On Certain Questions in the Theory of Optimal Control
On Certain Questions in the Theory of Optimal Control
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DOI：
10.1137/0301006
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发表时间：
1962
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影响因子：
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通讯作者：
A. F. Filippov
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文献类型：
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作者：
A. F. Filippov
X（0）-ix（0）12+ 1 A，其中X（t）lx（t）+ 1.因此，X（t）<-_ Ae for>= 0。因此，对于任何容许的u（t），具有初始条件x（0）x（t）的（1）的解在区间0<=<-_ t*（如果我们考虑C>= 0）上满足不等式[lx（t）]<= Atect*。我们将证明Q（t，x）对0<=<= t* 和lx <= A1/2 eC * 是一致有界的，即存在N使得对于任意u，Q（t，x），对于x和的值，ul< _ N.实际上，否则将存在序列tn-t，xn-x和un Q（tn，Xn），使得但是，通过我们所做的假设，对于足够大的n，Q（t，，xn）包含在有界集合Q（t，x）的e-邻域中。这种矛盾表明，Q（t，x）对于和x的指定值是一致有界的.因为f是连续的，如果（t，x，u）--< M如果0--< _-< t*，lx <= Aec*，u Q（t，x）
X (O)-ix () 12+ 1 A for almost all t, where X (t) Ix (t)+ 1. Consequently, X (t)<-_ Ae for> _-0. Hence, for every admissible u (t), the solution of (1) with initial condition x (0) x () satisfies the inequality Ix (t)]<= Atect* on the interval 0<=<-_ t*(if we consider that C>= 0). We shall show that the Q (t, x) are uniformly bounded for 0<=<= t* and for Ix<= A1/2eC*; ie, thatthere existsan N such that ul< __ N for every u Q (t, x) for the indicated values of and x. Indeed, otherwise there would be sequences tn--t, xn-x, and un Q (tn, Xn) such that But, by the assumptions we have made, Q (t,,, xn) is contained in an e-neighborhood of the bounded set Q (t, x), for sufficiently large n. This con-tradiction shows that the Q (t, x) are uniformly bounded for the indicated values of and x. Sincefis continuous, If (t, x, u)--< M if 0 _-< _-< t*, Ix<= Aec*, u Q (t, x)