Necessary integral optimality conditions for quasi-singular controls in one control problem

An optimal control problem with variable structure described by a system of nonlinear integral Volterra-type equations is considered in the article. A necessary optimality condition is obtained in the form of a linearized maximum condition. A case of degeneracy of the linearized maximum principle is further studied. Integral necessary optimality conditions of quasi-singular controls are proved.     

Necessary optimality conditions in a nonsmooth control problem with variable structure

An optimal control problem with nonsmooth performance criterion described by a system of ordinary differential equations is considered. Necessary first-order optimality conditions are obtained.     

Necessary first- and second-order optimality conditions in problems of control described by a system of Volterra difference equations

The terminal problem of optimal control described by a system of Volterra difference equations is considered. Constructively verifiable necessary first- and second-order optimality conditions are obtained under the assumption that the domain of control is open.     

Necessary conditions of optimality for vector-valued impulsive control problems

A vector-valued impulsive control problem is considered whose dynamics, defined by a differential inclusion, are such that the vector fields associated with the singular term do not satisfy the so-called Frobenius condition. A concept of robust solution based on a new reparametrization procedure is adopted in order to derive necessary conditions of optimality. These conditions are obtained by taking a limit of those for an appropriate sequence of auxiliary “standard” optimal control problems approximating the original one. An example to illustrate the nature of the new optimality conditions is provided.     

Necessary conditions of fractional optimal control problems with state constraints in the sense of Riemann‐Liouville

This paper is devoted to the optimal control problem of a fractional dynamic system with state constraints in the sense of Riemann‐Liouville. By means of the needle variation, we establish the Pontryagin's maximum principle for the optimal control problem. Moreover, when such a necessary condition is singular in some sense, we investigate the “second‐order” necessary conditions accordingly. As an application, two examples are presented to demonstrate the accuracy and efficiency of the result.     

Necessary and sufficient conditions for global optimality of eigenvalue optimization problems

The necessary and sufficient conditions for global optimality are derived for an eigenvalue optimization problem. We consider the generalized eigenvalue problem where real symmetric matrices on both sides are linear functions of design variables. In this case, a minimization problem with eigenvalue constraints can be formulated as Semi-Definite Programming (SDP). From the Karush-Kuhn-Tucker conditions of SDP, the necessary and sufficient conditions are derived for arbitrary multiplicity of the lowest eigenvalues for the case where important lower bound constraints are considered for the design variables. Received May 18, 2000     

Maximum principle for optimal control of vibrations of a dynamic Gao beam in contact with a rigid foundation

In our preceding paper, we studied an optimal control problem of vibrations of a dynamic Gao beam in contact with a reactive foundation and derived the Pontryagin maximum principle for the controlled system in fixed final horizon case. As a follow-up, in this paper, we focus on the investigation of the Gao beam that may come in contact with a rigid foundation underneath it. In this case, the nonlinear viscoelastic beam equation is equipped with the Signorini condition. By the Dubovitskii and Milyutin functional analytical approach, we investigate the new optimal control problem with multiple inequality constraints and present further original results of current interests.     

Optimal control of vibrations of a dynamic Gao beam in contact with a reactive foundation

In this paper, we investigate the optimal control of vibrations of a nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a reactive foundation underneath it. By the Dubovitskii and Milyutin functional analytical approach, we derive the Pontryagin maximum principle of the system governed by the Gao beam equation. And the first-order necessary optimality condition is presented for the optimal control problem in fixed final horizon case. Finally, we also sketch the numerical solution based on the obtained theoretical results.     

Singular controls in a given stepwise control problem

The Goursat-Darboux stepwise problem of systems control, i.e., the multi-stage problem of control of systems with distributed parameters, is considered. Second-order formulas for the increment in the quality functional are constructed. In investigating these formulas on needle variations in the control, a number of necessary Pontryagin maximum principle-type optimality conditions is obtained and a singular case is studied.     

Optimal control of mobile sources for heat conductivity processes

We consider a problem of optimal control of the mobile sources for heat conductivity processes, described by parabolic equation and systems of ordinary differential equations. Sufficient conditions for Frechet differentiability of the performance criterion and an expression for its gradient were determined. The necessary optimality condition was established in the form of the integral principles of maximum. The theoretical conclusions are illustrated by the solution of a numerical example.     

A multiple-criteria decision problem for optimal management of farm resources under uncertainty - a case study

An attempt is made to minimize the total operational cost and specific energy for rice cultivation, when the constraints relating to the availability of power sources and crop yield are supposed to be stochastic in nature. The Sequential Linear Goal Programming algorithm has been used to solve the resulting multiple objective optimization problem.     

High-order necessary conditions of optimality for nonlinear control systems

For nonlinear affine control systems with unbounded controls, necessary high-order optimality conditions are derived. These conditions are stated in the form of a high-order maximum principle and are expressed in terms of Lie brackets and Newton diagrams.     

Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel’s martingales and an independent multi-dimensional Brownian motion. Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables. As an application, the linear quadratic stochastic optimal control problem is studied.     

Optimal control on Lie groups with applications to attitude control

In this paper we solve a class of optimal control problems on Lie groups in the sense that we derive differential equations which the optimal controls must satisfy. These results are applied to the attitude control of a spacecraft modeled as a rigid body. Specifically, we derive control laws (both in open-loop and closed-loop form) to maneuver the spacecraft between two given rotational states in finite time. The laws are such that a cost functional measuring the over-all angular velocity during the spacecraft’s motion is minimized. They do not require recourse to numerical methods and hence can be easily implemented in an on-board attitude control system. After dealing with a three-axis controlled spacecraft we also discuss the case that only torques about two principal axes of an axisymmetric spacecraft can be exerted.     

Necessary first and second order optimality conditions in a discrete optimal control problem

An optimal control problem described by a system of two-dimensional nonlinear difference Volterra-type equations that are a discrete analogue of the two-dimensional integral Volterra equation is considered. The first and second variations of the quality functional are calculated assuming that the control domain is open. They are used to obtain an analogue of the Euler equation and to find the necessary second order optimality conditions that can be checked constructively.     

一类混杂系统的最优控制

研究了一类脉冲依赖于状态的混杂系统的最优控制问题.与传统的变分方法不同,通过将跳跃瞬间转化为一个新的待优化参数,得到了该混杂系统的必要最优性条件,从而将最优控制问题转化为一边界值问题,该边界值问题可由数值方法或解析方法解决.此外,利用广义微分的理论,将该必要最优性条件推广到Frechet微分形式.结论表明,在混杂动态系统运行的连续部分,最优解所满足的必要性条件和传统的连续系统相同.在混杂动态系统的脉冲点处,哈密尔顿函数满足连续性条件,协态变量则满足一定的跳跃条件.最后,通过两个实例分析,表明该方法是有效的.     

Necessary and sufficient conditions for a bounded solution to the optimality equation in average reward Markov decision chains

We consider average reward Markov decision processes with discrete time parameter and denumerable state space. We are concerned with the following problem: Find necessary and sufficient conditions so that, for arbitrary bounded reward function, the corresponding average reward optimality equation has a bounded solution. This problem is solved for a class of systems including the case in which, under the action of any stationary policy, the state space is an irreducible positive recurrent class.     

Solution of optimal control problem in class of piecewise-constant functions

A numerical method of solution of problems of optimal control of objects that may be described by systems of ordinary differential equations in the class of piecewise-constant controls is proposed in the article. At the same time, both the piecewise-constant values of the controls as well as the constancy intervals of these values are optimized. Analytic formulas of the gradient of the functional with respect to the optimized parameters are obtained. The formulas for the gradient of the functional obtained for an initial continuous optimal control problem and the corresponding discretized optimal control problem are compared and results of numerical experiments are presented.