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Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line. 相似文献
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S?awomir Plaskacz Magdalena Wi?niewska 《Central European Journal of Mathematics》2012,10(6):1940-1943
Filippov??s theorem implies that, given an absolutely continuous function y: [t 0; T] ?? ? d and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x??(t) ?? F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ where the function ??(·) is the estimation of dist(y??(t), F(t, y(t))) ?? ??(t). Setting P(t) = {x ?? ? n : |x ?y(t)| ?? r(t)}, we may formulate the conclusion in Filippov??s theorem as x(t) ?? P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ?? DP(t, x)(1) ?? ?. It allows to obtain Filippov??s theorem from a viability result for tubes. 相似文献
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The existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non-convex set of state constraints is established by the use a Lefschetz type fixed-point theorem for set-valued maps. 相似文献
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