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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

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1962. 8] Athans, M. , QP-timal Control,McGraw-Hill, N. , 1966. [3. timal Control, McGraw-Hill, N. , 1966. , A Sufficiency Theorem for Optimal Control J. Optim. , Vol. 8, No. 3, 1971. , Sufficient Conditions for the Optimal Control of Nonlinear Systems, SIAMJ. Control, Vol. 4, No. 1, 1966. , Some Sufficiency Conditions for Pareto-optimal Control, J. Dynamical Systems, Measurement and Control, Vol. 95, No. 3, 1963. , Sufficiency for Optimal Control, J. Optim. , Vol. 2, No. 5, 1968. , Sufficient Conditions for Optimal Control with State and Control Constraints, J.

5. 1. 23) 7T. (Ci) C A/"E. 24) 1T. 14 ). 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.

17) , * = * , V. )) =V. 16), • • 1 1 i 1 V. (x 0 ) -V. (x) ~V. 19) V. ,'(C)~. (C). 1 1 Since yi' is any point of 'll"i (ci), that concludes the proof of proposition (i). The second proposition follows at once from the definitions of 11': (c*) and l;i (C). )) o yi*(o) el;. (C) with trajectory • (C*). 11'. 1 1 : [ O,T 1 ] x· (O) = and From the definition of i* o • * x (r) + V. (x (r)) 1 • 11'. 21) But y i* (O) e ~i (C) implies ( 4. 22) we have at once that C* = C. 12), we have x~= x~(r) . This concludes the proof.

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