Existence Theorems for Lagrange and Pontryagin Problems of The Calculus of Variations and Optimal Control. More Dimensional Extensions in Sobolev Spaces

Let A be a closed subset of the tx-space E1 × En, t∈E1, x = (x1,…xn)∈En and for each (t, x)∈A, let U(t,x) be a closed subset of the u-space Em, u = (u1,…,um). 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) = (f0, f) = (f0, f1,…, fn) be a continuous vector function from M into En+1. Let Bbe a closed subset of points (t1x1,t2,x2)of E2n+2, x1 = (x1 1,…x1 n, x2 = (x2 1,…x2 n.We shall consider the class of all pairs x(t), u(t), t1 ≤t≤t2, of vector functions x(t), u(t) satisfying the following conditions : (a) x(t) is absolutely continuous (AC) in [t1, t2]; (b) u(t) is measurable in [t1, t2]; (c) (t,x(t))∈A for every t∈[t1, t2]; (d) u(t)∈U(t, x(t)) almost everywhere (a.e.) in [t1, t2]; (e) f0 (t,x(t), u(t)) is L-integrable in[t1, t2]; (f) dx/dt / f(t, x(t), u(t)) a.ein [t1, t2]; (g) (t 1,x(t 1), t 2, x(t 2))∈B.