to the optimal control, is obtained by the alternate update of control functions and multipliers.The authors wish to thank Dr. N. Fujii for his most valuable comments and suggestions.     

On sensitivity in an optimal control problem

We give a new criterion for the existence of value in differential games. The method of proof involves Lipschitz differential games and hence extends to games with more general dynamics. The connection between using measurable control functions or simply constants is clarified.     

Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of ₂-norms of wavelet expansion coefficients and the constraints are written in form of an ₂ automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in ₂. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns. AMS subject classification 65K10, 65N99, 93B40Angela Kunoth: This work has been supported partly by the Deutsche Forschungsgemeinschaft (SFB 611) at the Universität Bonn and by the European Communitys Human Potential Programme under contract HPRN-CT-2002-00286 Breaking Complexity.     

Active damping control for an underactuated multibody system

Light-weight robots and manipulators stand out due to their very good weight-to-load ratio and a low energy consumption. Unfortunately, the light-weight design yields a lower stiffness, which results in undesired elastic deformations, especially during high-speed working motion. One way to limit these unwanted oscillations can be implemented with modifications of the command signals, e.g. by input shaping or pre-computed feed-forward control. However, a feedback control-concept has the ability to provide a fast compensation of elastic deformations due to high-speed working motions or in case of unwanted environment contact as well as in the presence of other external disturbances. In this contribution, an active damping control (ADC) for fast moving manipulators with significant structural elasticities is presented. This approach does not require additional actuators and it is suitable for a large variety of manipulators that exhibit structural vibrations. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Galilei group in an optimal control problem

In the paper, results of studying an optimal control problem for the motion of a material particle under control constraints are presented. The invariance of this problem with respect to the extended Galilei group is used. From the viewpoint of calculations, the symmetry allows us to construct a family of solutions using an extremal determined numerically. From the analytical viewpoint, the symmetry gives an opportunity to reduce the system’s dimension and to investigate the properties of extremals.     

Allocation of an indivisible resource: Optimal control and prices

Under study is a dynamical system of economics with discrete time whose states at each moment correspond to nonnegative integer points of a two-dimensional vector space. There are two types of products and two facilities each of which can change the state of the system by random integer vectors with different collections of probabilities. By a control we understand the choice at each moment of time of one available collection of probabilities. The goal of control is to minimize the probability of leaving the positive quadrant. The question is considered of the existence of some prices that agree with an optimal control.     

Passive optimal control of an antisymmetric angle-ply laminate

Using the maximum principle of Ref. 1, a procedure to find numerical solutions of certain optimal control problems is given. As an application of this procedure, the optimal control of an antisymmetric angle-ply laminate is worked out in detail. Numerical solutions are given in the form of graphs.     

The Galilei group in an optimal control problem

In the paper, results of studying an optimal control problem for the motion of a material particle under control constraints are presented. The invariance of this problem with respect to the extended Galilei group is used. From the viewpoint of calculations, the symmetry allows us to construct a family of solutions using an extremal determined numerically. From the analytical viewpoint, the symmetry gives an opportunity to reduce the system’s dimension and to investigate the properties of extremals.