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A minimax problem for semilinear nonlocal competitive systems. (English) Zbl 0789.49009
A two-sided game for the control of a stationary semilinear competitive system with autonomous sources is considered, where the controls are the kernels of the nonlocal interaction terms. The saddle point (the optimal solution to the game) is characterized as the unique solution of the associated optimality system, which is solved by an iterative scheme.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35K10 Second-order parabolic equations
91A23 Differential games (aspects of game theory)
93C20 Control/observation systems governed by partial differential equations
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[1] Cosner C, Lenhart S, Protopopescu V (1990) Parabolic systems with nonlinear competitive interactions. IMA J Appl Math 44:285-298 · Zbl 0716.35033 · doi:10.1093/imamat/44.3.285
[2] Ekeland I, Teman R (1976) Convex Analysis and Variational Problems. North-Holland, Amsterdam
[3] Fife PC (1984) Mathematical Aspects of Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol 28. Springer-Verlag, Berlin
[4] Gilbarg D, Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin · Zbl 0562.35001
[5] Heinkenschloss M, Kelley CT, Tran HT (1992) Fast algorithms for nonsmooth compact fixed point problems. SIAM J Numer Anal 29:1769-1792 · Zbl 0763.65040 · doi:10.1137/0729099
[6] Leung A (1989) Systems of Nonlinear Partial Differential Equations, Applications to Biology and Engineering. Kluver, Dordrecht · Zbl 0691.35002
[7] Leung A, Stojanovic S (1990) Direct methods for some distributed games. Differential Integral Equations 3:1113-1125 · Zbl 0722.90093
[8] Maz’ja V (1985) Sobolev Spaces. Springer-Verlag, Berlin · Zbl 0727.46017
[9] Pao CV (1982) On nonlinear reaction-diffusion systems. J Math Anal Appl 87:165-198 · Zbl 0488.35043 · doi:10.1016/0022-247X(82)90160-3
[10] Protopopescu V, Santoro RT, Dockery J (1989) Combat modeling with partial differential equations. European J Oper Res 38:178-183 · Zbl 0658.90056 · doi:10.1016/0377-2217(89)90102-1
[11] Smoller J (1983) Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York · Zbl 0508.35002
[12] Stojanovic S (1989) Optimal damping control and nonlinear parabolic systems. Numer Funct Anal Optim 10:573-591 · Zbl 0674.49023 · doi:10.1080/01630568908816319
[13] Stojanovic S (1991) Optimal damping control and nonlinear elliptic systems. SIAM J Control Optim 29(3):594-608 · Zbl 0742.49017 · doi:10.1137/0329033
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