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Conservative and dissipative polymatrix replicators
On the hierarchical optimal control of a chain of distributed systems
1. | Department of Mechanical and Aerospace Engineering, University of Florida - REEF, 1350 N. Poquito Rd, Shalimar, FL 32579, United States |
2. | Munitions Directorate, Air Force Research Laboratory, 101 West Eglin Blvd, Eglin AFB, FL 32542, United States |
References:
[1] |
F. D. Araruna, E. Fernández-Cara and M. C. Santos, Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 21 (2015), 835-856.
doi: 10.1051/cocv/2014052. |
[2] |
E. Barucci, S. Polidoro and V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci., 11 (2001), 475-497.
doi: 10.1142/S0218202501000945. |
[3] |
G. K. Befekadu and P. J. Antsaklis, On the asymptotic estimates for exit probabilities and minimum exit rates of diffusion processes pertaining to a chain of distributed control systems, SIAM J. Contr. Optim., 53 (2015), 2297-2318.
doi: 10.1137/140990322. |
[4] |
T. Bodineau and L. Lefevere, Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats, J. Stat. Phys., 133 (2008), 1-27.
doi: 10.1007/s10955-008-9601-4. |
[5] |
F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577-1630.
doi: 10.1016/j.jfa.2010.05.002. |
[6] |
I. Ekeland and R. Témam, Convex Analysis and Variational Problems, North Holland, 1976. |
[7] |
D. L. Elliott, Diffusions on manifolds arising from controllable systems, in Geometric Methods in System Theory, (eds. D. Q. Mayne and R.W. Brockett), Reidel Publ. Co., Dordrecht, Holland, 3 (1973), 285-294.
doi: 10.1007/978-94-010-2675-8_19. |
[8] |
F. Guillén-González, F. Marques-Lopes and M. Rojas-Medar, On the approximate controllability of Stackelberg-Nash strategies for Stokes equations, Proc. Amer. Math. Soc., 141 (2013), 1759-1773.
doi: 10.1090/S0002-9939-2012-11459-5. |
[9] |
L. Hörmander, Hypoelliptic second order differential operators, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[10] |
K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheor. Verw. Geb., 30 (1974), 253-254.
doi: 10.1007/BF00533476. |
[11] |
G. Leitmann, On generalized Stackelberg strategies, J. Optim. Theor. Appl., 26 (1978), 637-643.
doi: 10.1007/BF00933155. |
[12] |
J. L. Lions, Some remarks on Stackelberg's optimization, Math. Models Methods Appl. Sci., 4 (1994), 477-487.
doi: 10.1142/S0218202594000273. |
[13] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[14] |
R. T. Rockafellar, Duality and stability in extremum problems involving convex functions, Pacific J. Math., 21 (1967), 167-187.
doi: 10.2140/pjm.1967.21.167. |
[15] |
J. C. Saut and B. Scheurer, Unique continuation for evolution equations, J. Diff. Equ., 66 (1987), 118-137.
doi: 10.1016/0022-0396(87)90043-X. |
[16] |
C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, Ser. Adv. Math. Appl. Sci., vol. 17, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
doi: 10.1142/9789814354110. |
[17] |
H. Von Stackelberg, Marktform und Gleichgewicht, Springer, Berlin, Germany, 1934. |
[18] |
D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 25 (1972), 651-713.
doi: 10.1002/cpa.3160250603. |
[19] |
D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-28999-2. |
[20] |
H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Diff. Equ., 12 (1972), 95-116.
doi: 10.1016/0022-0396(72)90007-1. |
show all references
References:
[1] |
F. D. Araruna, E. Fernández-Cara and M. C. Santos, Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 21 (2015), 835-856.
doi: 10.1051/cocv/2014052. |
[2] |
E. Barucci, S. Polidoro and V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci., 11 (2001), 475-497.
doi: 10.1142/S0218202501000945. |
[3] |
G. K. Befekadu and P. J. Antsaklis, On the asymptotic estimates for exit probabilities and minimum exit rates of diffusion processes pertaining to a chain of distributed control systems, SIAM J. Contr. Optim., 53 (2015), 2297-2318.
doi: 10.1137/140990322. |
[4] |
T. Bodineau and L. Lefevere, Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats, J. Stat. Phys., 133 (2008), 1-27.
doi: 10.1007/s10955-008-9601-4. |
[5] |
F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, J. Funct. Anal., 259 (2010), 1577-1630.
doi: 10.1016/j.jfa.2010.05.002. |
[6] |
I. Ekeland and R. Témam, Convex Analysis and Variational Problems, North Holland, 1976. |
[7] |
D. L. Elliott, Diffusions on manifolds arising from controllable systems, in Geometric Methods in System Theory, (eds. D. Q. Mayne and R.W. Brockett), Reidel Publ. Co., Dordrecht, Holland, 3 (1973), 285-294.
doi: 10.1007/978-94-010-2675-8_19. |
[8] |
F. Guillén-González, F. Marques-Lopes and M. Rojas-Medar, On the approximate controllability of Stackelberg-Nash strategies for Stokes equations, Proc. Amer. Math. Soc., 141 (2013), 1759-1773.
doi: 10.1090/S0002-9939-2012-11459-5. |
[9] |
L. Hörmander, Hypoelliptic second order differential operators, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[10] |
K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheor. Verw. Geb., 30 (1974), 253-254.
doi: 10.1007/BF00533476. |
[11] |
G. Leitmann, On generalized Stackelberg strategies, J. Optim. Theor. Appl., 26 (1978), 637-643.
doi: 10.1007/BF00933155. |
[12] |
J. L. Lions, Some remarks on Stackelberg's optimization, Math. Models Methods Appl. Sci., 4 (1994), 477-487.
doi: 10.1142/S0218202594000273. |
[13] |
J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[14] |
R. T. Rockafellar, Duality and stability in extremum problems involving convex functions, Pacific J. Math., 21 (1967), 167-187.
doi: 10.2140/pjm.1967.21.167. |
[15] |
J. C. Saut and B. Scheurer, Unique continuation for evolution equations, J. Diff. Equ., 66 (1987), 118-137.
doi: 10.1016/0022-0396(87)90043-X. |
[16] |
C. Soize, The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, Ser. Adv. Math. Appl. Sci., vol. 17, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
doi: 10.1142/9789814354110. |
[17] |
H. Von Stackelberg, Marktform und Gleichgewicht, Springer, Berlin, Germany, 1934. |
[18] |
D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 25 (1972), 651-713.
doi: 10.1002/cpa.3160250603. |
[19] |
D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-28999-2. |
[20] |
H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Diff. Equ., 12 (1972), 95-116.
doi: 10.1016/0022-0396(72)90007-1. |
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