Pytlak Radoslaw. Numerical methods for optimal control problems with state constraints

Springer. — ISSN 0075-8434; ISBN 3-540-66214-6.The Calculus of Variations
Optimal Control
Numerical Methods for Optimal Control Problems
Estimates on Solutions to Differential Equations and Their Approximations
Linear Approximations
Lagrangian, Hamiltonian and Reduced Gradients
First Order MethodRepresentation of Functional Directional Derivatives
Relaxed Controls
The Algorithm
Convergence Properties of the Algorithm
Proof of the Convergence Theorem, etc
Concluding Remarks
Implementation
Implementable Algorithm
Second Order Correction To the Line Search
Resetting the Penalty Parameter
Semi-Infinite Programming Problem
Numerical Examples
Second Order MethodFunction Space Algorithm
Semi-Infinite Programming Method
Bounding the Number of Constraints
Some Remarks on Direction Finding Subproblems
The Nonlinear Programming Problem
The Watchdog Technique for Redundant Constraints
Two-Step Superlinear Convergence
Numerical Experiments
Concluding Remarks
Runge—Kutta Based Procedure for Optimal Control of Differential — Algebraic EquationsThe Method
Implicit Runge-Kutta Methods
Calculation of the Reduced Gradients
Implementation of the Implicit Runge-Kutta Method
Simplified Newton Iterations
Stopping Criterion for the Newton Method
Stepsize Selection
Numerical Experiments
Some Remarks on Integration and Optimization Accuracies
Concluding Remarks
Primal Range—Space Method for Piecewise—Linear
Quadratic Programming
Software Implementation
A Range-Space Method-Introduction
The Basic Method
Efficient Implementation
Adding a Bound to the Working Set
Deleting a Bound from the Working Set
Adding a Vector a to the Working Set
Deleting a Vector a from the Working Set
Computation of the Lagrange Multipliers
Modifications and Extensions
Numerical ExperimentsОсвещенные понятия$\mathcal{L}$-stable integration procedure
Accessory problem
Active arc
Active set method
Adjoint equations
Adjoint equations, continuous time
Adjoint equations, continuous time for implicit systems
Adjoint equations, discrete time
Adjoint equations, discrete time for implicit systems
Adjoint operator
Adjoint operator, self-adjoint operator
Algebraic state
Approximation errors
Ascoli’s Theorem
Backoff problem
Backward differentiation formula (BDF)
Barrier function algorithm
Barycentric coordinates
Basic existence/uniqnuess theorem
Bellman’s dynamic programming equation
BFGS updating formula
BLAS (Basic Linear Algebra Subroutines)
Brachistochrone problem
Calculus of variations
Chebyshev functional
Cholesky factorization
Collocation method
Constraint qualification
Control function
Control system
Cycling
Descent function, nonpositive
Differential state
Differential-algebraic equations, implicit
Differential-algebraic equations, index one
Differential-algebraic equations, semi-explicit
Dini derivative
Direction finding subproblem
Discrete time system
Discrete time system, implicit
Dominated Convergence Theorem
Dual method
Endpoint constraints
Euler — Lagrange equation
Exact penalty function
Feasible directions algorithm
Gauss integration procedure
Givens matrix
Gradient algorithms
Gradient restoration algorithm
Gram — Schmidt factorization
Gronwall’s lemma
Hamilton — Jacobi equation
Hamiltonian
Hessenberg matrix
hessian
Hessian, reduced
Implicit function theorem
Lagrange multiplier
Lagrangian
Linearized equations
Maximum principle, Pontryagin’s
Maximum principle, weak
Maximum principle, weak, discrete time
Maximum set of linearly independent vectors
Mean value theorem
Minimax theorem
Minimum time problem
Multiple shooting method
Multipoint boundary value problem
Newton step
Nonlinear programming problem (NLP)
Null-space method
Optimality conditions, necessary, normal
Optimality conditions, sufficiency
Pathwise inequality constraint
Penalty test function
Piecewise constant approximation
Piecewise-linear quadratic programming problem
Plane rotation
Point of attraction
Powell-symmetric-Broyden update (PSB)
Primal method
Projection
Projection, matrix
Proximity algorithm
Radau IIA integration procedure
Radon probability measure
Range-space method
Relaxed controls
Relaxed dynamics
Relaxed necessary optimality conditions
Relaxed problem
Relaxed state trajectory
Reorthogonalization
Runge — Kutta integration procedure, explicit
Runge — Kutta integration procedure, implicit
Search directions, set
Second order correction
Semi-infinite programming (SIP)
Sensitivity equations
Sequential quadratic programming (SQP)
Sequential quadratic programming (SQP), reduced gradient
simplex
Singular subarc
State trajectory
Stepsize selection procedure
Stiff equations
Strict complementarity condition
Strong local minimum
Subgradient
Superlinear convergence
Superlinear convergence, two-step
Uniform approximation, $(R_{\varepsilon, \mu}, \xi)$
Watchdog technique
Windshear problem