Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

1978-11

AUTHORS

S. Gonzalez, A. Miele

ABSTRACT

This paper considers the numerical solution of two classes of optimal control problems, called Problem P1 and Problem P2 for easy identification.Problem P1 involves a functionalI subject to differential constraints and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter π so that the functionalI is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include nondifferential constraints to be satisfied everywhere along the interval of integration. Algorithms are developed for both Problem P1 and Problem P2.The approach taken is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase. The gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized.The principal property of both algorithms is that they produce a sequence of feasible suboptimal solutions: the functions obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the values of the functionalI corresponding to any two elements of the sequence are comparable.The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, while the stepsize of the restoration phase is obtained by a one-dimensional search on the constraint errorP. The gradient stepsize and the restoration stepsize are chosen so that the restoration phase preserves the descent property of the gradient phase. Therefore, the value of the functionalI at the end of any complete gradient-restoration cycle is smaller than the value of the same functional at the beginning of that cycle.The algorithms presented here differ from those of Refs. 1 and 2, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with Refs. 1 and 2, the present algorithms are capable of handling general final conditions; therefore, they are suited for the solution of optimal control problems with general boundary conditions. Their importance lies in the fact that many optimal control problems involve initial conditions of the type considered here.Six numerical examples are presented in order to illustrate the performance of the algorithms associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of these algorithms. More... »

PAGES

395-425

References to SciGraph publications

  • 1970-04. Sequential gradient-restoration algorithm for optimal control problems in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1973-07. Sequential gradient-restoration algorithm for optimal control problems with bounded state in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1968-07. Method of particular solutions for linear, two-point boundary-value problems in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1975-12. Recent advances in gradient algorithms for optimal control problems in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1974-02. Sequential gradient-restoration algorithm for optimal control problems with nondifferential constraints in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1975-02. Sequential conjugate-gradient-restoration algorithm for optimal control problems. Part 1. Theory in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 1975-02. Sequential conjugate-gradient-restoration algorithm for optimal control problems. Part 2. Examples in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf00933463

    DOI

    http://dx.doi.org/10.1007/bf00933463

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1007322541


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0102", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Applied Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Department of Electrical Engineering, Rice University, Houston, Texas", 
              "id": "http://www.grid.ac/institutes/grid.21940.3e", 
              "name": [
                "Department of Electrical Engineering, Rice University, Houston, Texas"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Gonzalez", 
            "givenName": "S.", 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Rice University, Houston, Texas", 
              "id": "http://www.grid.ac/institutes/grid.21940.3e", 
              "name": [
                "Rice University, Houston, Texas"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Miele", 
            "givenName": "A.", 
            "id": "sg:person.015552732657.49", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015552732657.49"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf00935541", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1049352093", 
              "https://doi.org/10.1007/bf00935541"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02665293", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1027124497", 
              "https://doi.org/10.1007/bf02665293"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00932781", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1017158292", 
              "https://doi.org/10.1007/bf00932781"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00934837", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1005976319", 
              "https://doi.org/10.1007/bf00934837"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00937371", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1008341596", 
              "https://doi.org/10.1007/bf00937371"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00927913", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1010952160", 
              "https://doi.org/10.1007/bf00927913"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02665294", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024534111", 
              "https://doi.org/10.1007/bf02665294"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "1978-11", 
        "datePublishedReg": "1978-11-01", 
        "description": "This paper considers the numerical solution of two classes of optimal control problems, called Problem P1 and Problem P2 for easy identification.Problem P1 involves a functionalI subject to differential constraints and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter \u03c0 so that the functionalI is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include nondifferential constraints to be satisfied everywhere along the interval of integration. Algorithms are developed for both Problem P1 and Problem P2.The approach taken is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase. The gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized.The principal property of both algorithms is that they produce a sequence of feasible suboptimal solutions: the functions obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the values of the functionalI corresponding to any two elements of the sequence are comparable.The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, while the stepsize of the restoration phase is obtained by a one-dimensional search on the constraint errorP. The gradient stepsize and the restoration stepsize are chosen so that the restoration phase preserves the descent property of the gradient phase. Therefore, the value of the functionalI at the end of any complete gradient-restoration cycle is smaller than the value of the same functional at the beginning of that cycle.The algorithms presented here differ from those of Refs. 1 and 2, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with Refs. 1 and 2, the present algorithms are capable of handling general final conditions; therefore, they are suited for the solution of optimal control problems with general boundary conditions. Their importance lies in the fact that many optimal control problems involve initial conditions of the type considered here.Six numerical examples are presented in order to illustrate the performance of the algorithms associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of these algorithms.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/bf00933463", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1044187", 
            "issn": [
              "0022-3239", 
              "1573-2878"
            ], 
            "name": "Journal of Optimization Theory and Applications", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "3", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "26"
          }
        ], 
        "keywords": [
          "optimal control problem", 
          "general boundary conditions", 
          "control problem", 
          "Problem P1", 
          "Problem P2", 
          "one-dimensional search", 
          "boundary conditions", 
          "gradient phase", 
          "initial conditions", 
          "sequential gradient-restoration algorithm", 
          "interval of integration", 
          "gradient-restoration algorithm", 
          "feasible suboptimal solution", 
          "functionalI subject", 
          "nondifferential constraints", 
          "differential constraints", 
          "numerical solution", 
          "descent property", 
          "norm squared", 
          "state vector", 
          "numerical examples", 
          "functionalI", 
          "stepsize", 
          "restoration phase", 
          "suboptimal solution", 
          "convergence characteristics", 
          "two-phase cycle", 
          "constraint satisfaction", 
          "more iterations", 
          "initial point", 
          "numerical results", 
          "initial state", 
          "present algorithm", 
          "first order", 
          "parameter \u03c0", 
          "final conditions", 
          "principal properties", 
          "iteration", 
          "algorithm", 
          "constraints", 
          "problem", 
          "functionalJ", 
          "solution", 
          "accuracy", 
          "Ref", 
          "properties", 
          "class", 
          "squared", 
          "analogy", 
          "parameters", 
          "order", 
          "conditions", 
          "vector", 
          "values", 
          "point", 
          "phase", 
          "function", 
          "approach", 
          "search", 
          "sequence", 
          "state", 
          "performance", 
          "fact", 
          "results", 
          "control", 
          "interval", 
          "integration", 
          "elements", 
          "feasibility", 
          "variation", 
          "end", 
          "easy identification", 
          "types", 
          "components", 
          "characteristics", 
          "P2", 
          "identification", 
          "cycle", 
          "P1", 
          "importance", 
          "beginning", 
          "subjects", 
          "satisfaction", 
          "example", 
          "paper"
        ], 
        "name": "Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions", 
        "pagination": "395-425", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1007322541"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/bf00933463"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/bf00933463", 
          "https://app.dimensions.ai/details/publication/pub.1007322541"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-10-01T06:26", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20221001/entities/gbq_results/article/article_141.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/bf00933463"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/bf00933463'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/bf00933463'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf00933463'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf00933463'


     

    This table displays all metadata directly associated to this object as RDF triples.

    178 TRIPLES      21 PREDICATES      117 URIs      102 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/bf00933463 schema:about anzsrc-for:01
    2 anzsrc-for:0102
    3 schema:author N688f630e78664c009c21f1215274e732
    4 schema:citation sg:pub.10.1007/bf00927913
    5 sg:pub.10.1007/bf00932781
    6 sg:pub.10.1007/bf00934837
    7 sg:pub.10.1007/bf00935541
    8 sg:pub.10.1007/bf00937371
    9 sg:pub.10.1007/bf02665293
    10 sg:pub.10.1007/bf02665294
    11 schema:datePublished 1978-11
    12 schema:datePublishedReg 1978-11-01
    13 schema:description This paper considers the numerical solution of two classes of optimal control problems, called Problem P1 and Problem P2 for easy identification.Problem P1 involves a functionalI subject to differential constraints and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter π so that the functionalI is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include nondifferential constraints to be satisfied everywhere along the interval of integration. Algorithms are developed for both Problem P1 and Problem P2.The approach taken is a sequence of two-phase cycles, composed of a gradient phase and a restoration phase. The gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations and is designed to force constraint satisfaction to a predetermined accuracy, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized.The principal property of both algorithms is that they produce a sequence of feasible suboptimal solutions: the functions obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the values of the functionalI corresponding to any two elements of the sequence are comparable.The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, while the stepsize of the restoration phase is obtained by a one-dimensional search on the constraint errorP. The gradient stepsize and the restoration stepsize are chosen so that the restoration phase preserves the descent property of the gradient phase. Therefore, the value of the functionalI at the end of any complete gradient-restoration cycle is smaller than the value of the same functional at the beginning of that cycle.The algorithms presented here differ from those of Refs. 1 and 2, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with Refs. 1 and 2, the present algorithms are capable of handling general final conditions; therefore, they are suited for the solution of optimal control problems with general boundary conditions. Their importance lies in the fact that many optimal control problems involve initial conditions of the type considered here.Six numerical examples are presented in order to illustrate the performance of the algorithms associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of these algorithms.
    14 schema:genre article
    15 schema:isAccessibleForFree true
    16 schema:isPartOf N0c1b284e168d4ac1a1e8b9a54505916a
    17 N2cf260e63c1c4923bf76402d19c9ef74
    18 sg:journal.1044187
    19 schema:keywords P1
    20 P2
    21 Problem P1
    22 Problem P2
    23 Ref
    24 accuracy
    25 algorithm
    26 analogy
    27 approach
    28 beginning
    29 boundary conditions
    30 characteristics
    31 class
    32 components
    33 conditions
    34 constraint satisfaction
    35 constraints
    36 control
    37 control problem
    38 convergence characteristics
    39 cycle
    40 descent property
    41 differential constraints
    42 easy identification
    43 elements
    44 end
    45 example
    46 fact
    47 feasibility
    48 feasible suboptimal solution
    49 final conditions
    50 first order
    51 function
    52 functionalI
    53 functionalI subject
    54 functionalJ
    55 general boundary conditions
    56 gradient phase
    57 gradient-restoration algorithm
    58 identification
    59 importance
    60 initial conditions
    61 initial point
    62 initial state
    63 integration
    64 interval
    65 interval of integration
    66 iteration
    67 more iterations
    68 nondifferential constraints
    69 norm squared
    70 numerical examples
    71 numerical results
    72 numerical solution
    73 one-dimensional search
    74 optimal control problem
    75 order
    76 paper
    77 parameter π
    78 parameters
    79 performance
    80 phase
    81 point
    82 present algorithm
    83 principal properties
    84 problem
    85 properties
    86 restoration phase
    87 results
    88 satisfaction
    89 search
    90 sequence
    91 sequential gradient-restoration algorithm
    92 solution
    93 squared
    94 state
    95 state vector
    96 stepsize
    97 subjects
    98 suboptimal solution
    99 two-phase cycle
    100 types
    101 values
    102 variation
    103 vector
    104 schema:name Sequential gradient-restoration algorithm for optimal control problems with general boundary conditions
    105 schema:pagination 395-425
    106 schema:productId N9f03b39045d4425ebd5d95dd58f238c8
    107 Nbeb085c8bd304c5f8c4a121cc819391a
    108 schema:sameAs https://app.dimensions.ai/details/publication/pub.1007322541
    109 https://doi.org/10.1007/bf00933463
    110 schema:sdDatePublished 2022-10-01T06:26
    111 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    112 schema:sdPublisher N0e4565fdaf3c449abdd204199d7a8eb1
    113 schema:url https://doi.org/10.1007/bf00933463
    114 sgo:license sg:explorer/license/
    115 sgo:sdDataset articles
    116 rdf:type schema:ScholarlyArticle
    117 N0c1b284e168d4ac1a1e8b9a54505916a schema:volumeNumber 26
    118 rdf:type schema:PublicationVolume
    119 N0e4565fdaf3c449abdd204199d7a8eb1 schema:name Springer Nature - SN SciGraph project
    120 rdf:type schema:Organization
    121 N2cf260e63c1c4923bf76402d19c9ef74 schema:issueNumber 3
    122 rdf:type schema:PublicationIssue
    123 N4c3a12ce74394c2cbe01223b42086292 rdf:first sg:person.015552732657.49
    124 rdf:rest rdf:nil
    125 N688f630e78664c009c21f1215274e732 rdf:first N7950c88dbaf84422bb1358b091ffd19e
    126 rdf:rest N4c3a12ce74394c2cbe01223b42086292
    127 N7950c88dbaf84422bb1358b091ffd19e schema:affiliation grid-institutes:grid.21940.3e
    128 schema:familyName Gonzalez
    129 schema:givenName S.
    130 rdf:type schema:Person
    131 N9f03b39045d4425ebd5d95dd58f238c8 schema:name doi
    132 schema:value 10.1007/bf00933463
    133 rdf:type schema:PropertyValue
    134 Nbeb085c8bd304c5f8c4a121cc819391a schema:name dimensions_id
    135 schema:value pub.1007322541
    136 rdf:type schema:PropertyValue
    137 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    138 schema:name Mathematical Sciences
    139 rdf:type schema:DefinedTerm
    140 anzsrc-for:0102 schema:inDefinedTermSet anzsrc-for:
    141 schema:name Applied Mathematics
    142 rdf:type schema:DefinedTerm
    143 sg:journal.1044187 schema:issn 0022-3239
    144 1573-2878
    145 schema:name Journal of Optimization Theory and Applications
    146 schema:publisher Springer Nature
    147 rdf:type schema:Periodical
    148 sg:person.015552732657.49 schema:affiliation grid-institutes:grid.21940.3e
    149 schema:familyName Miele
    150 schema:givenName A.
    151 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015552732657.49
    152 rdf:type schema:Person
    153 sg:pub.10.1007/bf00927913 schema:sameAs https://app.dimensions.ai/details/publication/pub.1010952160
    154 https://doi.org/10.1007/bf00927913
    155 rdf:type schema:CreativeWork
    156 sg:pub.10.1007/bf00932781 schema:sameAs https://app.dimensions.ai/details/publication/pub.1017158292
    157 https://doi.org/10.1007/bf00932781
    158 rdf:type schema:CreativeWork
    159 sg:pub.10.1007/bf00934837 schema:sameAs https://app.dimensions.ai/details/publication/pub.1005976319
    160 https://doi.org/10.1007/bf00934837
    161 rdf:type schema:CreativeWork
    162 sg:pub.10.1007/bf00935541 schema:sameAs https://app.dimensions.ai/details/publication/pub.1049352093
    163 https://doi.org/10.1007/bf00935541
    164 rdf:type schema:CreativeWork
    165 sg:pub.10.1007/bf00937371 schema:sameAs https://app.dimensions.ai/details/publication/pub.1008341596
    166 https://doi.org/10.1007/bf00937371
    167 rdf:type schema:CreativeWork
    168 sg:pub.10.1007/bf02665293 schema:sameAs https://app.dimensions.ai/details/publication/pub.1027124497
    169 https://doi.org/10.1007/bf02665293
    170 rdf:type schema:CreativeWork
    171 sg:pub.10.1007/bf02665294 schema:sameAs https://app.dimensions.ai/details/publication/pub.1024534111
    172 https://doi.org/10.1007/bf02665294
    173 rdf:type schema:CreativeWork
    174 grid-institutes:grid.21940.3e schema:alternateName Department of Electrical Engineering, Rice University, Houston, Texas
    175 Rice University, Houston, Texas
    176 schema:name Department of Electrical Engineering, Rice University, Houston, Texas
    177 Rice University, Houston, Texas
    178 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...