## Abstract

Optimal control has strongly influenced geometry since the early days of both subjects. In particular, it played a crucial role in the birth of differential geometry in the nineteenth century through the revolutionary ideas of redefining the notion of “straight line” (now renamed “geodesic”) by means of a curve minimization problem, and of emphasizing general invariance and covariance conditions. More recently, modern control theory has been heavily influenced by geometry. One aspect of this influence is the geometrization of the necessary conditions for optimality, which are recast as geometric conditions about reachable sets, thus becoming special cases of the broader question of the structure and properties of these sets. Recently, this has led to a new general version of the finite-dimensional maximum principle, stated here in full detail for the first time. A second aspect—in which Roger Brockett’s ideas have played a crucial role—is the use in control theory of concepts and techniques from differential geometry. In particular, this leads to regarding a control system as a collection of vector fields, and exploiting the algebraic structure given by the Lie bracket operation. This approach has led to new important developments on various nonlinear control problems. In optimal control, the vector-field view has produced invariant formulations of the maximum principle on manifolds—using either Poisson brackets or connections along curves—which, besides being more general and mathematically natural, actually have real advantages for the solution of concrete problems.

### Keywords

Manifold Covariance Propa Stratification Lime## Preview

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### References

- [1]Agrachev, A. A. and R. V. Gamkrelidze, “A second-order optimality principle for a time-optimal problem,”
*Math. Sbornik***100**, 142 (1976).Google Scholar - [2]Agrachev, A. A. and R. V. Gamkrelidze, “The exponential representation of flows and the chronological calculus,”
*Math. Sbornik***109**, 149 (1978).Google Scholar - [3]Boltyanskii, V. G.,
*The Homotopic Theory of Continuous Mappings and Vector Fields*, Trudy Matern. Instituta im. V.A. Steklova, Vol. XLVII, Izdat. Akad. Nauk SSSR, Moscow, 1955.Google Scholar - [4]Berkovitz, L. D.,
*Optimal Control Theory*, Springer-Verlag, New York, 1974.MATHGoogle Scholar - [5]Bressan, A., “On differential relations with lower continuous right-hand side. An existence theorem.”
*J. Diff. Equations***37**(1980), pp. 89–97.MathSciNetMATHCrossRefGoogle Scholar - [6]Bressan, A., “Solutions of lower semicontinuous differential inclusions on closed sets,”
*Rend. Sem. Mat. Univ. Padova***69**(1983), pp. 99–107.MathSciNetMATHGoogle Scholar - [7]Bressan, A., “Directionally continuous selections and differential inclusions,”
*Funk. Ekvac.***31**(1988), pp. 459–470.MathSciNetMATHGoogle Scholar - [8]Brockett, R. W., R. S. Millman and H. J. Sussmann (eds.),
*Differential Geometric Control Theory*, Progress in Mathematics No. 27, Birkhäuser, Boston 1983.MATHGoogle Scholar - [9]Browder, F. E., “On the fixed point index for continuous mappings of locally connected spaces,”
*Summa Brazil. Math.***4**(1960), pp. 253–293.MathSciNetGoogle Scholar - [10]Clarke, F. H., “The Maximum Principle under minimal hypotheses,”
*SIAM J. Control Optim.***14**(1976), pp. 1078–1091.MATHCrossRefGoogle Scholar - [11]Clarke, F. H.,
*Optimization and Nonsmooth Analysis*, Wiley-Interscience, New York, 1983.MATHGoogle Scholar - [12]Clarke, F. and R. B. Vinter, “Optimal multiprocesses,”
*SIAM J. Control Optim.***27**(1989), pp. 1072–1091.MathSciNetMATHCrossRefGoogle Scholar - [13]Colombo, R.M., A. Fryszkowski, T. Rzezuchowski, and V. Staicu, “Continuous selections of solution sets of Lipschitzean differential inclusions,”
*Funkcialaj Ekvacioj***34**(1991), pp. 321–330.MathSciNetMATHGoogle Scholar - [14]Do Carmo, M.,
*Riemannian Geometry*, Birkhäuser, Boston, 1992.MATHGoogle Scholar - [15]Ferreyra, G., H. Hermes, R. Gardner and H. J. Sussmann (eds.)
*Differential-Geometric Control*, Proceedings of the July 1997 Summer Research Institute of the American Mathematical Society (AMS), held in Boulder, CO, June–July 1997. To appear in the AMS*Symposia on Pure Mathematics*series.Google Scholar - [16]Fryszkowski, A., and T. Rzežuchowski, “Continuous version of the Filippov-Wažewski relaxation theorem,”
*J. Diff Equations***94**(1993), pp. 254–265.CrossRefGoogle Scholar - [17]Gabasov, V., and F. M. Kirillova, “High-order necessary conditions for optimality,”
*SIAM J. Control***10**(1972), pp. 127–168.MathSciNetMATHCrossRefGoogle Scholar - [18]Ioffe, A., “Euler-Lagrange and Hamiltonian formalisms for dynamic optimization,”
*Trans. Amer. Math. Society***349**(1997), pp. 2871–2900.MathSciNetMATHCrossRefGoogle Scholar - [19]Ioffe, A. D. and R. T. Rockafellar, “The Euler and Weierstrass conditions for non-smooth variational problems,”
*Calc. Var. Partial Differential Equations***4**(1996), pp. 59–87.MathSciNetMATHCrossRefGoogle Scholar - [20]Isidori, A.,
*Nonlinear Control Systems*Springer-Verlag, Berlin, 1995.MATHGoogle Scholar - [21]Jakubczyk, B., and W. Respondek (eds.),
*Geometry of Feedback and Optimal Control*, Marcel Dekker, New York, 1997.Google Scholar - [22]Jurdjevic, V.,
*Geometric Control Theory*, Cambridge University Press, 1997.Google Scholar - [23]B. Kaskosz, “A maximum principle in relaxed controls,”
*Nonlinear Analysis, TMA***14**(1990), pp. 357–367.MathSciNetMATHCrossRefGoogle Scholar - [24]Kaskosz, B., and S. Łojasiewicz Jr., “A Maximum Principle for generalized control systems,”
*Nonlinear Anal TMA***9**(1985), pp. 109–130.MATHCrossRefGoogle Scholar - [25]Kawski, M., and H J. Sussmann, “Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory.” In:
*Operators, Systems and Linear Algebra: Three Decades of Algebraic Systems Theory*, proceedings of the workshop held in Kaiserslautern, Germany, Sept. 24–26, 1997, in honor of Professor Paul Fuhrmann; U. Helmke, D. Praetzel-Wolters, E. Zerz (eds.); B. G. Teubner, Stuttgart, 1997, pp 111–129.Google Scholar - [26]Kelley, H. J., R. E. Kopp and H. G. Moyer, “Singular extremals,” in
*Topics in Optimization*, G. Leitman ed., Academic Press, New York, 1967.Google Scholar - [27]Knobloch, H. W.,
*High-Order Necessary Conditions in Optimal Control*, Springer-Verlag, New York, 1975.Google Scholar - [28]Krener, A. J., “The higher-order maximum principle and its application to singular extremals,”
*SIAM J. Control Opt.***15**(1977), pp. 256–293.MathSciNetMATHCrossRefGoogle Scholar - [29]Leray, J., and J. Schauder, “Topologie et équations fonctionnelles,”
*Ann. Sci. École Norm. Sup.***51**(1934), pp. 45–78.MathSciNetGoogle Scholar - [30]Loewen, P. D., and R. T. Rockafellar, “Optimal control of unbounded differential inclusions,”
*SIAM J. Control and Optimization***32**(1994), pp. 442–470.MathSciNetMATHCrossRefGoogle Scholar - [31]Loewen, P. D., and R. T. Rockafellar, “New necessary conditions for the generalized problem of Bolza,”
*SIAM J. Control and Optimization***34**(1996), pp. 1496–1511.MathSciNetMATHCrossRefGoogle Scholar - [32]Mordukhovich, B., and H. J. Sussmann (eds.),
*Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control*, Springer-Verlag, New York, 1996.Google Scholar - [33]Nijmeijer, H., and A. van der Schaft,
*Nonlinear Dynamical Control Systems*, Springer-Verlag, New York, 1990.MATHGoogle Scholar - [34]Pars, L.A.,
*An Introduction to the Calculus of Variations*, Heinemann, London, 1962.MATHGoogle Scholar - [35]Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mischenko,
*The Mathematical Theory of Optimal Processes*, Wiley, New York, 1962.MATHGoogle Scholar - [36]Rabinowitz, P. H.,
*Théorie du dégré topologique et applications à des problemes aux limites nonlinéaires.*Lect. Notes Université de Paris Sud (Orsay), 1973.Google Scholar - [37]Rabinowitz, P. H., “Some global results for nonlinear eigenvalue problems,”
*J. Functional Analysis***7**(1971), pp. 487–513.MathSciNetMATHCrossRefGoogle Scholar - [38]Rockafellar, R. T., “Equivalent subgradient versions of Hamiltonian and Euler-Lagrange equations in variational analysis,”
*SIAM J. Control and Optimization***34**(1996), pp. 1300–1314.MathSciNetMATHCrossRefGoogle Scholar - [39]Sussmann, H. J. “A weak regularity theorem for real analytic optimal control problems,”
*Revista Matemática Iberoamericana***2**, No. 3 (1986), pp. 307–317.MathSciNetMATHCrossRefGoogle Scholar - [40]Sussmann, H. J., “Trajectory regularity and real analyticity: some recent results,”
*Proc. 25th IEEE Conference on Decision and Control, Athens, Greece, 1986*, IEEE, New York, 1986, pp. 592–595.CrossRefGoogle Scholar - [41]Sussmann, H. J. (ed.),
*Nonlinear Controllability and Optimal Control*, Marcel Dekker, New York, 1990.MATHGoogle Scholar - [42]Sussmann, H.J., “A strong version of the Maximum Principle under weak hypotheses,”
*Proc. 33rd IEEE Conference on Decision and Control, Orlando, FL, 1994*, IEEE, New York, 1994, pp. 1950–1956.CrossRefGoogle Scholar - [43]Sussmann, H.J., “A strong version of the Łojasiewicz Maximum Principle,” in
*Optimal Control of Differential Equations*, Nicolai H. Pavel, Ed., Marcel Dekker, New York, 1994.Google Scholar - [44]Sussmann, H.J., “Some recent results on the maximum principle of optimal control theory.” In
*Systems and Control in the Twenty-First Century*, C. I. Byrnes, B. N. Datta, D. S. Gilliam, and C. F. Martin, Eds., Birkhäuser, Boston, 1997, pp. 351–372.Google Scholar - [45]Sussmann, H.J., “A strong maximum principle for systems of differential inclusions.” in
*Proceedings of the 35th IEEE Conference on Decision and Control*, Kobe, Japan, Dec. 1996. IEEE publications, 1996, pp. 1809–1814Google Scholar - [46]Sussmann, H. J., “An introduction to the coordinate-free maximum principle,” In [21] (1997), pp. 463–557.Google Scholar
- [47]Sussmann, H. J., “Multidifferential calculus: chain rule, open mapping and transversal intersection theorems,” to appear in
*Optimal control: theory, algorithms, and applications*, W. W. Hager and P. M. Pardalos, Editors, Kluwer Academic Publishers, 1997.Google Scholar - [48]Sussmann, H. J., “Some optimal control applications of real analytic stratifications and desingularization.” To appear in
*Singularities Symposium — Łojasiewiz 70*(proceedings of a conference held in Krakow, Poland, September 1996, in celebration of the 70th birthday of Stanislaw Łojasiewicz), B. Jakubczyk and W. Pawlucki, Eds., Banach Center Publications, Warsaw, Poland.Google Scholar - [49]Sussmann, H. J., and J. C. Willems, “300 years of optimal control: from the brachystochrone to the maximum principle,”
*IEEE Control Systems Magazine*, June 1997.Google Scholar - [50]Sussmann, H. J., and J. C. Willems, “Curve optimization from the brachystochrone to the maximum principle and beyond,” in preparation.Google Scholar
- [51]Tuan, H. D., “Controllability and extremality in nonconvex differential inclusions,”
*J. Optim. Theory Appl.***85**(1995), pp. 435–472.MathSciNetMATHCrossRefGoogle Scholar - [52]Warga, J., “Fat homeomorphisms and unbounded derivate containers,”
*J. Math. Anal. Appl.***81**(1981), pp. 545–560.MathSciNetMATHCrossRefGoogle Scholar - [53]Warga, J., “Controllability, extremality and abnormality in nonsmooth optimal control,”
*J. Optim. Theory Applic.***41**(1983), pp. 239–260.MathSciNetMATHCrossRefGoogle Scholar - [54]Warga, J., “Optimization and controllability without differentiability assumptions,”
*SIAM J. Control and Optimization***21**(1983), pp. 837–855.MathSciNetMATHCrossRefGoogle Scholar - [55]Warga, J., “Homeomorphisms and local
*C*^{1}approximations,”*Nonlinear Anal. TMA***12**(1988), pp. 593–597.MathSciNetMATHCrossRefGoogle Scholar - [56]Warga, J., “An extension of the Kaskosz maximum principle,”
*Applied Math. Optim.***22**(1990), pp. 61–74.MathSciNetMATHCrossRefGoogle Scholar - [57]Yourgrau, W., and S. Mandelstam,
*Variational Principles in Dynamics and Quantum Theory*, W. B. Saunders & Co., Philadelphia, 1968.Google Scholar - [58]Zhu. Q. J., “Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints,”
*J. Diff. Equations***124**(1996), pp. 186–204.MATHCrossRefGoogle Scholar