Bonne position au sens de Levitin--Polyak dans le cadre de la minimisation des fonctionnelles intégrales

D. Mentagui

Abstract: We investigate the relationship between the Levitin--Polyak well-posedness of the problem of minimization of the integral functional \[ I:x\in L^1(T)\to \int_Tf(t,x(t))\,dt \] on the set $U=\left\{ x:T\subset R^k\to R^m: x\in L^1(T);\ x(t)\in K(t)\text{ for a.e. }t\in T\right\}$ of integrable selections of a multifunction $K:t\in T\to K(t)\subset R^m$ and well-posedness of the minimization problem of $f(t,.)$ on $K(t)$. We show that well-posedness of problem $\inf(I,U)$ implies that of $\inf(f(t,.),K(t))$ for a.e.\ $t\in T$. The converse holds under another assumptions.

Classification (MSC2000): 49K40; 41A50, 49J15, 90C31

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