Asymptotics of a solution to an optimal boundary control problem

A problem of optimal boundary control of solutions to an elliptic-type equation with a small coefficient at the highest derivative and integral constraints on the control is considered. Asymptotic estimates for solutions to a problem that approximates the original problem are obtained.     

Robust optimal feedback for terminal linear-quadratic control problems under disturbances

We consider a linear dynamic system in the presence of an unknown but bounded perturbation and study how to control the system in order to get into a prescribed neighborhood of a zero at a given final moment. The quality of a control is estimated by the quadratic functional. We define optimal guaranteed program controls as controls that are allowed to be corrected at one intermediate time moment. We show that an infinite dimensional problem of constructing such controls is equivalent to a special bilevel problem of mathematical programming which can be solved explicitely. An easy implementable algorithm for solving the bilevel optimization problem is derived. Based on this algorithm we propose an algorithm of constructing a guaranteed feedback control with one correction moment. We describe the rules of computing feedback which can be implemented in real time mode. The results of illustrative tests are given.     

Asymptotic solution of singularly perturbed linear-quadratic optimal control problems with discontinuous coefficients

An asymptotic solution of a singularly perturbed linear-quadratic optimal control problem with discontinuous coefficients is constructed by directly substituting an boundary-layer asymptotic expansion of the solution into the condition of the problem and considering a series of problems for finding the asymptotic terms. The error in the approximate solution is estimated. It is shown that the values of the minimized functional do not increase when the next approximations of the optimal control are used.     

Explicit solution of an optimization problem

A simply statedn-variable constrained optimization problem, useful as a test problem, is explicitly solved. It has a large number of easily described local optima.This work was supported (in part) by the Office of Naval Research under contract number N00014-71-C-0112.     

Asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with integral constraint

A control problem for solutions of a boundary value problem for a second-order ordinary differential equation with a small parameter at the second derivative is considered on a closed interval. The control is scalar and subject to integral constraints. We construct a complete asymptotic expansion in powers of the small parameter in the Erdélyi sense.     

Optimal feedback control for a linear-quadratic control problem with a state inequality constraint

In this paper, we consider the linear-quadratic control problem (LQCP) for systems defined by evolution operators with an inequality constraint on the state. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closedloop structure. The synthesis of the feedback control requires one to solve two Riccati integral equations.     

Numerical solution of a minimax ergodic optimal control problem

In this work we consider an L^∞ minimax ergodic optimal control problem with cumulative cost. We approximate the cost function as a limit of evolutions problems. We present the associated Hamilton-Jacobi-Bellman equation and we prove that it has a unique solution in the viscosity sense. As this HJB equation is consistent with a numerical procedure, we use this discretization to obtain a procedure for the primitive problem. For the numerical solution of the ergodic version we need a perturbation of the instantaneous cost function. We give an appropriate selection of the discretization and penalization parameters to obtain discrete solutions that converge to the optimal cost. We present numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A complete asymptotic expansion of a solution to a singular perturbation optimal control problem on an interval with geometric constraints

We consider an optimal control problem for solutions of a boundary value problem on an interval for a second-order ordinary differential equation with a small parameter at the second derivative. The control is scalar and is subject to geometric constraints. Expansions of a solution to this problem up to any power of the small parameter are constructed and justified.     

The solution and sensitivity of a general optimal control problem

A solution method for the general optimal control problem is presented. This can be used to solve optimal control problems for which the system dynamics are not necessarily described by differential or difference equations. Having obtained the solution it is of theoretical and practical interest to investigate the sensitivity of the optimal trajectory to perturbations. Indicators of this sensitivity are the adjoint variables derived in the maximum principle. A method of deriving the adjoint variables from the solution is described. To illustrate the solution method and the determination of the adjoint variables a problem in the urban housing market is used.     

On an optimal control problem

In this paper, we applied the finite differences method to the solution of variational problem of an inverse problem for the Scrödinger equation with a final functional. These types of problems arise in various fields in quantum-mechanical, nuclear physics and modern physics [2, 11]. Also, we prove two estimates for the differences scheme and convergence speed of difference approximations according to the functional. The inverse problems for the Schrödinger equation having different variational formulation were investigated in [7, 12, 13].     

Approximate solution to a time optimal boundary control problem for the wave equation

The paper provides some examples of mutually dual unconstrained optimization problems originating from regularization problems for systems of linear equations and/or inequalities. The solution of each of these mutually dual problems can be found from the solution of the other problem by means of simple formulas. Since mutually dual problems have different dimensions, it is natural to solve the unconstrained optimization problem of the smaller dimension.     

On an optimal control problem with an integral functional of a rational control function

We present the solution of an optimization problem with integral performance functional that is nonlinear with respect to the control and contains a discounting parameter in the class of programmed controls under two-sided control constraints. The optimal control is found in the form of a function of time (a program). On the basis of the theoretical results, we perform numerical experiments with model and real data.     

Computational algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

A computational algorithm for optimal control problems with control and terminal inequality constraints involving first boundary-value problems of parabolic type is presented. The convergence properties are also studied.This work, which was partly supported by the Australian Research Grants Committee, was done during the period when Z. S. Wu was an Honorary Visiting Fellow in the School of Mathematics at the University of New South Wales, Australia.     

Computational algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

A corrigendum to Ref. 1 is given.