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Staticization and iterated staticization. (English) Zbl 1496.93062

Summary: DE075822538Conservative dynamical systems propagate as stationary points of the action functional. Using this representation, it has previously been demonstrated that one may obtain fundamental solutions for two-point boundary value problems for some classes of conservative systems via a solution of an associated dynamic program. It is also known that the gravitational and Coulomb potentials may be represented as stationary points of cubicly parameterized quadratic functionals. Hence, stationary points of the action functional may be represented via iterated “staticization” of polynomial functionals, where the staticization operator (introduced and discussed in [W. M. McEneaney and P. M. Dower, J. Differ. Equations 264, No. 2, 525–549 (2018; Zbl 1381.49036)] and [W. M. McEneaney and P. M. Dower, Automatica 81, 56–67 (2017; Zbl 1373.90169)] maps a function to the function value(s) at its stationary (i.e., critical) points. This leads to representations through operations on sets of solutions of differential Riccati equations. A key step in this process is the reordering of staticization operations. Conditions under which this reordering is allowed are obtained, and it is shown that the conditions are satisfied for an astrodynamics problem.

MSC:

93C20 Control/observation systems governed by partial differential equations
93C10 Nonlinear systems in control theory
35G20 Nonlinear higher-order PDEs
35D40 Viscosity solutions to PDEs
90C39 Dynamic programming
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[1] L. W. Baggett, Functional Analysis, Marcel Dekker, New York, 1992. · Zbl 0751.46001
[2] F. J. Beutler, The operator theory of the pseudo-inverse I. Bounded operators, J. Math. Anal. Appl., 10 (1965), pp. 451-470. · Zbl 0151.19205
[3] P. M. Dower, H. Kaise, W. M. McEneaney, T. Wang, and R. Zhao, Solution existence and uniqueness for degenerate SDEs with application to Schrödinger-equation representations, Commun. Inf. Syst., 21 (2021), pp. 297-315, https://doi.org/10.4310/CIS.2021.v21.n2.a6. · Zbl 1492.60162
[4] P. M. Dower and W. M. McEneaney, Representation of fundamental solution groups for wave equations via stationary action and optimal control, in Proceedings of the 2017 American Control Conference, Seattle, WA, 2017, pp. 2510-2515.
[5] P. M. Dower and W. M. McEneaney, Solving two-point boundary value problems for a wave equation via the principle of stationary action and optimal control, SIAM J. Control Optim., 55 (2017), pp. 2151-2205. · Zbl 1380.49020
[6] I. Ekeland, Legendre duality in nonconvex optimization and calculus of variations, SIAM J. Control Optim., 15 (1977), pp. 905-934. · Zbl 0377.90089
[7] C. G. Gray and E. F. Taylor, When action is not least, Amer. J. Phys., 75 (2007), pp. 434-458.
[8] W. R. Hamilton, On a general method in dynamics, Philos. Trans. Royal Soc., Part I (1835), pp. 95-144; Part II (1834), pp. 247-308.
[9] S. H. Han and W. M. McEneaney, Fundamental solutions for two-point boundary value problems in orbital mechanics, Applied Math. Optim., 77 (2018), pp. 129-172. · Zbl 1386.49054
[10] S. H. Han and W. M. McEneaney, The principle of least action and two-point boundary value problems in orbital mechanics, in Proceedings of the 2014 American Control Conference, Portland, OR, 2014, pp. 1939-1944.
[11] S. Izumino, Convergence of generalized inverse and spline projectors, J. Approx. Theory, 38 (1983), pp. 269-278. · Zbl 0519.41016
[12] S. Lang, Fundamentals of Differential Geometry, Springer, New York, 1999. · Zbl 0932.53001
[13] W. M. McEneaney, Stationarity-based representation for the Coulomb potential and a diffusion representation for solution of the Schrödinger equation, in Proceedings of the 23rd International Symposium on Mathematical Theory of Networks and Systems, Hong Kong, 2018, pp. 707-709.
[14] W. M. McEneaney and P. M. Dower, Static duality and a stationary-action application, J. Differential Equations, 264 (2018), pp. 525-549. · Zbl 1381.49036
[15] W. M. McEneaney and R. Zhao, Diffusion process representations for a scalar-field Schrödinger equation solution in rotating coordinates, in Numerical Methods for Optimal Control Problems, Springer INDAM Series 29, M. Falcone, R. Ferretti, L. Grune, and W. McEneaney, eds., Springer, New York, 2018, pp. 241-268. · Zbl 1416.49030
[16] W. M. McEneaney, A stochastic control verification theorem for the dequantized Schrödinger equation not requiring a duration restriction, Appl. Math. Optim., 79 (2019), pp. 427-452, https://doi.org/10.1007/s00245-017-9442-0. · Zbl 1420.35280
[17] W. M. McEneaney and P. M. Dower, Staticization, its dynamic program and solution propagation, Automatica J. IFAC, 81 (2017), pp. 56-67. · Zbl 1373.90169
[18] W. M. McEneaney and P. M. Dower, The principle of least action and fundamental solutions of mass-spring and \(n\)-body two-point boundary value problems, SIAM J. Control Optim., 53 (2015), pp. 2898-2933. · Zbl 1325.49055
[19] W. M. McEneaney, Max-Plus Methods for Nonlinear Control and Estimation, Birkhauser, Boston, 2006. · Zbl 1103.93005
[20] V. Rakocevic, On continuity of the Moore-Penrose and Drazin inverses, Matematicki Vesnik, 49 (1997), pp. 163-172. · Zbl 0999.47500
[21] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., John Wiley, New York, 1980. · Zbl 0501.46003
[22] R. Zhao and W. M. McEneaney, Iterated staticization and efficient solution of conservative and quantum systems, in Proceedings of the 2019 Conference on Control and its Applications, Chengdu, China, 2019, pp. 83-90.
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