Approximations of Relaxed Optimal Control Problems

In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations.The authors thank the referees for helpful comments and suggestions.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.     

Stochastic optimal control of ultradiffusion processes with application to dynamic portfolio management

We consider theoretical and approximation aspects of the stochastic optimal control of ultradiffusion processes in the context of a prototype model for the selling price of a European call option. Within a continuous-time framework, the dynamic management of a portfolio of assets is effected through continuous or point control, activation costs, and phase delay. The performance index is derived from the unique weak variational solution to the ultraparabolic Hamilton–Jacobi equation; the value function is the optimal realization of the performance index relative to all feasible portfolios. An approximation procedure based upon a temporal box scheme/finite element method is analyzed; numerical examples are presented in order to demonstrate the viability of the approach.     

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L ¹-norm and with order 1/2 w.r.t. the L ²-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.     

FINITE ELEMENT ANALYSIS OF OPTIMAL CONTROL PROBLEM GOVERNED BY STOKES EQUATIONS WITH L^2-NORM STATE-CONSTRAINTS

An optimal control problem governed by the Stokes equations with L2-norm state constraints is studied.Finite element approximation is constructed.The optimality...     

State-constrained optimal control of nonlinear elliptic variational inequalities

An optimization control problem for systems described by abstract variational inequalities with state constraints is considered. The solvability of this problem is proved. The problem is approximated by the penalty method. The convergence of this method is proved. Necessary conditions of optimality for the approximation problem are obtained. Its solution is an approximate optimal control of the initial problem.     

Optimal control of linear time-varying systems via Fourier series

The optimal control of linear time-varying systems with quadratic cost functional is obtained by Fourier series approximation. The properties of Fourier series are first briefly presented and the operational matrix of backward integration together with the product operational matrix are utilized to reduce the optimal control problem to a set of simultaneous linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.     

Asymptotic estimates for a solution of a singular perturbation optimal control problem on a closed interval under geometric constraints

An optimal control problem is considered for solutions of a boundary value problem for a second-order ordinary differential equation on a closed interval with a small parameter at the second derivative. The control is scalar and satisfies geometric constraints. General theorems on approximation are obtained. Two leading terms of an asymptotic expansion of the solution are constructed and an error estimate is obtained for these approximations.     

Extremal points and optimal control theory

Summary An optimal control problem is considered in a setting akin to that of the theory. of generalized curves. Rather than minimizing a functional depending on pairs of trajectories and controls subject to some constraints, a functional defined on a set of Radon measures is considered; the set of measures is determined by the constraints. An approximation scheme is developed, so that the solution of the optimal control problems can be effected by solving a sequence of nonlinear programming problems. Several existence theorems for this kind of generalized control problems are then proved; the most interesting is the one concerning problems in which the set of allowable controls is unbounded. Entrata in Redazione il 5 febbraio 1975.     

Higher-Order Convex Approximations of Young Measures in Optimal Control

The general theory of approximation of (possibly generalized) Young measures is presented, and concrete cases are investigated. An adjoint-operator approach, combined with quasi-interpolation of test integrands, is systematically used. Applicability is demonstrated on an optimal control problem for an elliptic system, together with one-dimensional illustrative calculations of various options.     

Calculation of optimal controls of specified accuracy

A useful approach to the calculation of optimal controls is to take a piecewise constant approximation to the control and to solve the resulting nonlinear program using available techniques. There is no way of specifying the required number of control intervals a priori, but this paper shows that the adjoint system used to calculate gradients for the optimization provides at each iteration sufficient information to assess the gain from increasing the number of intervals and to indicate the best locations for the appropriate switching times. An example is presented which shows the potential computational savings that can be realized when the number of control intervals is progressively increased until the desired accuracy of the approximation is achieved.     

Parametric continuation method with correction and its applications

The article discusses the parametric continuation method for nonlinear equations. A continuation algorithm with correction is proposed, an approximation accuracy theorem is proved, and issues of efficient numerical implementation are considered. An approach is described to the application of the continuation method for seeking the Pontryagin extremal solution in the optimal control problem. Algorithms developed by the author are applied to optimal control problems nonlinear in control, to problems with a nonsmooth control region, and to affine problems with mixed constraints. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 55–94.     

Asymptotic Solution of a Singularly Perturbed Linear-Quadratic Problem in Critical Case with Cheap Control

Using the direct scheme method, we construct an asymptotic expansion for the solution of a singularly perturbed optimal problem in critical case with cheap control and two fixed end-points. The asymptotic solution contains the outer series and two boundary-layer series in the vicinities of the two end-points. The error estimates for state and control variables and the functional are obtained. It is shown that the value of minimized functional does not increase when a higher-order approximation to the optimal control is used. An illustrative example is given.     

Taylor expansions of the value function associated with a bilinear optimal control problem

A general bilinear optimal control problem subject to an infinite-dimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated formal differentiation of the Hamilton–Jacobi–Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-hand sides. They form the basis for defining a suboptimal feedback law. The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker–Planck equation is also provided.     

A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations

In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.     

A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs

An optimal control problem governed by an elliptic variational inequality of the first kind and bilateral control constraints is studied. A smooth penalization technique for the variational inequality is applied and convergence of stationary points of the subproblems to an E-almost C-stationary point of the limit problem is shown. The subproblems are solved using a full approximation multigrid scheme (FAS) and alternatively a multigrid method of the second kind for which a convergence result is given. An overall algorithmic concept is provided and its performance is discussed by means of examples.     

On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming

An approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed. The approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem. Moreover, the approximate solutions are interpreted as value functions of some discrete time control problem. This allows to construct by dynamic programming a minimizing sequence of piecewise constant controls.     

A locally computed suboptimal control strategy for a class of interconnected systems

In this paper, a locally computed suboptimal control strategy for a class of interconnected systems is introduced. First, optimal statefeedback control equations are derived for a finite-horizon quadratic cost. Then, the control for each subsystem is separated into two portions. The first portion stabilizes the isolated subsystem, and the second portion corresponds to the interactions. To achieve a locally calculable control, an approximation to the optimal control equations is introduced, and two iterative suboptimal control algorithms are developed. In the first algorithm, the initial conditions of subsystems are assumed to be known; in the second algorithm, this information is replaced by statistical distributions. The orders of errors in the iterations of the algorithm and in the suboptimality are given in terms of interconnections. An example with comparisons is also included to show the performance of the approach.     

A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.