Class Of Optimal Control Problems Research Articles

不連続な状態遷移を考慮した学習最適制御による歩行軌道の生成手法

This paper is concerned with a gait generation framework for legged robots based on iterative learning control (ILC) of Hamiltonian systems. This method allows one to obtain solutions to a class of optimal control problems by iteration of laboratory experiments and, furthermore, precise knowledge of the plant model is not required for it by taking advantage of a symmetric property of Hamiltonian systems. Generally in walking motion, there are discontinuous state transitions caused by collision between the foot and the ground. The proposed framework can also deal with such state transitions without using the parameters of the transition model by combining ILC method and the least-squares. It is applied to a compass-like biped robot to generate optimal gait on the level ground. Some numerical examples demonstrate the effectiveness of the proposed method.

Clebsch optimal control formulation in mechanics

This paper introduces and studies a class of optimal control problems based on the Clebsch approach to Euler-Poincare dynamics. This approach unifies and generalizes a wide range of examples appearing in the literature: the symmetric formulation of N-dimensional rigid body and its generalization to other matrix groups; optimal control for ideal flow using the back-to-labels map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the N-Camassa-Holm equation and a new geodesic interpretation of its singular solutions.

Some Applications of Hamilton–Jacobi Inequalities for Classical and Impulsive Optimal Control Problems

This article is devoted to some applications of Hamilton–Jacobi inequalities for control problems of ordinary and impulsive dynamical systems. We focus on the study of necessary and sufficient global optimality conditions. The optimality conditions are obtained by using certain families of Lyapunov type functions, where Lyapunov type functions can be strongly or weakly monotone solutions of the corresponding Hamilton–Jacobi inequalities. For classical problems of the optimal control theory, new variants of the Caratheodory and Krotov types optimality conditions and conditions of the so-called Hamilton–Jacobi canonical optimality theory are proposed and compared. For impulsive dynamical systems with trajectories of bounded variation, conditions for the strong and weak monotonicity of the Lyapunov type functions are obtained. These monotonicity conditions are formulated in terms of generalized Hamilton–Jacobi inequalities and they allow to investigate various questions of the position and optimal control theory for impulsive systems.

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.

Optimal synthesis in an infinite-dimensional space

For a class of optimal control problems and Hamiltonian systems generated by these problems in the space l2, we prove the existence of extremals with a countable number of switchings on a finite time interval. The optimal synthesis that we construct in the space l2 forms a fiber bundle with piecewise smooth two-dimensional fibers consisting of extremals with a countable number of switchings over an infinite-dimensional basis of singular extremals.

Modified domain decomposition method for Hamilton-Jacobi-Bellman equations

This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.

Optimization of the velocity of motion for a flow according to Föppl-Lavrent’ev scheme

Problems concerning the optimal control over the advance of an object in a still water are considered in the framework of the nonstationary Euler hydrodynamical equation. It is assumed that the trail of the flow contains two point vortices of given intensity. The control parameter is the velocity of the advance, as a function of time. These optimization problems for a system of nonlinear partial differential equations having a free boundary (in the form of vortex centers that are no given a priory) are reduced to classical optimal control problems for a system of ordinary differential equations.

Nonlinear integro-differential equations and optimal control problems on time scales

This paper is mainly concerned with a class of optimal control problems of a system governed by the nonlinear integro-differential equation on time scales. Using the contraction mapping principle, the existence and uniqueness of a weak solution of the nonlinear integro-differential equation on time scales are proved. Discussing L T 1 -strong–weak lower semicontinuity of integral functional on time scales, we give sufficient conditions for the existence of optimal control. By the virtue of the integration by parts formula on time scales, the necessary conditions of optimality are derived.

Stochastic Optimal Control and Linear Programming Approach

We study a classical stochastic optimal control problem with constraints and discounted payoff in an infinite horizon setting. The main result of the present paper lies in the fact that this optimal control problem is shown to have the same value as a linear optimization problem stated on some appropriate space of probability measures. This enables one to derive a dual formulation that appears to be strongly connected to the notion of (viscosity sub) solution to a suitable Hamilton-Jacobi-Bellman equation. We also discuss relation with long-time average problems.

A necessary condition for optimal control of initial coupled forward-backward stochastic differential equations with partial information

This paper is concerned with a class of optimal control problems of forward-backward stochastic differential equations. One feature of these problems is that they are in the case of partial information and state equations are coupled at initial time. In terms of a classical convex variational technique, we establish a partial information maximum principle for the foregoing optimization problems. We also work out an example of partial information linear-quadratic optimal control to illustrate the application of the theoretical results; meanwhile, we find a forward-backward stochastic differential filtering equation, which is essentially different from classical forward stochastic filtering equations.

Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation

Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation

On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations

In this paper we study a classical Dirichlet optimal control problem for a nonlinear elliptic equation with the coefficients as controls in $L^\infty(\Omega)$. Since such problems have no solutions in general, we make an assumption on the coefficients of the state equation and introduce the class of so-called solenoidal controls. Using the direct method in the calculus of variations, we prove the existence of at least one optimal pair. We also study the stability of the above optimal control problem with respect to the domain perturbation. With this goal we introduce the concept of Mosco-stability for such problems and analyze the variational properties of Mosco-stable problems with respect to different types of domain perturbations.

Optimal stationary control of discrete processes and a polynomial time algorithm for stochastic control problem on networks

Abstract The stochastic version of classical discrete optimal control problems with a finite set of states and infinite time horizon is studied. An approach for determining the optimal stationary control in discrete processes with deterministic and random states’ transitions of the dynamical system is described. The main problem is formulated on networks and a polynomial time algorithm for its solving is proposed. An application of the proposed algorithm for a class of stochastic decision problems is described.

Nonlinear dynamic systems and optimal control problems on time scales

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

Algorithm 902

An algorithm is described to solve multiple-phase optimal control problems using a recently developed numerical method called the Gauss pseudospectral method . The algorithm is well suited for use in modern vectorized programming languages such as FORTRAN 95 and MATLAB. The algorithm discretizes the cost functional and the differential-algebraic equations in each phase of the optimal control problem. The phases are then connected using linkage conditions on the state and time. A large-scale nonlinear programming problem (NLP) arises from the discretization and the significant features of the NLP are described in detail. A particular reusable MATLAB implementation of the algorithm, called GPOPS , is applied to three classical optimal control problems to demonstrate its utility. The algorithm described in this article will provide researchers and engineers a useful software tool and a reference when it is desired to implement the Gauss pseudospectral method in other programming languages.

Optimal control of a class of fractional heat diffusion systems

In this paper, a solution procedure for a class of optimal control problems involving distributed parameter systems described by a generalized, fractional-order heat equation is presented. The first step in the proposed procedure is to represent the original fractional distributed parameter model as an equivalent system of fractional-order ordinary differential equations. In the second step, the necessity for solving fractional Euler–Lagrange equations is avoided completely by suitable transformation of the obtained model to a classical, although infinite-dimensional, state-space form. It is shown, however, that relatively small number of state variables are sufficient for accurate computations. The main feature of the proposed approach is that results of the classical optimal control theory can be used directly. In particular, the well-known “linear-quadratic” (LQR) and “Bang-Bang” regulators can be designed. The proposed procedure is illustrated by a numerical example.

Applying the Leitmann–Stalford sufficient conditions to maximization control problems with non-concave Hamiltonian

The main result in this short note is that the integral form of the Leitmann–Stalford sufficiency conditions can be verified for a class of optimal control problems whose Hamiltonian is not concave with respect to the state variable. The main requirement for this class of problems is that the dynamics is sufficiently dissipative. An application to a Stackelberg differential game between a producer and a developer is exemplified. Using our result we show that the necessary conditions implied by Pontryagin’s maximum principle are also sufficient. This allows a complete characterization of the solution.

A New Approach to Optimal Control Problem Constrained by Canonical Form

In this article, it is considered a class of optimal control problems constrained by differential and integral constraints are called canonical form. A modified measure theoretical approach is introduced to solve this class of optimal control problems. Keywords—Optimal control problem, Canonical form, Measure theory.

Optimal Control of Linear Impulsive Antiperiodic Boundary Value Problem on Infinite Dimensional Spaces

A class of optimal control problems for infinite dimensional impulsive antiperiodic boundary value problem is considered. Using exponential stabilizability and discussing the impulsive evolution operators, without compactness and exponential stability of the semigroup governed by original principle operator, we present the existence of optimal controls. At last, an example is given for demonstration.

Sensitivity Interpretations of the Costate Variable for Optimal Control Problems with State Constraints

In optimal control theory, it is well known that the costate arc and the associated maximized Hamiltonian function can be interpreted in terms of gradients of the value function, evaluated along the optimal state trajectory. Such relations have been referred to as “sensitivity relations” in the literature. We provide in this paper new sensitivity relations for state constrained optimal control problems. For the class of optimal control problems considered, there is no guarantee that the costate arc is unique; a key feature of the results is that they assert some choice of costate arc can be made for which the sensitivity relations are valid. The proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem. The introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer; the extra information thereby obtained yields the desired set of sensitivity relations.