Abstract
An open question contributed by Y. Orlov to a recently published volume Unsolved Problems in Mathematical Systems and Control Theory (V. D. Blondel and A. Megretski, eds.), Princeton Univ. Press (2004), concerns regularization of optimal control-affine problems. These noncoercive problems admit, in general, “cheap (generalized) controls” as minimizers; it has been questioned whether and under what conditions infima of the regularized problems converge to the infimum of the original problem. Starting from a study of this question, we show by simple functional-theoretic reasoning that it admits, in general, a positive answer. This answer does not depend on commutativity/noncommutativity of controlled vector fields. Instead, it depends on the presence or absence of a Lavrentiev gap.
We set an alternative question of measuring “singularity” of minimizing sequences for control-affine optimal control problems by socalled degree of singularity. It is shown that, in the particular case of singular linear-quadratic problems, this degree is tightly related to the “order of singularity” of the problem. We formulate a similar question for nonlinear control-affine problem and establish partial results. Some conjectures and open questions are formulated.
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References
A. A. Agrachev and A. V. Sarychev, On reduction of smooth control systems. Math. USSR Sb. 58 (1987), 15–30.
A. Bressan, On differential systems with impulsive controls. Rend. Sem. Mat. Univ. Padova 78 (1987), 227–235.
A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions. J. Optim. Theory Appl. 81 (1994), 435–457.
L. Cesari, Optimization. Theory and applications. Problems with ordinary differential equations. Springer-Verlag, New York–Heidelberg–Berlin (1983).
C.-W. Cheng and V. Mizel, On the Lavrentiev phenomenon for optimal control problems with second-order dynamics. SIAM J. Control Optim. 34 (1996), 2172–2179.
R. V. Gamkrelidze, On some extremal problems in the theory of differential equations with applications to the theory of optimal control. J. Soc. Ind. Appl. Math., Ser. A: Control 3 (1965), 106–128.
_____, Principles of optimal control theory. Plenum Press (1978).
J. P. Gauthier and V. M. Zakalyukin, On the motion planning problem. Complexity, entropy, and nonholonomic interpolation. J. Dynam. Control Systems 12 (2006), 371–404.
M. Guerra, Highly singular L-Q problems: Solutions in distribution spaces. J. Dynam. Control Systems 6 (2000), 265–309.
M. Guerra and A. Sarychev, Approximation of generalized minimizers and regularization of optimal control problems. In: Lagrangian and Hamiltonian Methods for Nonlinear Control, 2006 (F. Bullo and K. Fujimoto, eds.). Lect. Notes Control Inform. Sci. 366, Springer-Verlag (2007), pp. 269–279.
_____, Existence and Lipschitzian regularity for relaxed minimizers. In: Mathematical Control Theory and Finance (A. Sarychev et al., eds.). Springer-Verlag (2008), pp. 231–250.
F. Jean, Entropy and complexity of a path in sub-Riemannian geometry. ESAIM Control Optim. Calc. Var. 9 (2003), 485–508.
V. Jurdjevic, Geometric control theory. Cambridge University Press (1997).
V. Jurdjevic and I. A. K. Kupka, Linear systems with singular quadratic cost. Preprint (1992).
H. J. Kelley, R. E. Kopp, and H. G. Moyer, Singular extremals. In: Topics on Optimization (G. Leitmann, ed.). Academic Press, New York (1967), pp. 63–101.
A. J. Krener, The high order maximum principle and its application to singular extremals. SIAM J. Control Optim., 15 (1977), 256–293.
Y. Orlov, Vibrocorrect differential equations with measures. Mat. Zametki 38 (1985), 110–119,
_____, Theory of optimal systems with generalized controls [in Russian]. Nauka, Moscow (1988).
_____, Does cheap control solve a singular nonlinear quadratic problem? In: Unsolved Problems in Mathematical Systems and Control Theory (V. D. Blondel and A. Megretski, eds.). Princeton Univ. Press (2004), pp. 111–113.
A. V. Sarychev, Nonlinear systems with impulsive and generalized function controls. In: Proc. IIASA Workshop held in Sopron, Hungary, June 1989 (C. I. Byrnes and A. Kurzhansky, eds.). Birkhäuser, Boston (1991), pp. 244–257.
_____, First- and second-order integral functionals of the calculus of variations which exhibit the Lavrentiev phenomenon. J. Dynam. Control Systems 3 (1997), 565–588.
J. C. Willems, A. Kitapçi, and L. M. Silverman, Singular optimal control: a geometric approach. SIAM J. Control Optim. 24 (1986), No. 2, 323–337.
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The first author has been partially supported by Fundação para a Ciência e a Tecnologia (FCT), Portugal, co-financed by the European Community Fund FEDER/POCI via Research Center on Optimization and Control (CEOC) of the University of Aveiro, Portugal. The second author has been partially supported by MIUR, Italy via PRIN 2006019927.
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Guerra, M., Sarychev, A. Measuring singularity of generalized minimizers for control-affine problems. J Dyn Control Syst 15, 177–221 (2009). https://doi.org/10.1007/s10883-009-9065-0
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DOI: https://doi.org/10.1007/s10883-009-9065-0