Discontinuous differential equations. I.

*(English)*Zbl 0365.34017(Joint review with Zbl 0681.34009). The classical notions of solution, to an ordinary differential equation, are sometimes insufficient. Three generalisations have been proposed (Filippov, Krasovskij, and, implicitly, Hermes). The first part of the paper studies these concepts; it is shown that, in very general circumstances, the last two are equivalent (and sometimes one of the alternative descriptions has a decided advantage over the other). The concepts and results of the first part of this paper are applied in the second part [ibid. 32, 171–185 (1979; Zbl 0681.34009)] to linear differential games with hyperplane targets and feedback-type strategies; and to an example of Nash equilibrium in a non-linear game (the Leitman-Lin model of collective bargaining). Some uniqueness results are obtained, particularly in the context of Hermes’s notion of stability with respect to measurement; and applied to time-optimal feedback in linear control. The principal result (Theorem 9.4) is that, for controllable linear single-input systens, the corresponding feedback equation is stable with respect to measurement (while the assertion fails for multi-input systems).

Reviewer: Otomar Hájek

##### MSC:

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems, general |

91A23 | Differential games (aspects of game theory) |

91B26 | Auctions, bargaining, bidding and selling, and other market models |

##### Keywords:

control theory; time-optimal feedback; differential games; discontinuous differential equations
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##### References:

[1] | Aumann, R.J, Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301 |

[2] | Blaquière, A; Gérard, F; Leitmann, G, Quantitative and qualitative games, (1969), Academic Press New York · Zbl 0228.90061 |

[3] | Carathéodory, C, Vorlesungen über reele funktionen, (1918), Teubner Leipzig · JFM 46.0376.12 |

[4] | Filippov, A.F, Differential equations with discontinuous right-hand side (in Russian), Mat. sb., 51, (1960) · Zbl 0138.32204 |

[5] | Flügge-Lotz, I, Discontinuous automatic control, (1953), Princeton Univ. Press Princeton, N. J · Zbl 0051.15602 |

[6] | Friedman, A, Differential games, (1971), Wiley-Interscience New York · Zbl 0229.90060 |

[7] | Gamkrelidze, R.V, Theory of time-optimal processes in linear systems (in Russian), Izv. akad. nauk SSSR, 22, 449-477, (1958) · Zbl 0081.34703 |

[8] | Hájek, O, Dynamical systems in the plane, (1968), Academic Press London · Zbl 0169.54401 |

[9] | Hájek, O, Terminal manifolds and switching locus, Math. systems theory, 6, 289-301, (1973) · Zbl 0247.49002 |

[10] | \scO. Hájek, L1-Optimization in Linear systems with bounded controls, J. Optimization Theory Appl., in press. |

[11] | Hájek, O, Inverted paradoxes in optimal control theory, () |

[12] | Hájek, O, Pursuit games, (1975), Academic Press New York · Zbl 0324.90104 |

[13] | Halmos, P.R, Measure theory, (1950), Van Nostrand Reinhold New York · Zbl 0117.10502 |

[14] | Hermes, H, Discontinuous vector fields and feedback control, (), 155-165 · Zbl 0183.15905 |

[15] | Hermes, H, Calculus of set valued functions and control, J. math. mech., 8, 47-60, (1968) · Zbl 0175.05101 |

[16] | Hermes, H; LaSalle, J.P, Functional analysis and time optimal control, (1969), Academic Press New York · Zbl 0203.47504 |

[17] | Krasovskij, N.N, Game-theoretic problems of capture, (1970), Nauka Moscow, (in Russian) · Zbl 0232.90075 |

[18] | Krakovskij, N.N; Subbotin, A.I, Positional differential games, (1974), Nauka Moscow, (in Russian) · Zbl 0298.90067 |

[19] | Lee, E.B; Markus, L, Foundations of optimal control theory, (1967), Wiley New York · Zbl 0159.13201 |

[20] | Leitmann, G, Cooperative and non-cooperative many-player differential games, (1974), Springer-Verlag Vienna |

[21] | Wagner, D.H, Survey of measurable selection theorems, SIAM J. control optimization, 15, 859-903, (1977) · Zbl 0407.28006 |

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