# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics and Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187
##### References:
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##### 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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