On the efficiency of optimal grid synthesis in optimal control problems with fixed terminal time

In the paper, we consider nonlinear optimal control problems with the Bolza functional and with fixed terminal time. We suggest a construction of optimal grid synthesis. For each initial state of the control system, we obtain an estimate for the difference between the optimal result and the value of the functional on the trajectory generated by the suggested grid positional control. The considered feedback control constructions and the estimates of their efficiency are based on a backward dynamic programming procedure. We also use necessary and sufficient optimality conditions in terms of characteristics of the Bellman equation and the sub-differential of the minimax viscosity solution of this equation in the Cauchy problem specified for the fixed terminal time. The results are illustrated by the numerical solution of a nonlinear optimal control problem.     

Construction of a generalized solution to an equation that preserves the Bellman type in a given domain of the state space

A Cauchy problem is considered for a Hamilton-Jacobi equation that preserves the Bellman type in a spatially bounded strip. Sufficient conditions are obtained under which there exists a continuous generalized (minimax/viscosity) solution to this problem with a given structure in the strip. A construction of this solution is presented.     

Minimax production planning in failure-prone manufacturing systems

In this paper, we consider a minimax production planning model of a flexible manufacturing system with machines that are subject to random breakdown and repair. The objective is to choose the rate of production that minimizes the related minimax cost of production and inventory/shortage. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Isaacs equation. Under certain conditions, it is shown that the value function is continuously differentiable. A verification theorem is given to provide a sufficient condition for optimal control. Finally, two examples are solved explicitly.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grants OGP0036444 and A4169.     

On some properties of the control problem under a program disturbance in a formalization based on the minimax risk (regret) criterion

A control problem under uncertainty for a system described by an ordinary differential equation with a terminal performance index is considered. The control and disturbance are subject to geometric constraints. The problem is formalized in classes of nonanticipating control strategies and program disturbances with the use of constructive ideal motions and the Savage minimax risk (regret) criterion. The properties of the used motion bundles are described and a number of relations characterizing the optimal risk function, which is an element of the formalization, are presented.     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

Infinite-horizon minimax control with pointwise cost functional

This paper is devoted to the discussion of a minimax optimal control problem over an infinite-time horizon, where the functional to be minimized is the highest instantaneous cost that may occur under the worst combination of disturbances. The problem is formulated for a general stationary discrete-time dynamic system, and a dynamic programming algorithm is proposed for its solution. The relationship existing between the cost functional associated to a control law and the reachability properties of the resulting controlled system is discussed.     

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)     

Minimax solutions of Hamilton–Jacobi functional equations for neutral-type systems

A functional Hamilton–Jacobi equation with covariant derivatives which corresponds to neutral-type dynamical systems is obtained. The definition of a minimax solution of this equation is given. Conditions under which such a solution exists and is unique and well defined are found.     

Optimal guaranteed control of linear systems under disturbances

This paper is devoted to the problem of the minimax control of a dynamical system with quadratic performance functional under external disturbances and geometric control constraints. The optimal guaranteed control strategy is obtained in explicit form.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 198–205, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-000771a.     

Construction of a minimax solution for an eikonal-type equation

A formula for a minimax (generalized) solution of the Cauchy-Dirichlet problem for an eikonal-type equation is proved in the case of an isotropic medium providing that the edge set is closed; the boundary of the edge set can be nonsmooth. A technique of constructing a minimax solution is proposed that uses methods from the theory of singularities of differentiable mappings. The notion of a bisector, which is a representative of symmetry sets, is introduced. Special points of the set boundary—pseudovertices—are singled out and bisector branches corresponding to them are constructed; the solution suffers a “gradient catastrophe” on these branches. Having constructed the bisector, one can generate the evolution of wave fronts in smoothness domains of the generalized solution. The relation of the problem under consideration to one class of time-optimal dynamic control problems is shown. The efficiency of the developed approach is illustrated by examples of analytical and numerical construction of minimax solutions.     

Minimax Control of Parabolic Systems with Dirichlet Boundary Conditions and State Constraints

In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions. Accepted 7 June 1996     

Linear Formulation of a Distributed Boundary Control Problem

In this paper, we consider a distributed boundary control problem governed by an elliptic partial differential equation with state constraints and a minimax objective function. The continuous optimal control problem, discretized with the finite element method, is numerically approximated by a family of linear programming problems. Application to an optimal configuration problem is discussed.     

Deterministic Minimax Impulse Control

We prove the uniqueness of the viscosity solution of an Isaacs quasi-variational inequality arising in an impulse control minimax problem, motivated by an application in mathematical finance.     

On the Solution of a System of Hamilton–Jacobi Equations of Special Form

The paper is concerned with the investigation of a system of first-order Hamilton–Jacobi equations. We consider a strongly coupled hierarchical system: the first equation is independent of the second, and the Hamiltonian of the second equation depends on the gradient of the solution of the first equation. The system can be solved sequentially. The solution of the first equation is understood in the sense of the theory of minimax (viscosity) solutions and can be obtained with the help of the Lax–Hopf formula. The substitution of the solution of the first equation in the second Hamilton–Jacobi equation results in a Hamilton–Jacobi equation with discontinuous Hamiltonian. This equation is solved with the use of the idea of M-solutions proposed by A. I. Subbotin, and the solution is chosen from the class of multivalued mappings. Thus, the solution of the original system of Hamilton–Jacobi equations is the direct product of a single-valued and multivalued mappings, which satisfy the first and second equations in the minimax and M-solution sense, respectively. In the case when the solution of the first equation is nondifferentiable only along one Rankine–Hugoniot line, existence and uniqueness theorems are proved. A representative formula for the solution of the system is obtained in terms of Cauchy characteristics. The properties of the solution and their dependence on the parameters of the problem are investigated.     

Externality and Hamilton-Jacobi equations

The relationship between optimal control problems and Hamilton-Jacobi-Bellman equations is well known [9]. In fact the value function, defined as the infimum of the cost functional, satisfies in the viscosity sense an appropriate Hamilton-Jacobi-Bellman equation. In this paper we consider several control problems such that the cost functional associated to each problem depends explicitly on the value functions of the other problems. This leads to a system of Hamilton-Jacobi-Bellman equations. This is known, in economic context [14] cap XI, as an externality problem. In these problems may occur a lack of uniqueness of the value functions. We give conditions to ensure existence, uniqueness of the value functions and an implicit integral representation formula. Moreover, under uniqueness assumption, we prove that the variational solutions of the associated Hamilton-Jacobi system converge asymptotically to the value functions. We prove also an uniqueness theorem in the case of viscosity solutions of Hamilton-Jacobi-Bellman system.     

Indirect Obstacle Minimax Control for Elliptic Variational Inequalities

A minimax control problem for a coupled system of a semilinear elliptic equation and an obstacle variational inequality is considered. The major novelty of such problem lies in the simultaneous presence of a nonsmooth state equation (variational inequality) and a nonsmooth cost function (sup norm). In this paper, the existence of optimal controls and the optimality conditions are established.     

Optimal control for multi-phase fluid Stokes problems

Optimal control for a system consistent of the viscosity dependent Stokes equations coupled with a transport equation for the viscosity is studied. Motivated by a lack of sufficient regularity of the adjoint equations, artificial diffusion is introduced to the transport equation. The asymptotic behavior of the regularized system is investigated. Optimality conditions for the regularized optimal control problems are obtained and again the asymptotic behavior is analyzed. The lack of uniqueness of solutions to the underlying system is another source of difficulties for the problem under investigation.     

Second order Hamilton–Jacobi–Bellman equations with an unbounded operator

This work is devoted to the study of a class of Hamilton–Jacobi–Bellman equations associated to an optimal control problem where the state equation is a stochastic differential inclusion with a maximal monotone operator. We show that the value function minimizing a Bolza-type cost functional is a viscosity solution of the HJB equation. The proof is based on the perturbation of the initial problem by approximating the unbounded operator. Finally, by providing a comparison principle we are able to show that the solution of the equation is unique.     

Control optimization in a hyperbolic linear system

An optimal control problem is considered for a distributed-parameter Goursat-Darboux system with controlled boundary conditions. For the numerical solution of the problem, an algorithm based on separability and minimax theorems is constructed, which reduces the problem to finding the maximum of a concave functional defined in the class of one-variable functions.     

Optimality conditions for nondifferentiable minimax fractional programming with complex variables

We show that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. We establish the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities.