Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints
type: Item
instance of: scholarly article
title: Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints
zbMATH Open document ID: 1263.49001
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-29T00:00:00Z
review text: This article deals with necessary quadratic (second-order) conditions of extremum in optimal control problems with control appearing nonlinearly. The main difference from other publications is that here the necessary conditions take into account discontinuities of the first kind of the optimal control. The control problems under investigation are governed by ordinary differential equations with initial-final state constraints and mixed state-control constraints (of equality and inequality type). All the results included in this long article are based on the author's doctoral thesis, defended in 1988 at the Moscow Civil Engineering Institute. The author's necessary conditions are expressed in terms of positive semi-definiteness of the quadratic form on the so-called critical cone. In the case of discontinuous optimal control, the definitions of the quadratic form and critical cone are more complicated than in the continuous case. The natural strengthening of these necessary conditions turns them into sufficient conditions. The paper contains five chapters. In Chapter 1 the necessary quadratic conditions are derived under convenient assumptions. All data are assumed to be twice continuously differentiable and the gradients with respect to the control of active mixed constraints are assumed to be linearly independent. The reference control is assumed to be piecewise continuous on the corresponding time interval. The first- and second-order necessary conditions for optimality are formulated for the case of discontinuous optimal control. Chapter 2 is devoted to the abstract theory of higher-order conditions. In Chapters 3 and 4 the author provides proofs of the results presented in Chapter 1. Chapter 5 is devoted to some examples illustrating the applicability of the second-order necessary optimality conditions. The article is self-contained. It is definitely useful for advanced students and researchers interested in optimal control problems.
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 49K15
zbMATH DE Number: 6110970
zbMATH Keywords: optimal control problem
zbMATH Keywords: necessary second-order optimality conditions
zbMATH Keywords: discontinuous (piecewise continuous) control
zbMATH Keywords: quadratic form
zbMATH Keywords: critical cone
author: Nikolai P. Osmolovskii
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
full work available at URL: s10958-012-0824-1
OpenAlex ID: W4248095162
cites work: CONVEXITY OF VALUES OF VECTOR INTEGRALS, THEOREMS ON MEASURABLE CHOICE AND VARIATIONAL PROBLEMS
cites work: Q4923470
cites work: Second order optimality conditions for the \(L_ 1\)-minimization problem
cites work: Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints
cites work: Q3060964
cites work: Q4040624
cites work: Application of sliding mode method to optimal control problems with mixed constraints
cites work: Q4129369
cites work: Q3935031
cites work: Jacobi-type conditions for the problem of Bolza with inequalities
cites work: Q3752160
cites work: Q3915536
cites work: Q5509874
cites work: Second-Order Optimality Conditions in Sets of ${\text{L}}^\infty $ Functions with Range in a Polyhedron
cites work: On ${\text{L}}^2 $ Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems
cites work: Second-order necessary conditions in optimal control: Accessory-problem results without normality conditions
cites work: Q5518679
cites work: On second-order sufficient optimality conditions for c 1,1-optimization problems
cites work: Higher order necessary conditions in optimal control theory
cites work: Q4196953
cites work: Q4274229
cites work: Stability and sensitivity of solutions to nonlinear optimal control problems
cites work: Sufficient Optimality Conditions for Optimal Control Subject to State Constraints
cites work: Stability and sensitivity analysis for optimal control problems with control-state constraints
cites work: First and second order sufficient optimality conditions in mathematical programming and optimal control
cites work: Second Order Sufficient Conditions for Optimal Control Problems with Free Final Time: The Riccati Approach
cites work: Q5426478
cites work: Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control
cites work: Second-order sufficient conditions for control problems with mixed control-state constraints
cites work: Convex-valued Lipschitzian differential inclusions and the Pontryagin maximum principle
cites work: Q4001472
cites work: High-order conditions for a minimum on a set of sequences in the abstract problem with inequality constraints
cites work: High-order conditions for a minimum on a set of sequences in the abstract problem with inequality and equality constraints
cites work: High-order conditions with respect to a subsystem of constraints in the abstract minimization problem on a set of sequences
cites work: Necessary conditions and sufficient conditions of weak minimum for solutions with corner points
cites work: Another Jacobi sufficiency criterion for optimal control with smooth constraints
cites work: Q4133048
cites work: Q4126050
cites work: Investigation of extremals in the generalized isoperimetric problem using high-order conditions
cites work: Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations
cites work: Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints
cites work: Jacobi conditions and the Riccati equation for a broken extremal
cites work: Transformation of quadratic forms to perfect squares for broken extremals
cites work: Q5323189
cites work: Second Order Optimality Conditions for Controls with Continuous and Bang-Bang Components
cites work: Q5440842
cites work: Sufficiency conditions for weak local minima in multidimensional optimal control problems with mixed control-state restrictions
cites work: Q5727849
cites work: A NECESSARY AND SUFFICIENT CONDITION FOR A STRONG EXTREMUM IN A DEGENERATE PROBLEM OF THE CALCULUS OF VARIATIONS
cites work: Sufficiency Criteria via Focal Points and via Coupled Points
cites work: The Riccati Equation for Optimal Control Problems with Mixed State-Control Constraints: Necessity and Sufficiency
DOI: 10.1007/S10958-012-0824-1
description: scientific article; zbMATH DE number 6110970