The Pontryagin Maximum Principle and Sufficient Optimality Conditions for Nonlinear Problems

2003 ◽  
Vol 39 (12) ◽  
pp. 1671-1679 ◽  
Author(s):  
A. V. Arutyunov
Keyword(s):  
Maximum Principle ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Nonlinear Problems ◽  
Sufficient Optimality Conditions ◽  
The Pontryagin Maximum Principle

WHY REGULARIZATION OF LAGRANGE PRINCIPLE AND PONTRYAGIN MAXIMUM PRINCIPLE IS NEEDED AND WHAT IT GIVES

2018 ◽  
pp. 757-775
Author(s):  
Mikhail Iosifovich Sumin
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Mathematical Programming ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Lagrange Principle ◽  
Convex Problems ◽  
The Pontryagin Maximum Principle

We consider the regularization of the classical Lagrange principle and the Pontryagin maximum principle in convex problems of mathematical programming and optimal control. On example of the “simplest” problems of constrained infinitedimensional optimization, two main questions are discussed: why is regularization of the classical optimality conditions necessary and what does it give?

scholarly journals Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

2021 ◽  
Vol 31 (2) ◽  
pp. 265-284
Author(s):  
V.I. Sumin ◽  
M.I. Sumin
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Pontryagin Maximum Principle ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Lagrange Principle ◽  
The Pontryagin Maximum Principle

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

scholarly journals Maximum Principle for Stochastic Recursive Optimal Control Problems Involving Impulse Controls

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zhen Wu ◽  
Feng Zhang
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Linear Quadratic ◽  
Regular Control

We consider a stochastic recursive optimal control problem in which the control variable has two components: the regular control and the impulse control. The control variable does not enter the diffusion coefficient, and the domain of the regular controls is not necessarily convex. We establish necessary optimality conditions, of the Pontryagin maximum principle type, for this stochastic optimal control problem. Sufficient optimality conditions are also given. The optimal control is obtained for an example of linear quadratic optimization problem to illustrate the applications of the theoretical results.

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