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Continuous interpolation of solutions of Lipschitz inclusions. (English) Zbl 0984.34009

Let \(F\) be a set-valued map defined on \([0,T]\times \mathbb{R}^{n}\) and taking values on closed and nonempty subsets of \(\mathbb{R}^{n}\). The authors show that if \(F\) is measurable in \(t\) and Lipschitzian continuous in \(x\), then for a given finite set of trajectories to the problem \[ \dot x(t)\in F(t,x), \quad x(0)=\xi, \tag{1} \] starting from distinct initial points, there exists a continuous selection from the set of solutions to problem (1) that interpolates these trajectories. An argument from the paper of A. Cellina and A. Ornelas [Rocky Mt. J. Math. 22, 117-124 (1992; Zbl 0752.34012)] is used by the authors. A second result presented in this paper concerns the existence of Lipschitzian selections when \(F\) has compact and convex values.

MSC:

34A60 Ordinary differential inclusions

Citations:

Zbl 0752.34012
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References:

[1] Aubin, J.; Cellina, A., Differential Inclusions: Set-Valued Maps and Viability Theory (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0538.34007
[2] Bressan, A., Selections of Lipschitz multifunctions generating a continuous flow, Differential Integral Equations, 4, 483-490 (1991) · Zbl 0722.34009
[3] Broucke, M., Regularity of solutions and homotopic equivalence for hybrid systems, Proc. 37th IEEE CDC (1988), p. 4283-4288
[4] Cellina, A., On the set of solutions to lipschitzian differential inclusions, Differential Integral Equations, 1, 495-500 (1988) · Zbl 0723.34009
[5] Deimling, K., Multivalued Differential Equations (1992), de Gruyter: de Gruyter Berlin · Zbl 0760.34002
[6] Filippov, A. F., Differential Equations with Discontinuous Righthand Sides (1988), Kluwer Academic: Kluwer Academic Boston · Zbl 0664.34001
[7] Cellina, A.; Ornelas, A., Representation of the attainable set for lipschitzian differential inclusions, Rocky Mountain J. Math., 22, 117-124 (1992) · Zbl 0752.34012
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