Stability of solutions to extremal problems of boundary control for stationary heat convection equations

We study extremal problems of boundary control for stationary heat convection equations with Dirichlet boundary conditions on velocity and temperature. As the cost functional we choose the mean square integral deviation of the required temperature field from a given temperature field measured in some part of the flow region. The controls are functions appearing in the Dirichlet conditions on velocity and temperature. We prove the stability of solutions to these problems with respect to certain perturbations of both the quality functional and one of the known functions appearing in the original equations of the model.     

Existence of optimal stationary policies in deterministic optimal control

This paper considers deterministic discrete-time optimal control problems over an infinite horizon involving a stationary system and a nonpositive cost per stage. Various results are provided relating to existence of an ?-optimal stationary policy, and existence of an optimal stationary policy assuming an optimal policy exists.     

Boundedness of optimal matrices in extremal multigraph and digraph problems

We consider multigraphs in which any two vertices are joined by at mostq edges, and study the Turán-type problem for a given family of forbidden multigraphs. In the caseq=2, answering a question of Brown, Erds and Simonovits, we obtain an explicit upper bound on the size of the matrix generating an asymptotical solution of the problem. In the caseq>2 we show that some analogous statements do not hold, and so disprove a conjecture of Brown, Erds and Simonovits.     

On extremal solutions to stochastic control problems

We identify two solutions of a controlled diffusion if the corresponding one-dimensional marginals of the state and control process agree. The extreme points of the set of such equivalence classes are shown to correspond to Markov controls.     

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.     

Some extremal problems arising from discrete control processes

We consider a simple abstract model for a class of discrete control processes, motivated in part by recent work about the behavior of imperfect random sources in computer algorithms. The process produces a string ofn bits and is a “success” or “failure” depending on whether the string produced belongs to a prespecified setL. In an uninfluenced process each bit is chosen by a fair coin toss, and hence the probability of success is ¦L¦/2 ⁿ . A player called the controller, is introduced who has the ability to intervene in the process by specifying the value of some of the bits of the string. We answer the following questions for both worst and average case: (1) how much can the player increase the probability of success given a fixed number of interventions? (2) in terms of ¦L¦what is the expected number of interventions needed to guarantee success? In particular our results imply that if ¦L¦/2 ⁿ =1/Ω(n) where Ω(n) tends to infinity withn (so the probability of success with no interventions is 0(1)) then withO(√n logΩ(n)) interventions the probability of success is 1?0(1). Our main results and the proof techniques are related to well-known results of Kruskal, Katona and Harper in extremal set theory.     

Path-following for optimal control of stationary variational inequalities

Moreau-Yosida based approximation techniques for optimal control of variational inequalities are investigated. Properties of the path generated by solutions to the regularized equations are analyzed. Combined with a semi-smooth Newton method for the regularized problems these lead to an efficient numerical technique.     

On extremal solutions to stochastic control problems,II

The set of attainable laws of the joint state-control process of a controlled diffusion is analyzed from a convex analytic viewpoint. Various equivalence relations depending on one-dimensional marginals thereof are defined on this set and the corresponding equivalence classes are studied.     

Differential stability in optimal control problems

This paper deals with optimal control problems subject to differentiable perturbations in the objective function and constraints. The results of [9] are applied to obtain upper and lower bounds for the directional derivative of the extremal value function as well as necessary and sufficient conditions for the existence of the directional derivative. In particular, the results show the close connection between the multipliers of the Minimum Principle and the sensitivity of the optimal value with respect to perturbations.Partially supported by the Deutsche Forschungsgemeinschaft under No. Ma 691/2     

Singular manifolds in optimal control problems

We propose a method of solving an optimal control problem with constraints on the control. The method is applied to find an upper bound of the Hamilton-Pontryagin function and is based on the construction of a system of differential equations containing the constraints available in the problem as singular manifolds. Bibliography: 2 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 214–216.     

Averaging in optimal switching control problems

Abstract. The averaging in optimal switching control problems is considered under the following two cases: the switching cost does not depend on e and the switching cost vanishes as e tends to zero. The value function of the original fast problem converges locally uniformly to the value function of the averaged problem under both cases. The ways of averaging turn out to be different between both cases.     

On sensitivity in optimal control problems

Two types of interpretations of multipliers in both static and dynamic optimization problems are described. It is snown that the Lagrange multipliers encountered in mathematical programming problems and the auxiliary functions arising in Pontryagintype optimal control problems sometimes have highly analogous interpretations as rates of change of the optimal attainable value of an objective function, or in some cases as bounds on average rates of change.     

Discrete extremal problems

Precise and heuristic algorithms for solving various classes of discrete extremal problems are considered as are the relations between the class of discrete extremal problems and linear programming and are extremal problems from the point of view of the theory of polynomial completeness. A class of bottleneck optimization problems and stability in discrete extremal problems with a linear object function are also considered.S. P. Tarasov participated in the work of Secs. 3, 4, 5 and the bibliography.Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 16, pp. 39–101, 1979.     

Local stability of stationary states in discounted optimal control systems

This paper presents local optimality and stability conditions for stationary solutions of autonomous optimal control problems with a positive rate of discounting. The conditions do not require the Hamiltonian function to be concave with respect to the state variables. Two examples from optimal economic growth theory are discussed to show that our results can be applied in situations when other known stability conditions fail to be satisfied.Part of this paper was written while the author visited the Faculty of Management at the University of Toronto. Support from SSHRC Grant 410-83-9888 and from Grant P6601 of the Austrian Science Foundation is gratefully acknowledged.     

Sparse Dirichlet optimal control problems

Computational Optimization and Applications - In this paper, we analyze optimal control problems governed by an elliptic partial differential equation, in which the control acts as the Dirichlet...