On ε-optimal continuous selectors and their application in discounted dynamic programming

A quadratic regulator problem for a class of nonlinear systems is considered in which the control cost is multiplied by a small parameter, which becomes a so-called cheap control problem. Conditions are found under which the minimum cost becomes zero (perfect regulation) and the linear part in the optimal control law becomes dominant as the small parameter goes to zero. Near optimality of control laws truncated from the optimal control law in series form is also found.     

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.     

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H ₀.      

A computational method for solving optimal control of a system of parallel beams using Legendre wavelets

The optimal control of transverse vibration of two Euler–Bernoulli beams coupled in parallel by discrete springs is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. By use of the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of a linear time-invariant lumped-parameter systems. A computationally attractive method based on Legendre wavelets in time domain for solving the optimal control of the lumped parameter systems for any finite interval is proposed. Legendre wavelet integral operational matrix and the properties of a Kronecker product are used to find the approximated optimal trajectory and optimal law of the linear systems with respect to a quadratic cost function by only solving a linear system of algebraic equations. This method provides a straightforward and convenient approach for digital computation. A numerical example is provided to demonstrate the applicability and effectiveness of the proposed method.     

Trajectory generation with rich information content

Often, trajectories for mechanical systems are generated solving some optimization problem. Common approaches include time-optimal, energy optimal, etc., motion profiles. In order to decrease mechanical wear of real plants this profiles provide, e.g., a smooth movement (rest-to-rest) in accordance with restrictions in jerk, acceleration and velocity. There exists a number of methods, to calculate for a given trajectory the plant feed forward action and to design stabilizing controllers. In case of parameter uncertainty the control law often exhibits some adaptive part. Unfortunately, smooth trajectories tend to contain insufficient excitation for adaption and/or identification. Therefore, we propose to consider some measure for the information content concerning some unknown parameters in the trajectory optimization problem. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control with a Guide in the Guarantee Optimization Problem under Functional Constraints on the Disturbance

A motion control problem for a dynamic system under disturbances is considered on a finite time interval. There are compact geometric constraints on the values of the control and disturbance. The equilibrium condition in the small game is not assumed. The aim of the control is to minimize a given terminal performance index. The guaranteed result optimization problem is posed in the context of the game-theoretical approach. In the case when realizations of the disturbance belong to some a priori unknown compact subset of L₁ (the space of functions that are Lebesgue summable with the norm), we propose a new discrete-time control procedure with a guide. The proximity between the motions of the system and the guide is provided by the dynamic reconstruction of the disturbance. The quality of the control process is achieved by using an optimal counter-strategy in the guide. Conditions on the equations of motion under which this procedure ensures an optimal guaranteed result in the class of quasi-strategies are given. The scheme of the proof makes it possible to estimate the deviation of the realized value of the performance index from the value of the optimal result depending on the discretization parameter. Illustrative examples are given.     

Bounds on the micromagnetic energy of a uniaxial ferromagnet

We identify the optimal scaling law for a nonconvex, nonlocal variational problem representing the magnetic energy of a uniaxial ferromagnet. Our analysis is restricted to a certain parameter regime, in which the surface tension is sufficiently small relative to the other parameters of the problem. © 1998 John Wiley & Sons, Inc.     

Approximate solution of nonlinear problems of optimal control of oscillatory processes using the method of canonical separation of motions : PMM vol.40, n 6, 1976, pp. 1024–1033

An asymptotic method of solving certain problems of optimal control of motion of the standard type systems with rotating phase is developed. It is assumed that the controls enter only the small perturbing terms, and that the fixed time interval over which the process is being considered is long enough to ensure that the slow variables change essentially. Assuming also that the system and the controls satisfy the necessary requirements of smoothness, the method of canonical averaging [1] is used to construct a scheme for deriving a simplified boundary value problem of the maximum principle. The structure of the set of solutions of the boundary value problem is investigated and a scheme for choosing the optimal solution with the given degree of accuracy in the small parameter is worked out. The validity of the approximate method of solving the boundary value problem is proved. The method suggested in [2] for constructing a solution in the first approximation for similar problems of optimal control is developed.     

Estimation and control with cubic nonlinearities

The nonlinearities in a dynamic system and its measurement equations are assumed to be cubic and small, i.e., all proportional to a single scalar small parameter . The optimal digital nonlinear feedback control law is carried through the first power of , taking into account the non-Gaussian character of the state conditional distribution. The optimal law involves cubic and linear terms in the state estimate, as well as higher moments of the state conditional distribution.     

Optimal control of rigid body motion with the help of rotors using stereographic coordinates

This paper considers the problem of optimal controlling the rotational motion of a rigid body using three independent control torques developed by three rotors attached with the principal axes of inertia of the body and rotate with the help of electric motors rigidly mounted on the body. The optimal control law is given as non-linear function of new parameterizations of the rotation group derived by using the stereographic projection of the Euler parameters. Given a cost function we seek for a stabilizing feedback control law that minimizes this cost and asymptotically stabilizes the rotational motion of the body. The stabilizing properties of the proposed controllers are proved by using the optimal Liapunov function. Numerical examples and simulation study are presented.     

The energy-optimal motion of a vibration-driven robot in a resistive medium

The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections – one with a positive velocity and the other with a negative velocity of motion of the shell – is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. © Elsevier Ltd. All rights reserved.     

Reinsurance control in a model with liabilities of the fractional Brownian motion type

We propose a model for reinsurance control for an insurance firm in the case where the liabilities are driven by fractional Brownian motion, a stochastic process exhibiting long-range dependence. The problem is transformed to a nonlinear programming problem, the solution of which provides the optimal reinsurance policy. The effect of various parameters of the model, such as the safety loading of the reinsurer and the insurer, the Hurst parameter, etc. on the optimal reinsurance program is studied in some detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.     

Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.     

混合型随机微分方程的传输不等式

该文探讨一类由Wiener过程和Hurst参数1/2＜H＜1分数布朗运动驱动的混合型随机微分方程.通过使用一些变换技巧和逼近方法,这类方程的强解在d2度量和一致度量d∞下的二次传输不等式被建立.     

On compound Poisson processes arising in change-point type statistical models as limiting likelihood ratios

Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In a previous paper of one of the authors it was established that one of these likelihood ratios, which is an exponential functional of a two-sided Poisson process driven by some parameter, can be approximated (for sufficiently small values of the parameter) by another one, which is an exponential functional of a two-sided Brownian motion. In this paper we consider yet another likelihood ratio, which is the exponent of a two-sided compound Poisson process driven by some parameter. We establish, that similarly to the Poisson type one, the compound Poisson type likelihood ratio can be approximated by the Brownian type one for sufficiently small values of the parameter. We equally discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.     

Dynamics of an impact-progressive system

An impact oscillator with a frictional slider is considered. The basic function of the investigated system is to overcome the frictional force and move downwards. Based on the analysis of the oscillatory and progressive motions of the system, we introduce an impact Poincaré map with dynamical variables defined at the impact instants. The nonlinear dynamics of the impact system with a frictional slider is analyzed by using the impact Poincaré map. The stability and bifurcations of single-impact periodic motions are analyzed, and some information about the existence of other types of periodic-impact motions is provided. Since the system equilibrium is moving downwards, one way to monitor the progression rate is to calculate its progression in a finite time. The simulation results show that in a finite time, the largest progression of the system is found to occur for period-1 multi-impact motions existing in the regions of low forcing frequencies. Secondly, the progression of the period-1 single-impact motion with peak-impact velocity is also distinct enough. However, it is important to note, that the largest progression for period-1 multi-impact motion existing at a low forcing frequency is not an optimal choice for practical engineering applications. The greater the number of the impacts in an excitation period, the more distinct the adverse effects such as high noise levels and wear and tear caused by impacts. As a result, the progression of the period-1 single-impact motion with the peak-impact velocity is still optimal for practical applications. The influence of parameter variations on the oscillatory and progressive motions of the impact-progressive system are elucidated accordingly, and feasible parameter regions are provided.     

布朗运动在（r，p）－容度意义下的下极限性质

本文证明了布朗运动在（ｒ，ｐ）－容度意义下的一些基本下极限性质．     

An analytic–numerical method for the construction of the reference law of operation for a class of mechanical controlled systems

An analytic–numerical method for the construction of a reference law of operation for a class of dynamic systems describing vibrations in controlled mechanical systems is proposed. By the reference law of operation of a system, we mean a law of the system motion that satisfies all the requirements for the quality and design features of the system under permanent external disturbances. As disturbances, we consider polyharmonic functions with known amplitudes and frequencies of the harmonics but unknown initial phases. For constructing the reference law of motion, an auxiliary optimal control problem is solved in which the cost function depends on a weighting coefficient. The choice of the weighting coefficient ensures the design of the reference law. Theoretical foundations of the proposed method are given.