Proofs of the Necessary Condition for Control Problems and Related Topics
Let A denote the constraint set, a closed subset of the tx-space, with t in R, and the space variable x = (x1,…, x n ) in R n . Let U(t),the control set, be a subset of the u-space R m , u= (u1, …u m )the control variable. Let M = [(t,x,u)|(t,x)∈A, u ∈ U(t)] be a closed subset of R1+n+m, and let f = (f1,…,fn) be a continuous vector function from M into Rn. Let the boundary set B be a closed set of points (t1,x1,t2,x2) in R2n+2, x1 = (x 1 1 ,…x 1 n ), x2 = (x 2 1 ,…x 2 n ). Let g be a continuous function from B into R.
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