Approximations to Differential Inclusions by Discrete Inclusions

This report is devoted to second order discrete approximations to differential inclusions. The approximations are of the form of discrete inclusions with right-hand sides, which are explicitly described for some classes of differential inclusions. In the cases of linear differential inclusions or of differential inclusions with strongly convex right-hand sides, the approximating discrete inclusions are analogs of certain second order Runge-Kutta schemes. The approach can serve as a tool for numerical treatment of uncertain dynamical system and optimal control problems

Needle Variations in Infinite-Horizon Optimal Control

The paper develops the needle variations technique in application to a class of infinite-horizon optimal control problems in which an appropriate relation between the growth rate of the solution and the growth rate of the objective function is satisfied. The optimal objective value does not need to be finite. Based on the concept of weakly overtaking optimality we establish the normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable. A few illustrative examples are presented as well

Controllability of Discontinuous Systems

This report presents an approach to the local controllability problem for a discontinuous system. The approach is based on a concept of tangent vector field to a generalized dynamic system, which makes possible the differential geometry tools to be applied in the discontinuous case. Sufficient controllability conditions are derived

Maximum Principle for Infinite-horizon Optimal Control Problems under Weak Regularity Assumptions

The paper deals with first order necessary optimality conditions for a class of infinite-horizon optimal control problems that arise in economic applications. Neither convergence of the integral utility functional nor local boundedness of the optimal control is assumed. Using the classical needle variations technique we develop a normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable under weak regularity assumptions. The result generalizes some previous results in this direction. An illustrative economical example is presented

Best Approximations of Control/Uncertain Differential Systems by Means of Discrete-Time Systems

The author studies the exact (best possible) rate of approximability of an uncertain or control system by means of N-stage discrete-time systems. An ultimate solution is presented in the linear case and an estimate of the rate of approximability is given for a broad class of nonlinear systems. Some applications for numerical treatment of optimal control problems and of uncertain systems are indicated

Optimality Conditions for Discrete-Time Optimal Control on Infinite Horizon

The paper presents first order necessary optimality conditions of Pontrygin's type for a general class of discrete-time optimal control problems on infinite horizon. The main novelty is that the adjoint function, for which the (local) maximum condition in the Pontryagin principle holds, is explicitly defined for any given optimal state-control process. This is done based on ideas from previous papers of the first and the last authors concerning continuous-time problems. In addition, the obtained (local) maximum principle is in a normal form, and the optimality has the general meaning of weakly overtaking optimality (hence unbounded processes are allowed), which is important for many economic applications. Two examples are given, which demonstrate the applicability of the obtained results in cases where the known necessary optimality conditions fail to identify the optimal solutions