Multicriteria Decision Making by G. Leitmann, A. Marzollo

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By G. Leitmann, A. Marzollo

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Q. I (3) satisfying the terminal condition K. I The optimal control is -1 T R. B I K. )x , I X I 0 (6) and the minimal value of the performance criterion is J(u 0 , i ,x )- 21 XT K. (t )x 0 0 I 0 0 (7) 60 J. 2. Vector-valued I inear regulator problem Consider the vector valued optimization problem (1), (2). i ,x )is associated with player i and the controls of 0 alI players have been aggregated into the control u since the players have agreed to play cooperatively. The Pareto optimal strategy (see notes by Leitmann) is the usnal solution concept in vector valued optimization problems and in cooperative control problems.

0. , Balakrishnan, A. V. and Neustadt, L. , 1967. Ho, Y. c. , 1969. Starr, A. W. and Ho, Y. , Nonzero-sum differential games, J. , 3, 184, 1969. Starr, A. , Further properties of nonzero-sum differential games, J. , 3, 207, 1969. , Y. , Differential games, dynamic optimization and general- ized control theory, J. of. , 6, 179, 1970. Ciletti, M. , New results in the theory of differential games with information time lag, J. , 8, 287, 1971. , Leitmann, G. and Wan, H. Y. , A differential game model of duopoly, Econometrica, 39, 911, 1971.

X oa' x) a. From ca. ) , Let us define the Peaahable set E (yi) from yi a a. a { y and by : and r Cy,yi ,s)} Cl. Cl. (x-J oa. ,x-J) E . * r (/,yl,s ), be a nice point OfL (C). a a a a If yj + t E T(E (i),yj), then there exists an infinite sequence a a a a a ln Z , namely ct v = 1 ,2, . . and II t v - t ct Cl. II -+ 0 as v -+ oo } and an infinite sequence of strictly positive numbers { £v v = 1,2, ... l + £vtv E E ( i ) ct for v sufficiently large ct 47 Vector-Valued Optimization Hence 'l'hen, from Lemmas 2 and 3 we have Theorem 3.

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