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By George Leitmann

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Extra resources for Cooperative and Non-Cooperative Many Players Differential Games: Course Held at the Department of Automation and Information July 1973

Example text

5, function 4>i(. )is of class C2 on R 1 x X*. 24) result in 4. 25) grad ~ (yi )gi (yi, ip(x))~ O 1 i i i • grad ~ (y )g (y ,p (x)) =O 1 1 Ifui (x) =constant 'V x EX*, this condition is satisfied. 'Jiui ed (x) Regular Optimal Trajectories: Necessary Conditions for an Equilibrium for ali yi € 11': (C) , yi (/;. ®, . grad <1>. 3 x1 1 -- ' avi* (x) ... ' ax n ) . 5, point y i = y i* (T) , T < T~, is an interior point of R 1 X x*' and hence possesses a neighborhood all of whose points belong to optimal trajectories.

54 4. J J where Ai =Ai (T), = O,l, ... j .. 4. Piecewise Regular Optimal Trajectories: Necessary Conditions for an Equilibrium. 5. 6. { X 1 , X 2 , XL} is a finite decomposition of X if and only if (i) X i> i = 1, 2, ... , L, is a domain of Rn , (ii)Xin>s = cJ> for i :#j, ( iii) X. 6, L x = i~l xi De finition 4. 7. l:f(C), ... (C) 1 = i "" i {y: ~ (y) 1 = C, X € Now we introduce the following X. } J for j E {1 , 2 , ... , L} • Piecewise Regular Optimal Trajectories 55 Assumptions. (i) (*)Set X is a domain of R n with a ftnite decomposition { X 1 , X2, ...

Grad <1>. 3 x1 1 -- ' avi* (x) ... ' ax n ) . 5, point y i = y i* (T) , T < T~, is an interior point of R 1 X x*' and hence possesses a neighborhood all of whose points belong to optimal trajectories. 25) is valid at every point of such a neighborhood. so that it is an identity in yi; thus, it can be differentiated with respect to yi. 5. Omitting the arguments for the sake of brevity, let o d2 <1>. *1 axn axl o ~ ayi = o ai ~ i}f n dx;" 50 4. Non-Cooperative Differential Games o * ~ dyi 3p~ ~ X1 dx n 3 * PK 3p~ "' o dx7 dx n allevaluatedatyi =yi*(r),ui = pi* (x*(r)).

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