Journal ArticleDOI
Second-order sufficient conditions for control problems with mixed control-state constraints
Helmut Maurer,Sabine Pickenhain +1 more
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In this article, the second-order sufficient conditions for local minima of optimal control problems with state and control constraints were extended to include general boundary conditions and a direct sufficiency criterion based on a quadratic function that satisfies a Hamilton-Jacobi inequality.Abstract:
References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.read more
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Journal ArticleDOI
Optimization Problems with Perturbations: A Guided Tour
TL;DR: The emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty.
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Second order gradient ascent pulse engineering
TL;DR: Improvements to the gradient ascent pulse engineering (GRAPE) algorithm are reported, including more accurate gradients, convergence acceleration using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm as well as faster control derivative calculation algorithms.
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Optimal Control and Applications to Aerospace: Some Results and Challenges
TL;DR: This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems.
Journal ArticleDOI
A SEIR model for control of infectious diseases with constraints
TL;DR: This paper proposes the introduction of constraints involving state variables on an optimal control problem applied to a compartmental SEIR (Susceptible. Exposed, Infectious and Recovered) model to study the solution of problems when mixed state control constraints are used to impose upper bounds on the available vaccines.
Journal ArticleDOI
Optimization methods for the verification of second order sufficient conditions for bang-bang controls
TL;DR: In this article, the authors present an optimization method for finding optimal bang-bang controls and verifying second-order sufficient conditions (SSC) which have been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii.
References
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Book
Nonlinear Programming: Sequential Unconstrained Minimization Techniques
TL;DR: This report gives the most comprehensive and detailed treatment to date of some of the most powerful mathematical programming techniques currently known--sequential unconstrained methods for constrained minimization problems in Euclidean n-space--giving many new results not published elsewhere.
Book ChapterDOI
Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems
TL;DR: In this article, second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems with constraints defined by closed convex cones are given, which are an immediate generalization of those known for the finite-dimensional case.
Book
Methods of dynamic and nonsmooth optimization
TL;DR: In this article, the basic problem in the Calculus of Variations Verification Functions and Dynamic Programming Optimal Control is discussed, as well as the problem of verifying the correctness of variations.
Book ChapterDOI
First and second order sufficient optimality conditions in mathematical programming and optimal control
TL;DR: First and second order sufficient conditions for infinite-dimensional programming problems with constraints defined by arbitrary closed convex cones are given in this article, which are applicable to optimal control problems with state constraints where the definiteness conditions can only hold in a weaker norm than that in which the functions involved are differentiable.
Book
Optimization: A Theory of Necessary Conditions
TL;DR: In this paper, a comprehensive treatment of necessary conditions for general optimization problems is presented in the context of a general theory for extremal problems in a topological vector space setting, and generalized Lagrange multiplier rules are derived for optimization problems with equality and generalized "inequality" constraints.