On variational perturbations of control problems: Minimum-time problem and minimum-effort problem

In order to obtain numerical solutions for an abstract optimal control problem, one approximates the abstract operations in a computationally feasible manner. After having found an approximate optimal solution, the question is whether a sequence of these approximate optimal solutions converges to an optimal solution of the original problem. In this work, we are concerned with this type of convergence on the time-optimal control problem for a class of linear systems with distributed parameters and on the minimum-effort problem.     

Classical characteristics of the bellman equation in constructions of grid optimal synthesis

We consider optimal control problems with fixed final time and terminal-integral cost functional, and address the question of constructing a grid optimal synthesis (a universal feedback) on the basis of classical characteristics of the Bellman equation. To construct an optimal synthesis, we propose a numerical algorithm that relies on the necessary optimality conditions (the Pontryagin maximum principle) and sufficient conditions in the Hamiltonian form. We obtain estimates for the efficiency of the numerical method. The method is illustrated by an example of the numerical solution of a nonlinear optimal control problem.     

On the asymptotics of the optimal value of the performance functional in a linear optimal control problem

We consider the asymptotics of the optimal value of the performance functional in an optimal control problem for a linear system with rapid and slow variables, with convex terminal performance functional depending on the slow variables, and with smooth geometric constraints on the control. We find sufficient conditions for the regularity of asymptotics and construct an algorithm for finding the complete asymptotics of the optimal value of the performance functional.     

An Objective Penalty Function of Bilevel Programming

Penalty methods are very efficient in finding an optimal solution to constrained optimization problems. In this paper, we present an objective penalty function with two penalty parameters for inequality constrained bilevel programming under the convexity assumption to the lower level problem. Under some conditions, an optimal solution to a bilevel programming defined by the objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based on the objective penalty function, an algorithm is developed to obtain an optimal solution to the original bilevel programming, with its convergence proved under some conditions.     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.     

Existence and Structure of Optimal Solutions of Infinite-Dimensional Control Problems

In this work we analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval [0,τ] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t ∈ [0,τ] \backslash E , where E \subset [0,τ] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on τ . Accepted 26 July 2000. Online publication 13 November 2000.     

Problem of optimal endogenous growth with exhaustible resources and possibility of a technological jump

In 2013, S. Aseev, K. Besov, and S. Kaniovski (“The problem of optimal endogenous growth with exhaustible resources revisited,” Dyn. Model. Econometr. Econ. Finance 14, 3–30) considered the problem of optimal dynamic allocation of economic resources in an endogenous growth model in which both production and research sectors require an exhaustible resource as an input. The problem is formulated as an infinite-horizon optimal control problem with an integral constraint imposed on the control. A full mathematical study of the problem was carried out, and it was shown that the optimal growth is not sustainable under the most natural assumptions about the parameters of the model. In the present paper we extend the model by introducing an additional possibility of “random” transition (jump) to a qualitatively new technological trajectory (to an essentially unlimited backstop resource). As an objective functional to be maximized, we consider the expected value of the sum of the objective functional in the original problem on the time interval before the jump and an evaluation of the state of the model at the moment of the jump. The resulting problem also reduces to an infinite-horizon optimal control problem, and we prove an existence theorem for it and write down an appropriate version of the Pontryagin maximum principle. Then we characterize the optimal transitional dynamics and compare the results with those for the original problem (without a jump).     

Optimal control of robots under stochastic uncertainty: robust feedback control

The most important aspect in the optimal control and design of manipulators is the determination of the basic movement, i.e. the calculation of the optimal trajectory on which the robot has to move. Having an optimal reference trajectory and an optimal open-loop control, there is the need of control corrections by applying a certain feedback control. Different attempts exist for this. In this article a method will be shown which is based on classical control theory, that works with cost functions being minimized. The aim is to take into account stochastic parameter variations in order to obtain robust optimal feedback controls. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the existence of dual solutions for Lorentzian cost functions

The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided.     

Optimal Enforcement on a Pure Seller's Market of Illicit Drugs

An optimal control problem for the dynamic enforcement (crackdown) of dealers on a pure seller's market for illicit drugs is explored. Theorems on existence and uniqueness of the optimal synthesis are proved. Using a technique of resolution of singularities for degenerate differential equations, we design analytically an optimal enforcement policy.