Turnpike Solutions in the Procedures Seeking Optimal Controls

Many applied problems of optimal control feature degeneracy, that is, have among their conditions some latent passive differential relations or discrete chains that can be eliminated to simplify the problem by reducing its order without affecting the desired solution. Its corresponding trajectory is a turnpike, that is, an invariant under all or part of the boundary conditions. The turnpike solutions can be used as efficient initial approximations in iterative methods of search for optimal controls. Consideration was given to the existence conditions for precise or approximate turnpike solutions, a scheme to seek and refine them was suggested, and efficiency of this approach on the whole was illustrated by representative examples. The class of control on models related with solution of topical problems of stable development where the turnpikes are pivotal was discussed in detail.     

Large time control and turnpike properties for wave equations

In the last decades mathematical control theory has been extensively developed to handle various models, including Ordinary and Partial Differential Equations (ODE and PDE), both of deterministic and stochastic nature, discrete and hybrid systems.However, little attention has been paid to the length of the time horizon of control, which is necessarily long in many applications, and to how it affects the nature of controls and controlled trajectories. The turnpike property refers precisely to those aspects and stresses the fact that, often, optimal controls and trajectories, in long time intervals, undergo some relevant asymptotic simplification property ensuring that, during most of the time-horizon of control, optimal pairs remain close to the steady-state optimal one.Due to the intrinsic finite velocity of propagation and the oscillatory nature of solutions of the free wave equation, optimal controls for waves are typically of oscillatory nature. But, despite this, as we shall see, under suitable coercivity conditions on the cost functional to be minimised and when controllability holds, the turnpike property is also fulfilled for the wave equation.When this occurs, the approximation of the time-depending control problem by the steady-state one is justified, a fact that is often employed in applications to reduce the computational cost.We present some recent results of this nature for the wave equation and other closely related conservative systems, and discuss some other related issues and a number of relevant open problems that arise in this field.     

Turnpike solutions in optimal control problems for quantum-mechanical systems

We consider the problem of optimal impulse control for a quantum-mechanical spin sequence as a system of oscillators. We study this problem with a double transformation to a derivative problem in which phase variables are oscillator amplitudes. As a result, we get the turnpike solutions. For special (turnpike) sets of boundary conditions they correspond exactly to generalized solutions of the original problem, while for other conditions they can be used as initial approximations for iterative procedures. This problem is a generalization of the special case of two oscillators which we study exhaustively and use as an illustrative example.     

Control of singularly perturbed hybrid stochastic systems

We study a class of optimal stochastic control problems involving two different time scales. The fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process. Under a fundamental “ergodicity” property for a class of “infinitesimal control systems” associated with the fast mode, we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system. Both the finite- and infinite-discounted horizon cases are considered. We show how an approximate optimal control law can be constructed from the solution of the limit control problem. In the particular case where the infinitesimal control systems possess the so-called turnpike property, i.e., characterized by the existence of global attractors, the limit control problem can be given an interpretation related to a decomposition approach     

证券组合的快车道问题研究

本文建立了新的证券组合最优控制模型，分析了证券市场长期投资过程中证券组合的快车道问题；提出了隐式规划的模拟退火方法，进行了快车道的仿真，这对于证券市场长期稳定运行和投资具有实际意义。     

Turnpike theorem in the problem of continuous optimization with phase restrictions

The turnpike theorem in the problem of optimal control by ordinary differential equations with phase restrictions is proved. An example is given.     

经济转轨的最终态势：快车道理论应用研究

本文针对计划经济向社会主义市场经济过度中以国有和非国有企业为代表的经济转轨最终态势问题，建立了经济转轨最优控制模型。应用快车道理论分析了经典和隐式规划结构，提出了隐式规划的模拟退火方法，进行了快车道的仿真。     

An approximate optimal control for discrete models: Its design

Constructive schemes for designing approximate optimal controls based on the Krotov sufficient conditions of optimality, extension principle, and global estimation are described. Different approximations for Bellman-type relations and effective indirect schemes based on degeneracy and turnpike nature of solutions of applied problems are studied. Irrespective of the initial formulation of a problem, discrete-time models (exact or approximate models obtained from discretization of the initial model) are used in the final design stage. Examples are given.     

Degenerate problem of stabilization of a specified trajectory

Consideration is given to the problem of stabilization of a specified (supporting) mode as a degenerate problem of minimization of the root-mean-square deviation from it where there are no deviations of control actions. This formulation makes it possible to completely use available limited control resources for stabilization. We propose a method of approximate optimal synthesis in the neighborhood of a turnpike manifold that is obtained as a solution in the form of optimal synthesis of linearly quadratic problem with unbounded linear control. The investigation is performed for a linearized discrete model of the controlled system; however, the obtained stabilizing control can be applied in the initial nonlinear system directly or after the correction by regular iterations.     

Consistent approximation of a nonlinear optimal control problem with uncertain parameters

This paper focuses on a non-standard constrained nonlinear optimal control problem in which the objective functional involves an integration over a space of stochastic parameters as well as an integration over the time domain. The research is inspired by the problem of optimizing the trajectories of multiple searchers attempting to detect non-evading moving targets. In this paper, we propose a framework based on the approximation of the integral in the parameter space for the considered uncertain optimal control problem. The framework is proved to produce a zeroth-order consistent approximation in the sense that accumulation points of a sequence of optimal solutions to the approximate problem are optimal solutions of the original problem. In addition, we demonstrate the convergence of the corresponding adjoint variables. The accumulation points of a sequence of optimal state-adjoint pairs for the approximate problem satisfy a necessary condition of Pontryagin Minimum Principle type, which facilitates assessment of the optimality of numerical solutions.     

Characterization of the smoothness of optimal periodic control problem solutions

Optimal periodic control problems with integrable or square-integrable control variables are considered. Necessary conditions for optimality, specific features of the set of feasible solutions and the system complex structure are exploited to characterize the smoothness of optimal solutions or all feasible solutions. It is shown that the above devices and the Hardy-Littlewood theorems on Lipschitz classes allow the integral characterization of the control and state continuity for basic practical situations. The importance of such characterization for the trigonometric approximation of optimal periodic control problems is emphasized. The theoretical considerations are illustrated by examples of control problems from the field of chemical engineering.     

Difference approximation of optimal periodic control problems

The constrained optimal periodic control problem is approximated by a sequence of discretized problems in which the system of differential equations of the basic continuous problem is replaced by a system of one–step difference equations. Two kinds of approximate optimal controls are derived from the optimal solutions of discretized problems: the first in the form of a step function and the second in the form of a special trigonometric polynomial generated by a positive kernel. Sufficient conditions for approximate solutions to be weakly convergent to the optimal solution of the basic problem are given. Certain improvements in the difference approximation considered are discussed and potential applications given.     

Suboptimal control of singularly perturbed systems and periodicoptimization

A traditional approach to singularly perturbed optimal control problems is based on an approximation of these problems by reduced problems which are obtained via the formal replacement of the fast variables by the states of equilibrium of the fast subsystems considered with frozen slow variables and controls. It is shown that such an approximation is valid if and only if certain families of periodic optimization problems admit steady state solutions. It is also shown how the solutions of these problems can be used to construct suboptimal controls for singularly perturbed problems when approximation by reduced problems is not possible     

Optimal control problem via neural networks

This paper attempts to propose a new method based on capabilities of artificial neural networks, in function approximation, to attain the solution of optimal control problems. To do so, we try to approximate the solution of Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. In the proposed error function, we used trial solutions for the trajectory function, control function and the Lagrange multipliers. These trial solutions are constructed by using neurons. Then, we minimize the error function that contains just the weights of the trial solutions. Substituting the optimal values of the weights in the trial solutions, we obtain the optimal trajectory function, optimal control function and the optimal Lagrange multipliers.     

Multicomponent model-based optimization of the regional development strategies

Consideration was given to optimization of the development strategy for the multi-component socio-ecologic-economic model with the innovation block oriented to practical modeling of particular regions. Under the idealizing assumptions reflecting the model specificity (domination of the linear control actions), the turnpike solution which is directly independent of the boundary conditions was determined and used as an efficient initial approximation in the multistage optimization procedure with successive refinement of the model and the scenario computational experiments for estimation of the impact of various real factors.     

On the relation between strict dissipativity and turnpike properties

For discrete time nonlinear systems we study the relation between strict dissipativity and so called turnpike-like behavior in optimal control. Under appropriate controllability assumptions we provide several equivalence statements involving these two properties. The relation of strict dissipativity to an exponential variant of the turnpike property is also studied.     

Control optimization in bilinear systems

We consider optimization problems for continuous systems linear with respect to state and control. Due to the specifics of such problems (degeneracy, turnpike character of solutions), we propose to use nonlocal Krotov’s iterative method together with the Gurman-Dykhta transformation into a regular derivative problem, which significantly improves the efficiency of this method. The proposed procedure is illustrated with an example.     

Approximate discrete maximum principle for the difference approximation of optimal periodic control problems

Distinctive features of the difference approximation for optimal periodic control (OPC) problems are considered. It is shown that the convergence conditions on approximate solutions imply consistency conditions on the time grid step and the accuracy of the periodicity constraint approximation such that the approximate discrete maximum principle holds for a sequence of discretized problems. A specific form of this principle when applied to the difference approximation method for OPC problems is analysed and computational methods are discussed. Application of this approach to a class of systems non-linearly dependent on control variables, and occurring in the chemical industry, is suggested.     

离散时滞系统的最优滑模控制

针对离散线性时滞系统,设计了最优滑模面和最优离散变结构控制律.在设计最优滑模面的过程中,利用逐次逼近算法,将既含有时滞项又含有超前项的两点边值问题转化为不含时滞项和超前项的线性两点边值问题,并证明了其解序列一致收敛于原系统的最优滑模.进一步给出了原系统的变结构控制律.     

Control of single-sector economy over a finite time interval with allowance for employer consumption

The problem of optimal control of a single-sector economics over a finite time interval was studied for the class of the linear uniform production functions in terms of the criterion for maximum employer consumption under exponential growth of the manpower resources. Solution was obtained in the form of the “turnpike theorem.” Its properties were considered. The results obtained were specified for the case of the Cobb-Douglas production function.