## Optimization and Multiobjective Control of Time-Discrete Systems: Dynamic Networks and Multilayered StructuresRichard Bellmann developed a theory of dynamic programming which is for many reasons still in the center of great interest. The authors present a new approach in the ?eld of the optimization and multi-objective control of time-discrete systems which is closely related to the work of Richard Bellmann. They develop their own concept and their extension to the optimization and multi-objective control of time-discrete systems as well as to dynamic networks and multilayered structures are very stimulating for further research. Di?erent perspectives of discrete control and optimal dynamic ?ow problems on networks are treated and characterized. Together with the algorithmic solutions a framework of multi-objective control problems is - rived. The conclusion with a real world example underlines the necessity and - portance of their theoretic framework. As they come back to the classical Bellmann concept of dynamic programming they stress and honor his basic concept without debase their own work. Multilayereddecisionprocessesaspartofthedesignandanalysisofcomplexsystems and networks will be essential in many ways and ?elds in the future. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

1 | |

2 | |

4 | |

113 Hierarchical Control and Stackelbergs Optimization Principle | 7 |

Pareto Optima | 8 |

115 Stationary and NonStationary Control of TimeDiscrete Systems | 10 |

13 Alternate Players Control Condition and Nash Equilibria for Dynamic Games in Positional Form | 11 |

14 Algorithms for Solving SingleObjective Control Problems on Networks | 15 |

32 The Control Problem on a Network with TransitionTime Functions on the Edges | 133 |

322 An Algorithm for Solving the Problem on a Network with TransitionTime Functions on the Edges | 134 |

33 MultiObjective Control of TimeDiscrete Systems with Varying Time of States Transitions | 141 |

332 A Dynamic cGame on Networks with TransitionTime Functions on the Edges | 146 |

333 Remark on Determining Pareto Optima for the MultiObjective Control Problem with Varying Time of States Transitions | 149 |

34 An Algorithm for Solving the Discrete Optimal Control Problem with Inﬁnite Time Horizon and Varying Time of the States Transitions | 150 |

342 An Algorithm for Determining an Optimal Stationary Control for Dynamical Systems with Inﬁnite Time Horizon | 152 |

35 A General Approach for Algorithmic Solutions of Discrete Optimal Control Problems and its GameTheoretic Extension | 154 |

142 An Extension of Dijkstras Algorithm for Optimal Control Problems with a Free Number of Stages | 18 |

15 MultiObjective Control and NonCooperative Games on Dynamic Networks | 22 |

152 The Problem of Determining the Optimal NonStationary Strategies in a Dynamic cGame | 25 |

16 Main Results for Dynamic cGames with Constant Costs of the Edges and Determining Optimal Stationary Strategies of the Players | 26 |

17 Computational Complexity of the Problem of Determining Optimal Stationary Strategies in a Dynamic cGame | 45 |

19 Determining Nash Equilibria for NonStationary Dynamic cGames | 53 |

192 Determining Nash Equilibria | 55 |

110 Application of the Dynamic cGame for Studying and Solving MultiObjective Control Problems | 57 |

111 MultiObjective Control and Cooperative Games on Dynamic Networks | 58 |

1112 A Pareto Solution for the Problem with NonStationary Strategies on Networks | 59 |

112 Determining Pareto Solutions for MultiObjective Control Problems on Networks | 60 |

1122 Pareto Solution for the NonStationary Case of the Problem | 65 |

113 Determining Pareto Optima for MultiObjective Control Problems | 66 |

114 Determining a Stackelberg Solution for Hierarchical Control Problems | 67 |

1141 A Stackelberg Solution for Static Games | 68 |

1142 Hierarchical Control on Networks and Determining Stackelberg Stationary Strategies | 69 |

1143 An Algorithm for Determining Stackelberg Stationary Strategies on Acyclic Networks | 73 |

1144 An Algorithm for Solving Hierarchical Control Problems | 78 |

MaxMin Control Problems and Solving ZeroSum Games on Networks | 80 |

22 MaxMin Control Problem with Inﬁnite Time Horizon | 82 |

23 ZeroSum Games on Networks and a Polynomial Time Algorithm for MaxMin Paths Problems | 83 |

231 Problem Formulation | 84 |

232 An Algorithm for Solving the Problem on Acyclic Networks | 86 |

233 Main Results for the Problem on an Arbitrary Network | 88 |

234 A Polynomial Time Algorithm for Determining Optimal Strategies of the Players in a Dynamic cGame | 90 |

235 A PseudoPolynomial Time Algorithm for Solving a Dynamic cGame | 95 |

24 A Polynomial Time Algorithm for Solving Acyclic lGames on Networks | 101 |

242 Main Properties of Optimal Strategies in Acyclic lGames | 102 |

243 A Polynomial Time Algorithm for Finding the Value and the Optimal Strategies in an Acyclic lGame | 103 |

Algorithms for Finding the Value and the Optimal Strategies of the Players | 105 |

251 Problem Formulation and Main Properties | 106 |

252 Determining the Best Response of the First Player for a Fixed Strategy of the Second Player | 107 |

253 Some Preliminary Results | 110 |

254 The Reduction of Cyclic Games to Ergodic Games | 111 |

256 A Polynomial Time Algorithm for Solving Cyclic Games Based on the Reduction to Acyclic lGames | 113 |

257 An Approach for Solving Cyclic Games Based on a Dichotomy Method and Solving Dynamic cGames | 116 |

26 Cyclic Games with Random States Transitions of the Dynamical System | 117 |

27 A Nash Equilibria Condition for Cyclic Games with p Players | 118 |

28 Determining Pareto Optima for Cyclic Games with p Players | 122 |

Extension and Generalization of Discrete Control Problems and Algorithmic Approaches for its Solving | 125 |

311 The SingleObjective Control Problem with Varying Time of States Transitions of the Dynamical System | 126 |

312 An Algorithm for Solving a SingleObjective Control Problem with Varying Time of States Transitions of the Dynamical System | 127 |

313 The Discrete Control Problem with Cost Functions of Systems Passages that Depend on the TransitionTime of States Transitions | 132 |

352 An Algorithm for Determining an Optimal Solution of the Problem with Fixed Starting and Final States | 156 |

353 The Discrete Optimal Control Problem on a Network | 159 |

354 The GameTheoretic Control Model with p Players | 160 |

355 The GameTheoretic Control Problem on Networks and an Algorithm for its Solving | 161 |

Pareto Optima | 169 |

36 ParetoNash Equilibria for MultiObjective Games | 171 |

361 Problem Formulation | 172 |

362 Main Results | 173 |

363 Discrete and Matrix MultiObjective Games | 177 |

364 Some Comments on and Interpretations of MultiObjective | 179 |

Discrete Control and Optimal Dynamic Flow Problems on Networks | 180 |

411 The Minimum Cost Dynamic Flow Problem | 182 |

412 The Main Results | 183 |

413 The Dynamic Model with Flow Storage at Nodes | 186 |

414 The Dynamic Model with Flow Storage at Nodes and Integral Constant DemandSupply Functions | 188 |

415 The Algorithm | 189 |

416 Constructing the TimeExpanded Network and its Size | 190 |

417 Approaches for Solving the Minimum Cost Flow Problem with Diﬀerent Types of Cost Functions on the Edges | 200 |

418 Determining the Minimum Cost Flows in Dynamic Networks with Transition Time Functions that Depend on Flow and Time | 208 |

419 An Algorithm for Solving the Maximum Dynamic Flow Problem | 212 |

42 MultiCommodity Dynamic Flow Problems and Algorithms for their Solving | 214 |

422 The Main Results | 216 |

423 The Algorithm | 220 |

425 The Dynamic MultiCommodity Minimum Cost Flow Problem with Transition Time Functions that Depend on Flows and on Time | 224 |

426 Generalizations | 229 |

43 The GameTheoretic Approach for Dynamic Flow Problems on Networks | 231 |

Applications and Related Topics | 233 |

511 Motivation | 234 |

513 Control Theoretic Part | 237 |

514 Problem of Fixed Point Controllability and NullControllability | 238 |

515 Optimal Investment Parameter | 240 |

516 A GameTheoretic Extension Relation to Multilayered Decision Problems | 244 |

The Kyoto Game | 250 |

522 A Second Cooperative Treatment of the TEM Model | 259 |

523 Comments | 268 |

53 An Emission Reduction Process The MILAN Model | 269 |

532 Sequencing and Dynamic Programming | 271 |

Optimal Solutions on kLayered Graphs | 274 |

Conclusion | 275 |

277 | |

283 | |

### Other editions - View all

Optimization and Multiobjective Control of Time-Discrete Systems: Dynamic ... Dmitrii Lozovanu No preview available - 2010 |

Optimization and Multiobjective Control of Time-Discrete Systems: Dynamic ... Dmitrii Lozovanu No preview available - 2009 |

### Common terms and phrases

acyclic directed graph acyclic network Algorithm for Solving arbitrary control parameters corresponding cost ﬂow problem cost functions cyclic game deﬁned diﬀerent directed cycle directed graph directed path dynamic c-game dynamic network dynamic programming dynamical system easy to observe ergodic ﬁnd ﬁnite ﬁrst ﬁxed G there exists game on network game-theoretic grand coalition graph G integral-time cost Lemma max-min paths minimum cost ﬂow multi-commodity ﬂow multi-commodity flow problem multi-objective control problem multi-objective game Nash equilibria network G non-cooperative game obtain optimal control optimal paths optimal solution optimal stationary strategies optimal strategies Pareto solution position x0 problem on network reduced network saddle point satisﬁed set of edges sink vertex xf Stackelberg solution starting position system’s passage T(xf Theorem time-expanded network NT time-moment transition time functions tree v(xq vectors of control x(tj x0 to xf zero-sum