Extremal selections of multifunctions generating a continuous flow

• Autorzy: Alberto Bressan , Graziano Crasta
• Języki publikacji: EN

Abstrakty

EN

Let $F:[0,T] × ℝ^n → 2^{ℝ^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if F satisfies the following Lipschitz Selection Property:
(LSP) For every t,x, every y ∈ c̅o̅F(t,x) and ε > 0, there exists a Lipschitz selection ϕ of c̅o̅F, defined on a neighborhood of (t,x), with |ϕ(t,x)-y| < ε,
then there exists a measurable selection f of ext F such that, for every x₀, the Cauchy problem
ẋ(t) = f(t,x(t)), x(0) = x₀,
has a unique Carathéodory solution, depending continuously on x₀.
We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class for which (LSP) holds consists of those continuous multifunctions F whose values are compact and have convex closure with nonempty interior.

Słowa kluczowe

EN

differential inclusion; extremal selection

Kategorie tematyczne

• 34A60: Differential inclusions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1994-1995
• Tom: 60
• Numer: 2
• Strony: 101-117

Daty

wydano

1994

otrzymano

1992-10-28

poprawiono

1993-11-10

Twórcy

autor

Alberto Bressan

• S.I.S.S.A., via Beirut 4, Trieste 34014, Italy

autor

Graziano Crasta

• S.I.S.S.A., via Beirut 4, Trieste 34014, Italy

Bibliografia

• [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.
• [2] A. Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31 (1988), 459-470.
• [3] A. Bressan, On the qualitative theory of lower semicontinuous differential inclusions, J. Differential Equations 77 (1989), 379-391.
• [4] A. Bressan, The most likely path of a differential inclusion, ibid. 88 (1990), 155-174.
• [5] A. Bressan, Selections of Lipschitz multifunctions generating a continuous flow, Differential Integral Equations 4 (1991), 483-490.
• [6] A. Bressan and G. Colombo, Boundary value problems for lower semicontinuous differential inclusions, Funkcial. Ekvac. 36 (1993), 359-373.
• [7] A. Cellina, On the set of solutions to Lipschitzean differential inclusions, Differential Integral Equations 1 (1988), 495-500.
• [8] F. S. De Blasi and G. Pianigiani, On the solution set of nonconvex differential inclusions, J. Differential Equations, to appear.
• [9] F. S. De Blasi and G. Pianigiani, Topological properties of nonconvex differential inclusions, Nonlinear Anal. 20 (1993), 871-894.
• [10] A. LeDonne and M. V. Marchi, Representation of Lipschitz compact convex valued mappings, Atti Accad. Naz. Lincei Rend. 68 (1980), 278-280.
• [11] A. F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control Optim. 1 (1962), 76-84.
• [12] A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova 83 (1990), 33-44.
• [13] A. A. Tolstonogov, Extreme continuous selectors of multivalued maps and their applications, preprint, 1992.