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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
91A23 Differential games (aspects of game theory)
91B26 Auctions, bargaining, bidding and selling, and other market models
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