On singularities of value function for Bolza optimal control problem

We study the set of points of nondifferentiability, called the singular set, of the value function of a Bolza optimal control problem. The value function is a viscosity solution to an associated Hamilton-Jacobi equation. The method of characteristics associates to this equation a Hamiltonian system, that in turn can be used to study the propagation of singularities of the value function. In particular, we obtain an extension of the Rankine-Hugoniot type condition, which is well-known in the conservation law theory.     

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.     

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.     

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].     

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.     

Mordukhovich subgradients of the value function to a parametric discrete optimal control problem

This paper is devoted to the study of the first-order behavior of the value function of a parametric discrete optimal control problem with nonconvex cost functions and control constraints. By establishing an abstract result on the Mordukhovich subdifferential of the value function of a parametric mathematical programming problem, we derive a formula for computing the Mordukhovich subdifferential of the value function to a parametric discrete optimal control problem.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.     

The Galilei group in an optimal control problem

In the paper, results of studying an optimal control problem for the motion of a material particle under control constraints are presented. The invariance of this problem with respect to the extended Galilei group is used. From the viewpoint of calculations, the symmetry allows us to construct a family of solutions using an extremal determined numerically. From the analytical viewpoint, the symmetry gives an opportunity to reduce the system’s dimension and to investigate the properties of extremals.     

The Galilei group in an optimal control problem

In the paper, results of studying an optimal control problem for the motion of a material particle under control constraints are presented. The invariance of this problem with respect to the extended Galilei group is used. From the viewpoint of calculations, the symmetry allows us to construct a family of solutions using an extremal determined numerically. From the analytical viewpoint, the symmetry gives an opportunity to reduce the system’s dimension and to investigate the properties of extremals.     

The inverse optimal value problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost vectors, determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value. The above problem, referred here as the inverse optimal value problem, is significantly different from standard inverse optimization problems that involve determining a cost vector for a linear program such that a pre-specified solution vector is optimal. In this paper, we show that the inverse optimal value problem is NP-hard in general. We identify conditions under which the problem reduces to a concave maximization or a concave minimization problem. We provide sufficient conditions under which the associated concave minimization problem and, correspondingly, the inverse optimal value problem is polynomially solvable. For the case when the set of feasible cost vectors is polyhedral, we describe an algorithm for the inverse optimal value problem based on solving linear and bilinear programming problems. Some preliminary computational experience is reported.Mathematics Subject Classification (1999):49N45, 90C05, 90C25, 90C26, 90C31, 90C60Acknowledgement This research has been supported in part by the National Science Foundation under CAREER Award DMII-0133943. The authors thank two anonymous reviewers for valuable comments.     

Balance function for the optimal control problem

The balance-function concept for transforming constrained optimization problems into unconstrained optimization problems, for the purpose of finding numerical iterative solutions, is extended to the optimal control problem. This function is a combination orbalance between the penalty and Lagrange functions. It retains the advantages of the penalty function, while eliminating its numerical disadvantages. An algorithm is developed and applied to an orbit transfer problem, showing the feasibility and usefulness of this concept.These results are part of the author's doctoral thesis written under Professors H. Lo and D. Alspaugh of Purdue University.     

The value function of an infinite-horizon single-item lot-sizing problem

We characterize the value function of a discounted infinite-horizon version of the single-item lot-sizing problem. As corollaries, we show that this value function inherits several properties of finite, mixed-integer program value functions; namely, it is subadditive, lower semicontinuous, and piecewise linear.     

On sensitivity in an optimal control problem

We give a new criterion for the existence of value in differential games. The method of proof involves Lipschitz differential games and hence extends to games with more general dynamics. The connection between using measurable control functions or simply constants is clarified.     

The problem of controlling resources on network graphs as an optimal control problem

The problem of optimal distribution of resources dedicated to a certain complex of inter-related tasks according to the criterion of minimum execution time of all tasks is described. A reenterable (reusable) resource is considered instead of a traditional separable-type resource supposing a fixed distribution among tasks. The problem is formalized in direct static and dynamic settings. The latter is a classical performance optimal control problem. The correctness of the formalization is substantiated.     

Sufficient conditions for asymptotic optimal control

For a selected family of Lagrange-type control problems involving a nonnegative integral costJ _T (y,u) over the interval [0,T], 0<T<, with system conditions consisting of differential inequalities and/or equalities, the following material is treated: (i) a resumé of relevant necessary conditions and sufficient conditions for a pair (y _T ,u _T ) to minimizeJ _T (y,u); (ii) conditions sufficient for the convergence asT of minimizing pairs (y _T ,u _T ) over [0,T] to a limit pair (y ,u ) over the infinite-time interval [0, ); (iii) conditions sufficient for (y ,u ) to minimize the costJ (y,u) over [0, ); and (iv) conditions sufficient for the optimal cost per unit timeJ _T (y _T ,u _T )/T to have a limit asT.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

The variational stability of an optimal control problem for Volterra-type equations

We study the variational stability of an optimal control problem for a Volterra-type nonlinear functional-operator equation. This means that for this optimal control problem (P _? ) with a parameter ? we study how its minimum value min(P _? ) and its set of minimizers argmin(P _? ) depend on ?. We illustrate the use of the variational stability theorem with a series of particular problems.