Anzeige
Erschienen in:
2011 | OriginalPaper | Buchkapitel
Existence Theorems for Lagrange and Pontryagin Problems of The Calculus of Variations and Optimal Control. More Dimensional Extensions in Sobolev Spaces
verfasst von : Lamberto Cesari
Erschienen in: Calculus of Variations, Classical and Modern
Verlag: Springer Berlin Heidelberg
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Let A be a closed subset of the tx-space E
1
× E
n
, t∈E
1
, x = (x
1
,…x
n
)∈E
n
and for each (t, x)∈A, let U(t,x) be a closed subset of the u-space E
m
, u = (u
1
,…,u
m
). We do not exclude that A coincides with the whole tx-space and that U coincides with the whole u-space. Let M denote the set of all (t, x, u) with (t, x)∈A. u∈U(t, x). Let f(t, x, u) = (f
0
, f) = (f
0
, f
1
,…, f
n
) be a continuous vector function from M into E
n+1
. Let Bbe a closed subset of points (t
1
x
1
,t
2
,x
2
)of E
2n+2
, x
1
= (x
1
1
,…x
1
n
, x
2
= (x
2
1
,…x
2
n
.We shall consider the class of all pairs x(t), u(t), t
1
≤t≤t
2
, of vector functions x(t), u(t) satisfying the following conditions :
(a)
x(t) is absolutely continuous (AC) in [t
1
, t
2
];
(b)
u(t) is measurable in [t
1
, t
2
];
(c)
(t,x(t))∈A for every t∈[t
1
, t
2
];
(d)
u(t)∈U(t, x(t)) almost everywhere (a.e.) in [t
1
, t
2
];
(e)
f
0
(t,x(t), u(t)) is L-integrable in[t
1
, t
2
];
(f)
dx/dt / f(t, x(t), u(t)) a.ein [t
1
, t
2
];
(g)
(
t
1
,x(
t
1
),
t
2
, x(
t
2
))∈B.