ref_calculus_variations_optimal_control.bib

@article{arutyunovInvestigationDegeneracyPhenomenon1997,
  title = {Investigation of the {{Degeneracy Phenomenon}} of the {{Maximum Principle}} for {{Optimal Control Problems}} with {{State Constraints}}},
  author = {Arutyunov, Aram V. and Aseev, Sergei M.},
  year = {1997},
  month = may,
  journal = {SIAM Journal on Control and Optimization},
  volume = {35},
  number = {3},
  pages = {930--952},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/S036301299426996X},
  url = {https://epubs.siam.org/doi/10.1137/S036301299426996X},
  urldate = {2023-04-14},
  abstract = {Standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest, e.g., when the left endpoint of state trajectories is fixed at x0  and x0  lies in the boundary of the state constraint set; in these cases a nonzero, but nevertheless trivial, set of multipliers exists. We give conditions for the existence of nontrivial multipliers. A feature of these conditions is that they allow nonconvex velocity sets and measurably time-dependent data. The proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint, originating in the dynamic programming literature.}
}

@article{arutyunovStateConstraintsOptimal1995,
  title = {State Constraints in Optimal Control. {{The}} Degeneracy Phenomenon},
  author = {Arutyunov, A. V. and Aseev, S. M.},
  year = {1995},
  month = nov,
  journal = {Systems \& Control Letters},
  volume = {26},
  number = {4},
  pages = {267--273},
  issn = {0167-6911},
  doi = {10.1016/0167-6911(95)00021-Z},
  url = {https://www.sciencedirect.com/science/article/pii/016769119500021Z},
  urldate = {2023-04-14},
  abstract = {In this paper we study the degeneracy phenomenon arising in optimal control problems with state constraints. It is shown that this phenomenon occurs because of the incompleteness of the standard variants of Pontryagin's maximum principle for problems with state constraints. The new maximum principle containing some additional information about the behavior of the Hamiltonian at the endtimes is developed. As application we obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle. The results obtained envelope the optimal control problems with systems described by differential inclusions and ordinary differential equations.},
  langid = {english}
}

@article{arutyunovSurveyRegularityConditions2020,
  title = {A {{Survey}} on {{Regularity Conditions}} for {{State-Constrained Optimal Control Problems}} and the {{Non-degenerate Maximum Principle}}},
  author = {Arutyunov, Aram and Karamzin, Dmitry},
  year = {2020},
  month = mar,
  journal = {Journal of Optimization Theory and Applications},
  volume = {184},
  number = {3},
  pages = {697--723},
  issn = {1573-2878},
  doi = {10.1007/s10957-019-01623-7},
  url = {https://doi.org/10.1007/s10957-019-01623-7},
  urldate = {2023-05-02},
  abstract = {A survey on the theory of maximum principle for state-constrained optimal control problems is presented. The focus is on such issues as regularity and controllability conditions, non-degeneracy and normality of the maximum principle, and on the continuity of the measure multiplier.},
  langid = {english}
}

@article{bettiolSensitivityInterpretationsCostate2010,
  title = {Sensitivity {{Interpretations}} of the {{Costate Variable}} for {{Optimal Control Problems}} with {{State Constraints}}},
  author = {Bettiol, Piernicola and Vinter, Richard B.},
  year = {2010},
  month = jan,
  journal = {SIAM Journal on Control and Optimization},
  volume = {48},
  number = {5},
  pages = {3297--3317},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/080732614},
  url = {https://epubs.siam.org/doi/10.1137/080732614},
  urldate = {2023-04-14},
  abstract = {This paper provides necessary conditions  of optimality for optimal control problems, in which the pathwise constraints comprise both ``pure'' constraints on the state variable and ``mixed'' constraints on control and state variables. The proofs are along the lines of earlier analysis for mixed constraint problems, according to which Clarke's theory of ``stratified'' necessary conditions is applied to a modified optimal control problem resulting from absorbing the mixed constraint into the dynamics; the difference here is that necessary conditions which now take into account the presence of pure state constraints are applied to the modified problem. Necessary conditions are given for a rather general formulation of the problem containing both forms of the constraints, and then these are specialized to problems having special structure. While combined pure state and mixed control/state problems have been previously treated in the literature, the necessary conditions in this paper are proved under less restrictive hypotheses and for novel formulations of the constraints.}
}

@article{bocciaOptimalControlProblems2016,
  title = {Optimal {{Control Problems}} with {{Mixed}} and {{Pure State Constraints}}},
  author = {Boccia, A. and {de Pinho}, M. D. R. and Vinter, R. B.},
  year = {2016},
  month = jan,
  journal = {SIAM Journal on Control and Optimization},
  volume = {54},
  number = {6},
  pages = {3061--3083},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/15M1041845},
  url = {https://epubs.siam.org/doi/10.1137/15M1041845},
  urldate = {2023-04-14},
  abstract = {We develop necessary conditions of broad applicability for optimal control problems in which the state and control are subject to mixed constraints. We unify, subsume, and significantly extend most of the results on this subject, notably in the three special cases that comprise the bulk of the literature: calculus of variations, differential-algebraic systems, and mixed constraints specified by equalities and inequalities. Our approach also provides a new and unified calibrated formulation of the appropriate constraint qualifications, and shows how to extend them to nonsmooth data. Other features include a very weak hypothesis concerning the type of local minimum, nonrestrictive hypotheses on the data, and stronger conclusions, notably as regards the maximum (or Weierstrass) condition. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. This leads to local, intermediate, and global versions of the necessary conditions, according to how the hypotheses are formulated.}
}

@article{brogliatoResultsOptimalControl2009,
  title = {Some Results on Optimal Control with Unilateral State Constraints},
  author = {Brogliato, Bernard},
  year = {2009},
  month = may,
  journal = {Nonlinear Analysis: Theory, Methods \& Applications},
  volume = {70},
  number = {10},
  pages = {3626--3657},
  issn = {0362-546X},
  doi = {10.1016/j.na.2008.07.020},
  url = {https://www.sciencedirect.com/science/article/pii/S0362546X08004161},
  urldate = {2023-04-14},
  abstract = {In this paper, we study the problem of quadratic optimal control with state variables unilateral constraints, for linear time-invariant systems. The necessary conditions are formulated as a linear invariant system with complementary slackness conditions. Some structural properties of this system are examined. Then it is shown that the problem can benefit from the higher order Moreau's sweeping process, that is, a specific distributional differential inclusion, and from ten Dam's geometric theory [A.A. ten Dam, K.F. Dwarshuis, J.C. Willems, The contact problem for linear continuous-time dynamical systems: A geometric approach, IEEE Trans. Automat. Control 42 (4) (1997) 458{\textendash}472; A.A. ten Dam, Unilaterally Constrained Dynamical Systems, Ph.D. Thesis, Rijsuniversiteit Groningen, NL, available at http://irs.ub.rug.nl/ppn/159407869, 1997] for partitioning of the admissible domain boundary (in particular for the case of multivariable systems). In fact, the first step may be also seen as follows: does the higher order Moreau's sweeping process (developed in Acary et~al. [V. Acary, B. Brogliato, D. Goeleven, Higher order Moreau's sweeping process: Mathematical formulation and numerical simulation, Math. Programm. A 113 (2008) 133{\textendash}217]) correspond to the necessary conditions of some optimal control problem with an extended integral action? The knowledge of the qualitative behaviour of optimal trajectories at junction times is improved with the approach, which also paves the way towards efficient time-stepping numerical algorithms to solve the optimal control boundary value problem.},
  langid = {english}
}

@article{brysonOptimalControl195019851996,
  title = {Optimal Control-1950 to 1985},
  author = {Bryson, A.E.},
  year = {1996},
  month = jun,
  journal = {IEEE Control Systems Magazine},
  volume = {16},
  number = {3},
  pages = {26--33},
  issn = {1941-000X},
  doi = {10.1109/37.506395},
  abstract = {Optimal control had its origins in the calculus of variations in the 17th century. The calculus of variations was developed further in the 18th century by Euler and Lagrange and in the 19th century by Legendre, Jacobi, Hamilton, and Weierstrass. In the early 20th century, Bolza and Bliss put the final touches of rigor on the subject. In 1957, Bellman gave a new view of Hamilton-Jacobi theory which he called dynamic programming, essentially a nonlinear feedback control scheme. McShane (1939) and Pontryagin (1962) extended the calculus of variations to handle control variable inequality constraints, the latter enunciating his elegant maximum principle. The truly enabling element for use of optimal control theory was the digital computer, which became available commercially in the 1950s. In the 1980s research began, and continues today, on making optimal feedback logic more robust to variations in the plant and disturbance models; one element of this research is worst-case and H-infinity control, which developed out of differential game theory.}
}

@article{buskensSQPmethodsSolvingOptimal2000,
  title = {{{SQP-methods}} for Solving Optimal Control Problems with Control and State Constraints: Adjoint Variables, Sensitivity Analysis and Real-Time Control},
  shorttitle = {{{SQP-methods}} for Solving Optimal Control Problems with Control and State Constraints},
  author = {B{\"u}skens, Christof and Maurer, Helmut},
  year = {2000},
  month = aug,
  journal = {Journal of Computational and Applied Mathematics},
  volume = {120},
  number = {1},
  pages = {85--108},
  issn = {0377-0427},
  doi = {10.1016/S0377-0427(00)00305-8},
  url = {https://www.sciencedirect.com/science/article/pii/S0377042700003058},
  urldate = {2023-05-03},
  abstract = {Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.},
  langid = {english}
}

@article{clarkeOptimalControlProblems2010,
  title = {Optimal {{Control Problems}} with {{Mixed Constraints}}},
  author = {Clarke, Francis and {de Pinho}, M. R.},
  year = {2010},
  month = jan,
  journal = {SIAM Journal on Control and Optimization},
  volume = {48},
  number = {7},
  pages = {4500--4524},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/090757642},
  url = {https://epubs.siam.org/doi/abs/10.1137/090757642},
  urldate = {2023-04-14},
  abstract = {This paper provides necessary conditions  of optimality for optimal control problems, in which the pathwise constraints comprise both ``pure'' constraints on the state variable and ``mixed'' constraints on control and state variables. The proofs are along the lines of earlier analysis for mixed constraint problems, according to which Clarke's theory of ``stratified'' necessary conditions is applied to a modified optimal control problem resulting from absorbing the mixed constraint into the dynamics; the difference here is that necessary conditions which now take into account the presence of pure state constraints are applied to the modified problem. Necessary conditions are given for a rather general formulation of the problem containing both forms of the constraints, and then these are specialized to problems having special structure. While combined pure state and mixed control/state problems have been previously treated in the literature, the necessary conditions in this paper are proved under less restrictive hypotheses and for novel formulations of the constraints.}
}

@article{cotsDirectIndirectMethods2018,
  title = {Direct and Indirect Methods in Optimal Control with State Constraints and the Climbing Trajectory of an Aircraft},
  author = {Cots, Olivier and Gergaud, Joseph and Goubinat, Damien},
  year = {2018},
  journal = {Optimal Control Applications and Methods},
  volume = {39},
  number = {1},
  pages = {281--301},
  issn = {1099-1514},
  doi = {10.1002/oca.2347},
  url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/oca.2347},
  urldate = {2023-05-02},
  abstract = {In this article, the minimum time and fuel consumption of an aircraft in its climbing phase are studied. The controls are the thrust and the lift coefficient and state constraints are taken into account: air slope and speed limitations. The application of the maximum principle leads to parameterize the optimal control and the multipliers associated with the state constraints with the state and the costate and leads to describe a multipoint boundary value problem, which is solved by multiple shooting. This indirect method is the numerical implementation of the maximum principle with state constraints and it is initialized by the direct method, both to determine the optimal structure and to obtain a satisfying initial guess. The solutions of the boundary value problems we define give extremals, which satisfy necessary conditions of optimality with at most 2 boundary arcs. Note that the aircraft dynamics has a singular perturbation but no reduction is performed.},
  langid = {english}
}

@article{ferreiraNondegenerateNecessaryConditions1999,
  title = {Nondegenerate {{Necessary Conditions}} for {{Nonconvex Optimal Control Problems}} with {{State Constraints}}},
  author = {Ferreira, M. M. A and Fontes, F. A. C. C and Vinter, R. B},
  year = {1999},
  month = may,
  journal = {Journal of Mathematical Analysis and Applications},
  volume = {233},
  number = {1},
  pages = {116--129},
  issn = {0022-247X},
  doi = {10.1006/jmaa.1999.6270},
  url = {https://www.sciencedirect.com/science/article/pii/S0022247X99962704},
  urldate = {2023-04-14},
  abstract = {Standard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set, and so give no useful information about optimal controls. Recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions. In these papers it is typically shown that, if a constraint qualification is imposed, requiring existence of inward pointing velocities, then sets of multipliers exist in addition to the trivial ones. A simple, new approach for deriving nondegenerate necessary conditions is presented, which permits relaxation of hypotheses previously imposed, concerning data regularity and convexity of the velocity set.},
  langid = {english}
}

@article{gollanOptimalControlProblems1980,
  title = {On Optimal Control Problems with State Constraints},
  author = {Gollan, B.},
  year = {1980},
  month = sep,
  journal = {Journal of Optimization Theory and Applications},
  volume = {32},
  number = {1},
  pages = {75--80},
  issn = {1573-2878},
  doi = {10.1007/BF00934843},
  url = {https://doi.org/10.1007/BF00934843},
  urldate = {2023-05-02},
  abstract = {This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.},
  langid = {english}
}

@book{gregoryConstrainedOptimizationCalculus2017,
  title = {Constrained {{Optimization In The Calculus Of Variations}} and {{Optimal Control Theory}}},
  author = {Gregory, J.},
  year = {2017},
  month = dec,
  publisher = {{Chapman and Hall/CRC}},
  address = {{New York}},
  doi = {10.1201/9781351070867},
  abstract = {The major purpose of this book is to present the theoretical ideas and the analytical and numerical methods to enable the reader to understand and efficiently solve these important optimizational problems.The first half of this book should serve as the major component of a classical one or two semester course in the calculus of variations and optimal control theory. The second half of the book will describe the current research of the authors which is directed to solving these problems numerically. In particular, we present new reformulations of constrained problems which leads to unconstrained problems in the calculus of variations and new general, accurate and efficient numerical methods to solve the reformulated problems. We believe that these new methods will allow the reader to solve important problems.},
  isbn = {978-1-351-07086-7}
}

@article{hartlSurveyMaximumPrinciples1995a,
  title = {A {{Survey}} of the {{Maximum Principles}} for {{Optimal Control Problems}} with {{State Constraints}}},
  author = {Hartl, Richard F. and Sethi, Suresh P. and Vickson, Raymond G.},
  year = {1995},
  month = jun,
  journal = {SIAM Review},
  volume = {37},
  number = {2},
  pages = {181--218},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0036-1445},
  doi = {10.1137/1037043},
  url = {https://epubs.siam.org/doi/abs/10.1137/1037043},
  urldate = {2023-04-13},
  abstract = {This paper suggests some further developments in the theory of first-order necessary optimality conditions for problems of optimal control with infinite time horizons. We describe an approximation technique involving auxiliary finite-horizon optimal control problems and use it to prove new versions of the Pontryagin maximum principle. Special attention is paid to the behavior of the adjoint variables and the Hamiltonian. Typical cases, in which standard transversality conditions hold at infinity, are described. Several significant earlier results are generalized.}
}

@book{hestenesCalculusVariationsOptimal1980,
  title = {Calculus of Variations and Optimal Control Theory},
  author = {Hestenes, Magnus Rudolph},
  year = {1980},
  month = jan,
  abstract = {(from the author's Preface): This book is a record of my lectures on the calculus of variations and optimal control theory given at the University of CA, Los Angeles. Variational theory is presented from three points of view.}
}

@article{hymasNeighboringExtremalsOptimal1973,
  title = {Neighboring Extremals for Optimal Control Problems.},
  author = {Hymas, C. E. and Cavin, R. K. and Colunga, D.},
  year = {1973},
  journal = {AIAA Journal},
  volume = {11},
  number = {8},
  pages = {1101--1109},
  publisher = {{American Institute of Aeronautics and Astronautics}},
  issn = {0001-1452},
  doi = {10.2514/3.6882},
  url = {https://doi.org/10.2514/3.6882},
  urldate = {2023-04-13}
}

@book{chachuatNonlinearDynamicOptimization2007,
  title = {Nonlinear and {{Dynamic Optimization}}: {{From Theory}} to {{Practice}}},
  shorttitle = {Nonlinear and {{Dynamic Optimization}}},
  editor = {Chachuat, Beno{\i}t C.},
  year = {2007},
  publisher = {{EPFL}},
  address = {{Lausanne, CH}},
  url = {https://infoscience.epfl.ch/record/111939?ln=en}
}

@article{jacobsonNewNecessaryConditions1971,
  title = {New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints},
  author = {Jacobson, D. H and Lele, M. M and Speyer, J. L},
  year = {1971},
  month = aug,
  journal = {Journal of Mathematical Analysis and Applications},
  volume = {35},
  number = {2},
  pages = {255--284},
  issn = {0022-247X},
  doi = {10.1016/0022-247X(71)90219-8},
  url = {https://www.sciencedirect.com/science/article/pii/0022247X71902198},
  urldate = {2023-05-02},
  abstract = {Necessary conditions of optimality for state-variable inequality constrained problems are derived which differ from those of Bryson, Denham, and Speyer with regard to the behavior of the adjoint variables at junctions of interior and boundary arcs. In particular, it is shown that the earlier conditions under-specify the behavior of the adjoint variables at the junctions. An example is used to demonstrate that the earlier conditions may yield non-stationary trajectories. For a certain class of problems, it is shown that only boundary points, as opposed to boundary arcs, are possible. An analytic example illustrates this behavior.},
  langid = {english}
}

@article{karamzinExtensionOptimalControl2014,
  title = {On Some Extension of Optimal Control Theory},
  author = {Karamzin, Dmitry Yu. and {de Oliveira}, Valeriano A. and Pereira, Fernando L. and Silva, Geraldo N.},
  year = {2014},
  month = nov,
  journal = {European Journal of Control},
  volume = {20},
  number = {6},
  pages = {284--291},
  issn = {0947-3580},
  doi = {10.1016/j.ejcon.2014.09.003},
  url = {https://www.sciencedirect.com/science/article/pii/S0947358014000776},
  urldate = {2023-04-14},
  abstract = {Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.},
  langid = {english}
}

@article{karamzinFewQuestionsRegarding2019,
  title = {On a {{Few Questions Regarding}} the {{Study}} of {{State-Constrained Problems}} in {{Optimal Control}}},
  author = {Karamzin, Dmitry and Pereira, Fernando Lobo},
  year = {2019},
  month = jan,
  journal = {Journal of Optimization Theory and Applications},
  volume = {180},
  number = {1},
  pages = {235--255},
  issn = {1573-2878},
  doi = {10.1007/s10957-018-1394-2},
  url = {https://doi.org/10.1007/s10957-018-1394-2},
  urldate = {2023-05-02},
  abstract = {The article is focused on the investigation of the necessary optimality conditions in the form of Pontryagin's maximum principle for optimal control problems with state constraints. A number of results on this topic, which refine the existing ones, are presented. These results concern the nondegenerate maximum principle under weakened controllability assumptions and also the continuity of the measure Lagrange multiplier.},
  langid = {english}
}

@article{kelleyGuidanceTheoryExtremal1962,
  title = {Guidance Theory and Extremal Fields},
  author = {Kelley, H.},
  year = {1962},
  month = oct,
  journal = {IRE Transactions on Automatic Control},
  volume = {7},
  number = {5},
  pages = {75--82},
  issn = {1558-3651},
  doi = {10.1109/TAC.1962.1105503},
  abstract = {A guidance concept employing properties of optimal flight paths is developed on the basis of Jacobi's accessory minimum problem for the second variation. The analysis is equivalent to construction of a field of extremals in the neighborhood of a predetermined extremal serving as a "nominal" trajectory. In the absence of inequality constraints on the control variables, a linear terminal control scheme with time-varying gains is realized. The addition of inequality constraints leads to nonlinear control behavior. Certain propulsion system parameters are characterized as state variables as a convenient means for providing adaptive behavior in respect to in-flight changes in propulsion system performance. An application is given to an intercept problem sufficiently simple to allow analytical solution, and some numerical results comparing optimal and approximately optimal guidance in their effects on flight performance are presented. Treatment of a certain type of problem arising in rocket applications is discussed.}
}

@book{leviClassicalMechanicsCalculus2014,
  title = {Classical {{Mechanics With Calculus}} of {{Variations}} and {{Optimal Control}}: {{An Intuitive Introduction}}},
  shorttitle = {Classical {{Mechanics With Calculus}} of {{Variations}} and {{Optimal Control}}},
  author = {Levi, Mark},
  year = {2014},
  month = mar,
  series = {Student {{Mathematical Library}}},
  number = {69},
  publisher = {{American Mathematical Society}},
  address = {{Providence, Rhode Island : University Park, Pennsylvania : Mathematics Advanced Study Semesters}},
  url = {https://bookstore.ams.org/stml-69/},
  isbn = {978-0-8218-9138-4},
  langid = {english}
}

@book{liberzonCalculusVariationsOptimal2011,
  title = {Calculus of {{Variations}} and {{Optimal Control Theory}}: {{A Concise Introduction}}},
  shorttitle = {Calculus of {{Variations}} and {{Optimal Control Theory}}},
  author = {Liberzon, Daniel},
  year = {2011},
  month = dec,
  publisher = {{Princeton University Press}},
  url = {http://liberzon.csl.illinois.edu/teaching/cvoc/cvoc.html},
  isbn = {0-691-15187-3}
}

@book{maccluerCalculusVariationsMechanics2012,
  title = {Calculus of {{Variations}}: {{Mechanics}}, {{Control}} and {{Other Applications}}},
  shorttitle = {Calculus of {{Variations}}},
  author = {MacCluer, Charles R.},
  year = {2012},
  month = nov,
  edition = {Reprint edition},
  publisher = {{Dover Publications}},
  address = {{Mineola, N.Y.}},
  abstract = {The first truly up-to-date treatment of the calculus of variations, this text is also the first to offer a simple introduction to such key concepts as optimal control and linear-quadratic control design. Suitable for junior/senior{\textendash}level students of math, science, and engineering, this volume also serves as a useful reference for engineers, chemists, and forest/environmental managers. Its broad perspective features numerous exercises, hints, outlines, and comments, plus several appendixes, including a practical discussion of MATLAB.Students will appreciate the text's reader-friendly style, which features gradual advancements in difficulty and starts by developing technique rather than focusing on technical details. The examples and exercises offer many citations of engineering-based applications, and the exercises range from elementary to graduate-level projects, including longer projects and those related to classic papers.},
  isbn = {978-0-486-49837-9},
  langid = {english}
}

@article{maurerOptimalControlProblems1977,
  title = {On {{Optimal Control Problems}} with {{Bounded State Variables}} and {{Control Appearing Linearly}}},
  author = {Maurer, H.},
  year = {1977},
  month = may,
  journal = {SIAM Journal on Control and Optimization},
  volume = {15},
  number = {3},
  pages = {345--362},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/0315023},
  url = {https://epubs.siam.org/doi/abs/10.1137/0315023},
  urldate = {2023-05-03},
  abstract = {The generalized Legendre{\textendash}Clebsch higher order tests for optimality of singular arcs in optimal control problems depend upon the orders of the arcs involved. To date three distinct definitions of order have been given but many authors do not distinguish among them. The features of each definition are discussed with special reference to the applicability of the higher order tests and of the conditions at junctions between singular and nonsingular arcs; only in terms of one of the definitions are the junction conditions generally valid. An illustrative example is presented.}
}

@misc{maurerTutorialControlState2011,
  title = {Tutorial on {{Control}} and {{State Constrained Optimal Control Problems}}},
  author = {Maurer, Helmut},
  year = {2011},
  month = sep,
  address = {{Imperial College London, UK}},
  url = {https://inria.hal.science/inria-00629518}
}

@book{mesterton-gibbonsPrimerCalculusVariations2009,
  title = {A {{Primer}} on the {{Calculus}} of {{Variations}} and {{Optimal Control Theory}}},
  author = {{Mesterton-Gibbons}, Mike},
  year = {2009},
  month = jul,
  series = {Student {{Mathematical Library}}},
  edition = {First Edition},
  number = {50},
  publisher = {{American Mathematical Society}},
  address = {{Providence, R.I}},
  url = {https://bookstore.ams.org/stml-50},
  abstract = {The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.},
  isbn = {978-0-8218-4772-5},
  langid = {english}
}

@article{mieleNumericalSolutionsSimplest1972,
  title = {Numerical {{Solutions}} in the {{Simplest Problem}} of the {{Calculus}} of {{Variations}}},
  author = {Miele, A. and Pritchard, R.},
  year = {1972},
  month = jul,
  journal = {SIAM Review},
  volume = {14},
  number = {3},
  pages = {385--398},
  issn = {0036-1445},
  doi = {10.1137/1014066},
  url = {https://epubs.siam.org/doi/10.1137/1014066},
  urldate = {2018-04-06},
  abstract = {This is an expository paper dealing with the numerical techniques for solving the simplest problem in the calculus of variations. Both first-variation methods and second-variation methods are reviewed.Concerning first-variation methods, a direct approach and an indirect approach are given. They both cause a reduction of the integral at each iteration.Concerning second-variation methods, two possibilities are explored: (i) reduction of the integral, and (ii) reduction of the performance index. In case (i), two viewpoints are employed: (a) minimization of the sum of the first variation and the second variation, and (b) minimization of the first variation for a given second variation. In case (ii), two viewpoints are employed : (a) quasi-linearization of the Euler equation, and (b) descent process on the performance index.}
}

@article{murrayExistenceTheoremsOptimal1986,
  title = {Existence {{Theorems}} for {{Optimal Control}} and {{Calculus}} of {{Variations Problems Where}} the {{States Can Jump}}},
  author = {Murray, J. M.},
  year = {1986},
  month = may,
  journal = {SIAM Journal on Control and Optimization},
  volume = {24},
  number = {3},
  pages = {412--438},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/0324024},
  url = {https://epubs.siam.org/doi/abs/10.1137/0324024},
  urldate = {2023-04-13},
  abstract = {The concluding result of the paper states that variational problems are generically solvable (and even well-posed in a strong sense) without the convexity and growth conditions always present in individual existence theorems. This and some other generic well-posedness theorems are obtained as realizations of a general variational principle extending the variational principle of Deville--Godefroy--Zizler.}
}

@book{osmolovskiiApplicationsRegularBangBang2014,
  title = {Applications to {{Regular}} and {{Bang-Bang Control}}: {{Second-Order Necessary}} and {{Sufficient Optimality Conditions}} in {{Calculus}} of {{Variations}} and {{Optimal Control}}},
  shorttitle = {Applications to {{Regular}} and {{Bang-Bang Control}}},
  author = {Osmolovskii, Nikolai P. and Maurer, Helmut},
  year = {27 {\'u}nora 2014},
  series = {Advances in {{Design}} and {{Control}}},
  publisher = {{SIAM}},
  address = {{Philadelphia}},
  abstract = {This book is devoted to the theory and applications of second-order necessary and sufficient optimality conditions in the calculus of variations and optimal control. The authors develop theory for a control problem with ordinary differential equations subject to boundary conditions of equality and inequality type, and for mixed state-control constraints of equality type. The book has several distinctive features: necessary and sufficient conditions are given in the form of no-gap conditions; the theory covers broken extremals where the control has finitely many points of discontinuity; and a number of numerical examples in various areas of application are fully solved. This book will be of interest to academic researchers in calculus of variations and optimal control. It will also be a useful resource to researchers and engineers who use applications of optimal control in areas such as mechanics, mechatronics, physics, economics, or chemical, electrical and biological engineering.},
  isbn = {978-1-61197-235-1}
}

@inproceedings{podmajerskyOnLineNeighbouringExtremalController2009,
  title = {On-{{Line Neighbouring-Extremal Controller Design}} for {{Setpoint Transition}} in {{Presence}} of {{Uncertainty}}},
  booktitle = {Proc. of 17th {{Internationial Conference}} on {{Process Control}}},
  author = {Podmajersk{\'y}, Mari{\'a}n and Fikar, Miroslav},
  year = {2009},
  publisher = {{Slovak University of Technology in Bratislava, Institute of Information Engineering, Automation, and Mathematics}},
  address = {{{\v S}trbsk{\'e} Pleso, Slovakia}},
  url = {https://www.uiam.sk/pc09/data/papers/048.pdf}
}

@article{rampazzoDegenerateOptimalControl2000,
  title = {Degenerate {{Optimal Control Problems}} with {{State Constraints}}},
  author = {Rampazzo, Franco and Vinter, Richard},
  year = {2000},
  month = jan,
  journal = {SIAM Journal on Control and Optimization},
  volume = {39},
  number = {4},
  pages = {989--1007},
  publisher = {{Society for Industrial and Applied Mathematics}},
  issn = {0363-0129},
  doi = {10.1137/S0363012998340223},
  url = {https://epubs.siam.org/doi/10.1137/S0363012998340223},
  urldate = {2023-04-14},
  abstract = {This paper provides necessary conditions  of optimality for optimal control problems, in which the pathwise constraints comprise both ``pure'' constraints on the state variable and ``mixed'' constraints on control and state variables. The proofs are along the lines of earlier analysis for mixed constraint problems, according to which Clarke's theory of ``stratified'' necessary conditions is applied to a modified optimal control problem resulting from absorbing the mixed constraint into the dynamics; the difference here is that necessary conditions which now take into account the presence of pure state constraints are applied to the modified problem. Necessary conditions are given for a rather general formulation of the problem containing both forms of the constraints, and then these are specialized to problems having special structure. While combined pure state and mixed control/state problems have been previously treated in the literature, the necessary conditions in this paper are proved under less restrictive hypotheses and for novel formulations of the constraints.}
}

@incollection{rockafellarConvexAnalysisCalculus2001,
  title = {Convex {{Analysis}} in the {{Calculus}} of {{Variations}}},
  booktitle = {Advances in {{Convex Analysis}} and {{Global Optimization}}},
  author = {Rockafellar, R. T.},
  year = {2001},
  series = {Nonconvex {{Optimization}} and {{Its Applications}}},
  pages = {135--151},
  publisher = {{Springer, Boston, MA}},
  doi = {10.1007/978-1-4613-0279-7_7},
  url = {https://link.springer.com/chapter/10.1007/978-1-4613-0279-7_7},
  urldate = {2018-08-04},
  abstract = {Convexity properties are vital in the classical calculus of variations, and many notions of convex analysis, such as the Legendre-Fenchel transform, have their origin there. Conceptual developments in contemporary convex analysis have in turn enriched that venerable subject by making it possible to treat a vastly larger class of problems effectively in a ``neoclassical'' framework of extended-real-valued functions and their subgradients.},
  isbn = {978-0-7923-6942-4 978-1-4613-0279-7},
  langid = {english}
}

@article{russakGeneralProblemsBounded1970,
  title = {On General Problems with Bounded State Variables},
  author = {Russak, I. Bert},
  year = {1970},
  month = dec,
  journal = {Journal of Optimization Theory and Applications},
  volume = {6},
  number = {6},
  pages = {424--452},
  issn = {1573-2878},
  doi = {10.1007/BF00932720},
  url = {https://doi.org/10.1007/BF00932720},
  urldate = {2023-05-02},
  abstract = {This paper serves as the sequel to a recent article by the author concerned with a certain canonical control problem. The present paper extends the first-order necessary conditions obtained there to a general form of the control problem of Bolza with inequality state constraints of the type {$\psi\alpha$}(t, x){$\leq$}0, {$\alpha$}=1,...,m.},
  langid = {english}
}

@article{seierstadSufficientConditionsOptimal1977,
  title = {Sufficient {{Conditions}} in {{Optimal Control Theory}}},
  author = {Seierstad, Atle and Sydsaeter, Knut},
  year = {1977},
  journal = {International Economic Review},
  volume = {18},
  number = {2},
  eprint = {2525753},
  eprinttype = {jstor},
  pages = {367--391},
  publisher = {{[Economics Department of the University of Pennsylvania, Wiley, Institute of Social and Economic Research, Osaka University]}},
  issn = {0020-6598},
  doi = {10.2307/2525753},
  url = {https://www.jstor.org/stable/2525753},
  urldate = {2023-04-24}
}

@article{sussmann300YearsOptimal1997,
  title = {300 Years of Optimal Control: From the Brachystochrone to the Maximum Principle},
  shorttitle = {300 Years of Optimal Control},
  author = {Sussmann, H.J. and Willems, J.C.},
  year = {1997},
  month = jun,
  journal = {IEEE Control Systems},
  volume = {17},
  number = {3},
  pages = {32--44},
  issn = {1066-033X},
  doi = {10.1109/37.588098},
  abstract = {An historical review of the development of optimal control from the publication of the brachystochrone problem by Johann Bernoulli in 1696. Ideas on curve minimization already known at the time are briefly outlined. The brachystochrone problem is stated and Bernoulli's solution is given. Bernoulli's personality and his family are discussed. The article then traces the development of the necessary conditions for a minimum, from the Euler-Lagrange equations to the work of Legendre and Weierstrass and, eventually, the maximum principle of optimal control theory}
}

@book{troutmanVariationalCalculusOptimal1995,
  title = {Variational {{Calculus}} and {{Optimal Control}}: {{Optimization}} with {{Elementary Convexity}}},
  shorttitle = {Variational {{Calculus}} and {{Optimal Control}}},
  author = {Troutman, John L.},
  year = {1995},
  month = dec,
  edition = {2nd edition},
  publisher = {{Springer}},
  address = {{New York}},
  abstract = {An introduction to the variational methods used to formulate and solve mathematical and physical problems, allowing the reader an insight into the systematic use of elementary (partial) convexity of differentiable functions in Euclidian space. By helping students directly characterize the solutions for many minimization problems, the text serves as a prelude to the field theory for sufficiency, laying as it does the groundwork for further explorations in mathematics, physics, mechanical and electrical engineering, as well as computer science.},
  isbn = {978-0-387-94511-8},
  langid = {english}
}

@article{zermeloUberNavigationsproblemBei1931,
  title = {{\"U}ber Das {{Navigationsproblem}} Bei Ruhender Oder Ver{\"a}nderlicher {{Windverteilung}}},
  author = {Zermelo, E.},
  year = {1931},
  journal = {ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik},
  volume = {11},
  number = {2},
  pages = {114--124},
  issn = {1521-4001},
  doi = {10.1002/zamm.19310110205},
  url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19310110205},
  urldate = {2023-04-23},
  langid = {english}
}