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.     

Asymptotics of the solution in a problem of optimal boundary control of a flow through a part of the boundary

We consider a problem of optimal control through a part of the boundary of solutions to an elliptic equation in a bounded domain with smooth boundary with a small parameter at the Laplace operator and integral constraints on the control. A complete asymptotic expansion of the solution to this problems in powers of the small parameter is constructed.     

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.     

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.     

Analysis of Numerical Differentiation Formulas in a Boundary Layer on a Shishkin Grid

A problem of numerical differentiation of functions with large gradients in a boundary layer is investigated. The problem is that for functions with large gradients and a uniform grid the relative error of the classical difference formulas for derivatives may be considerable. It is proposed to use a Shishkin grid to obtain a relative error of the formulas that is independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform with respect to the small parameter. In the case of a uniform grid, a boundary layer region is indicated outside of which the numerical differentiation formulas have an error that is uniform with respect to the small parameter. The results of numerical experiments are presented.     

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.     

Steplike contrast structures for a second-order ordinary differential equation with a small parameter multiplying the highest derivative

A boundary value problem is considered for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative. The limit equation has three solutions, of which two are stable and are separated by the third unstable one. For the original problem, an asymptotic expansion of a solution is studied that undergoes a jump from one stable root of the limit equation to the other in the neighborhood of a certain point. A uniform asymptotic approximation of this solution is constructed up to an arbitrary power of the small parameter.     

Stabilization of objects using a constant control

A special class of nonlinear control systems describing the process of combustion of carbon in an oxygen-rich vessel with complete mixing is considered. The control parameter is the consumption of carbon supplied to the vessel. Differential equations describing the complicated reaction of combustion are suggested. The method of successive approximations for determination of a steady control is shown to converge; the steady-state solution is shown to be stabilized using this control. In the presence of a small external disturbance the original problem is reduced to the problem of stability of the general system under permanent disturbance. A control algorithm has been developed. Simulation experiments for the combustion process have been carried out. The results of these experiments suggest that the algorithm for control of the carbon combustion process is capable of being used in practice.     

Reduction from a Semi-Infinite Interval to a Finite Interval of a Nonlinear Boundary Value Problem for a System of Second-Order Equations with a Small Parameter

We consider a boundary value problem over a semi-infinite interval for a nonlinear autonomous system of second-order ordinary differential equations with a small parameter at the leading derivatives. We impose certain constraints on the Jacobian under which a solution to the problem exists and is unique. To transfer the boundary condition from infinity, we use the well-known approach that rests on distinguishing the variety of solutions satisfying the limit condition at infinity. To solve an auxiliary Cauchy problem, we apply expansions of a solution in the parameter.     

On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case

A boundary-value problem of finding eigenvalues is considered for the negative Laplace operator in a disk with Neumann boundary condition on almost all the circle except for a small arc of vanishing length, where the Dirichlet boundary condition is imposed. A complete asymptotic expansion with respect to a parameter (the length of the small arc) is constructed for an eigenvalue of this problem that converges to a double eigenvalue of the Neumann problem.     

Power-series approximations for the optimal feedback control of noisy nonlinear systems

Power-series methods are developed for designing approximately optimal state regulators for a nonlinear system subject to white Gaussian random disturbances. The performance index of the control is an ensemble average of a quadratic form. A perfect observation of the system state is assumed. When the system nonlinearity is small and it is characterized by a polynomial function of the state, a definite method is presented to construct a suboptimal feedback control of a power-series form in a small nonlinearity parameter. If the variance of noise is small, an alternative method is also applicable which yields a suboptimal control in a power series with respect to a variance parameter. A simple one-dimensional problem is examined to make comparison between controls of the two different forms.     

Asymptotic behavior of the optimal cost functional for a rapidly stabilizing indirect control in the singular case

The optimal control problem for a linear system with fast and slow variables in the form of indirect control with a convex terminal cost functional and a smooth geometric constraint on the control is studied. An asymptotic expansion of the cost functional up to any power of a small parameter is constructed.     

一个带有小参数的二维椭圆方程的渐近性分析

通过数值方法研究在边界充分(逐段)光滑区域上的带有小参数的二维椭圆方程在部分Dirichlet边界控制下的渐近性问题.对于一维的情形求解析解的结果,对高维问题提出类似的问题.但高维问题解析求解一般不可能,因此采用数值分析的方法.数值结果表明,在所选的条件下,边界值对小常数仍然不是解析的.     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang-bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.     

Robust error estimates for the finite element approximation of elliptic optimal control problems

In this paper, we consider the finite element approximation of an elliptic optimal control problem. Based on an assumption on the adjoint state of the continuous problem with a small parameter, which represents a regularization of the bang–bang type control problem, we derive robust a priori error estimates for optimal control and state and a posteriori error estimate is also presented. Numerical experiments confirm our theoretical results.     

An algorithm for the asymptotic solution of a singularly perturbed linear time-optimal control problem

An algorithm for the approximate solution (in the asymptotic sense) of a singularly perturbed linear time-optimal control problem is proposed. A computational procedure is outlined, which permits the use of the resulting asymptotic approximation for. the exact solution of the problem with a prescribed value of the small parameter.     

Optimal control for systems described by hyperbolic equations with strong nonlinearity

An optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Thus the optimal control for the regularized problem with a small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.     

Solution of a periodic optimal control problem by asymptotic series

In order to understand the numerical behavior of a certain class of periodic optimal control problems, a relatively simple problem is posed. The complexity of the extremal paths is uncovered by determining an analytic approximation to the solution by using the Lindstedt-Poincaré asymptotic series expansion. The key to obtaining this series is in the proper choice of the expansion parameter. The resulting expansion is essentially a harmonic series in which, for small values of the expansion parameter and a few terms of the series, excellent agreement with the numerical solution is obtained. A reasonable approximation of the solution is achieved for a relatively large value of the expansion parameter.This work was sponsored partially by the National Science Foundation, Grant No. ECS-84-13745.     

Interval optimal control problem in a Hilbert space

An optimal control problem for a system involving an interval parameter is considered. The concepts of a universal optimal state and a universal optimal control are introduced. The existence and uniqueness of a universal solution to the interval optimal control problem is proved, and an algorithm for its determination is presented. The interval optimal control problem for a system described by the boundary value problem for a second-order ordinary differential equation is solved as an example.     

Singular perturbations in a class of problems of optimal control with integral convex criterion

The problem of optimal control is investigated with a linear law of motion and convex quality criterion. A small positive parameter appears in front of the derivatives of some of the unknowns in the law of motion. The behaviour of the optimal solution is studied when the small parameter approaches zero with some assumptions that are different from thos encountered in the literature.