The numerical simulation is carried out using Matlab. The Pontryagin maximum principle (PMP), established at the end of the 1950s for ﬁnite dimensional general nonlinear continuous-time dynamics (see , and see  for the history of this discovery), is a milestone of classical optimal control theory. Introduction A maximum principle of the Pontryagin type for systems described by nonlinear difference equations (1966) by H Halkin Venue: SIAM J. The analysis shows that only five modes are required to achieve minimum energy consumption: full propulsion, cruising, coasting, full regeneration, and full regeneration with conventional braking. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then solved to extract optimal control trajectories. Overview I Derivation 1: Hamilton-Jacobi-Bellman equation I Derivation 2: Calculus of Variations I Properties of Euler-Lagrange Equations I Boundary Value Problem (BVP) Formulation I Numerical Solution of BVP I Discrete Time Pontryagin Principle First, Pontryagin's maximum principle (PMP) is applied to derive necessary conditions and to determine the possible operating modes. On stability of the pontryagin maximum principle with respect to time discretization ... important role in both qualitative and numerical aspects of optimal control and occupy an intermediate position between discrete-time and continuous-time control systems. We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. Pontryagin’s Maximum Principle has been widely discussed and used in Optimal Control Theory since the second half of the 20th century. Parallel to the Pontryagin theory, in the USA an alter- The economic interpretations of major variables are also discussed. Assuming differentiability, the maximum principle is obtained in the usual form. In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. cal solution by using the Pontryagin's maximum principle, and rewrites it for a discrete-time system, using the inverse Laplace transform, and, finally, gives a solu­ tion for it. [4 1 This paper is to introduce a discrete version of Pontryagin's maximum principle. 1. 2. ... Computer time and storage requirements are moderate and increase roughly linearly with the problem dimension. The discrete maximum principle of Pontryagin is obtained in a straightforward manner from the strong Lagrange duality theorem, first in a new form in which the Lagrangian is minimized both with respect to the state and to the control variables. We give a simple proof of the Maximum Principle for smooth hybrid control sys-tems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle. On Stability of the Pontryagin Maximum Principle with respect to Time Discretization Boris S. Mordukhovich and Ilya Shvartsman Abstract—The paper deals with optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time, wherein the discretization step tends to zero. Very little has been published on the application of the maximum principle to industrial management or operations-research problems. Traditionally, the time domains that are widely used in mathematical descriptions are limited to real numbers for the case of continuous-time optimal control problems or to integers for the case of discrete-time optimal control problems. An optimal control algorithm based on the discrete maximum principle is applied to multireservoir network control. An example is solved to illustrate the use of the algorithm. Control problems posed on smooth manifolds requirements are moderate and increase roughly linearly the. Several assumptions of continuity and of linear independence a discrete version of the type! 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