American Institute of Mathematical Sciences

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April  2015, 2(2): 187-199. doi: 10.3934/jdg.2015.2.187

On the hierarchical optimal control of a chain of distributed systems

Getachew K. Befekadu 1, and  Eduardo L. Pasiliao 2,

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

Received  September 2015 Revised  November 2015 Published  December 2015

We consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak Hörmander type condition, with a control distributed over an open subdomain. In particular, we consider two objectives that we would like to accomplish. The first one being of a controllability type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition; while the second one is keeping the state trajectory of the overall system close to a given reference trajectory over a finite time interval. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains that are compatible with the strategy subspaces of the leader and that of the follower, respectively. Then, using the notion of Stackelberg's optimization (which is a hierarchical optimization framework), we provide a new result on the existence of optimal control strategies for such an optimization problem, where the follower (which corresponds to the second criterion) is required to respond optimally, in the sense of best-response correspondence to the strategy of the leader (which is associated to the controllability-type problem) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the target set on the strategy of the follower with respect to the direction of leader-follower (and vice-versa) information flow.
Keywords: distributed systems, Stackelberg's optimization., hierarchical systems, Degenerate parabolic equations.
Mathematics Subject Classification: Primary: 35K10, 35K65, 93A13, 93E20, 91A3.
Citation: Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, North Holland, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. Hörmander, Hypoelliptic second order differential operators, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.  Google Scholar

[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.  Google Scholar

[11]

G. Leitmann, On generalized Stackelberg strategies, J. Optim. Theor. Appl., 26 (1978), 637-643. doi: 10.1007/BF00933155.  Google Scholar

[12]

J. L. Lions, Some remarks on Stackelberg's optimization， Math. Models Methods Appl. Sci., 4 (1994), 477-487. doi: 10.1142/S0218202594000273.  Google Scholar

[13]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Von Stackelberg, Marktform und Gleichgewicht, Springer, Berlin, Germany, 1934. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, North Holland, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. Hörmander, Hypoelliptic second order differential operators, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.  Google Scholar

[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.  Google Scholar

[11]

G. Leitmann, On generalized Stackelberg strategies, J. Optim. Theor. Appl., 26 (1978), 637-643. doi: 10.1007/BF00933155.  Google Scholar

[12]

J. L. Lions, Some remarks on Stackelberg's optimization， Math. Models Methods Appl. Sci., 4 (1994), 477-487. doi: 10.1142/S0218202594000273.  Google Scholar

[13]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

H. Von Stackelberg, Marktform und Gleichgewicht, Springer, Berlin, Germany, 1934. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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