September  2015, 35(9): 4455-4475. doi: 10.3934/dcds.2015.35.4455

Optimal control of differential inclusions on manifolds

1. 

Department of Mathematics and Statistics, Queen's University, Kingston, ON, K7L 3N6, Canada

2. 

Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008-5248, United States

Received  April 2014 Revised  October 2014 Published  April 2015

Dynamic optimization problems for differential inclusions on manifolds are considered. A mathematical framework for derivation of optimality conditions for generalized dynamical systems is proposed. We obtain optimality conditions in form of generalized Euler-Lagrange relations and in form of partially convexified Hamiltonian inclusions by using metric regularity of terminal and dynamic constraints.
Citation: Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
References:
[1]

A. Agrachev and Yu. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Heidelberg, 2004. doi: 10.1007/978-3-662-06404-7.

[2]

M. Barbero-Liñán and M. Muñoz-Lecanda, Geometric approach to Pontryagins maximum principle, Acta applicandae mathematicae, 108 (2009), 429-485. doi: 10.1007/s10440-008-9320-5.

[3]

A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511734939.

[4]

D. Bessis, Yu. Ledyaev and R. Vinter, Dualization of the Euler and Hamiltonian inclusions, Nonlinear Analysis, 43 (2001), 861-882. doi: 10.1016/S0362-546X(99)00238-2.

[5]

J. Bismut, Large Deviations and the Malliavin Calculus, Birkhäuser, Boston, 1984.

[6]

J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl., 48 (1986), 9-52. doi: 10.1007/BF00938588.

[7]

F. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.

[8]

F. Clarke, Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173 (2005), x+113 pp. doi: 10.1090/memo/0816.

[9]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[10]

E. N. Devdariani and Yu. S. Ledyaev, Maximum principle for implicit control systems, Appl. Math. Optim., 40 (1999), 79-103. doi: 10.1007/s002459900117.

[11]

R. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, New York, 1978.

[12]

A. D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc., 251 (1979), 61-69. doi: 10.1090/S0002-9947-1979-0531969-6.

[13]

A. Ioffe, Euler-Lagrange and Hamiltonian formalisms in dynamic optimization, Transactions of the American Mathematical Society, 349 (1997), 2871-2900. doi: 10.1090/S0002-9947-97-01795-9.

[14]

A. Ioffe and R. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems, Calculus of Variations and Partial Differential Equations, 4 (1996), 59-87. doi: 10.1007/BF01322309.

[15]

V. Jurdjevic, Geometric control theory, Cambridge University Press, Cambridge, 1997.

[16]

I. Kolar and P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.

[17]

Yu. S. Ledyaev, Theorems on an implicitly given set-valued mapping, Dokl. Akad. Nauk SSSR, 276 (1984), 543-546.

[18]

Yu. Ledyaev and Q. Zhu, Nonsmooth analysis on smooth manifolds, Transactions of the American Mathematical Society, 359 (2007), 3687-3732. doi: 10.1090/S0002-9947-07-04075-5.

[19]

J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2013.

[20]

P. Loewen and R. Rockafellar, Optimal control of unbounded differential inclusions, SIAM Journal of Control and Optimization, 32 (1994), 442-470. doi: 10.1137/S0363012991217494.

[21]

M. Modugno and G. Stefani, Some results on second tangent and cotangent spaces, Quaderni del Dipartimento di Matematica dell'Università del Salento, 16 (1978).

[22]

B. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM Journal of Control and Optimization, 33 (1995), 882-915. doi: 10.1137/S0363012993245665.

[23]

B. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 331, Springer, Berlin, 2006.

[24]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, 2002.

[25]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962.

[26]

G. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002.

[27]

H. Sussman, Symmetries and integrals of motion in optimal control, Geometry in nonlinear control and differential inclusions. Mathematics institute of Polish academy of sciences. Banach center publications, 32 (1995), 379-393.

[28]

R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems, SIAM journal on control and optimization, 35 (1997), 56-77. doi: 10.1137/S0363012995283133.

[29]

R. Vinter and P. Woodford, On the occurrence of intermediate local minimizers that are not strong local minimizers, Systems & Control Letters, 31 (1997), 235-242. doi: 10.1016/S0167-6911(97)00041-8.

[30]

R. Vinter, The Hamiltonian inclusion for nonconvex velocity sets, SIAM Journal on Optimization and Control, 52 (2014), 1237-1250. doi: 10.1137/130917417.

[31]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.

show all references

References:
[1]

A. Agrachev and Yu. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Heidelberg, 2004. doi: 10.1007/978-3-662-06404-7.

[2]

M. Barbero-Liñán and M. Muñoz-Lecanda, Geometric approach to Pontryagins maximum principle, Acta applicandae mathematicae, 108 (2009), 429-485. doi: 10.1007/s10440-008-9320-5.

[3]

A. Bennett, Lagrangian Fluid Dynamics, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511734939.

[4]

D. Bessis, Yu. Ledyaev and R. Vinter, Dualization of the Euler and Hamiltonian inclusions, Nonlinear Analysis, 43 (2001), 861-882. doi: 10.1016/S0362-546X(99)00238-2.

[5]

J. Bismut, Large Deviations and the Malliavin Calculus, Birkhäuser, Boston, 1984.

[6]

J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl., 48 (1986), 9-52. doi: 10.1007/BF00938588.

[7]

F. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.

[8]

F. Clarke, Necessary conditions in dynamic optimization, Mem. Amer. Math. Soc., 173 (2005), x+113 pp. doi: 10.1090/memo/0816.

[9]

F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer, London, 2013. doi: 10.1007/978-1-4471-4820-3.

[10]

E. N. Devdariani and Yu. S. Ledyaev, Maximum principle for implicit control systems, Appl. Math. Optim., 40 (1999), 79-103. doi: 10.1007/s002459900117.

[11]

R. Gamkrelidze, Principles of Optimal Control Theory, Plenum Press, New York, 1978.

[12]

A. D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc., 251 (1979), 61-69. doi: 10.1090/S0002-9947-1979-0531969-6.

[13]

A. Ioffe, Euler-Lagrange and Hamiltonian formalisms in dynamic optimization, Transactions of the American Mathematical Society, 349 (1997), 2871-2900. doi: 10.1090/S0002-9947-97-01795-9.

[14]

A. Ioffe and R. Rockafellar, The Euler and Weierstrass conditions for nonsmooth variational problems, Calculus of Variations and Partial Differential Equations, 4 (1996), 59-87. doi: 10.1007/BF01322309.

[15]

V. Jurdjevic, Geometric control theory, Cambridge University Press, Cambridge, 1997.

[16]

I. Kolar and P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.

[17]

Yu. S. Ledyaev, Theorems on an implicitly given set-valued mapping, Dokl. Akad. Nauk SSSR, 276 (1984), 543-546.

[18]

Yu. Ledyaev and Q. Zhu, Nonsmooth analysis on smooth manifolds, Transactions of the American Mathematical Society, 359 (2007), 3687-3732. doi: 10.1090/S0002-9947-07-04075-5.

[19]

J. Lee, Introduction to Smooth Manifolds, Springer-Verlag, New York, 2013.

[20]

P. Loewen and R. Rockafellar, Optimal control of unbounded differential inclusions, SIAM Journal of Control and Optimization, 32 (1994), 442-470. doi: 10.1137/S0363012991217494.

[21]

M. Modugno and G. Stefani, Some results on second tangent and cotangent spaces, Quaderni del Dipartimento di Matematica dell'Università del Salento, 16 (1978).

[22]

B. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions, SIAM Journal of Control and Optimization, 33 (1995), 882-915. doi: 10.1137/S0363012993245665.

[23]

B. Mordukhovich, Variational Analysis and Generalized Differentiation, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 331, Springer, Berlin, 2006.

[24]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, Providence, 2002.

[25]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962.

[26]

G. Smirnov, Introduction to the Theory of Differential Inclusions, American Mathematical Society, Providence, 2002.

[27]

H. Sussman, Symmetries and integrals of motion in optimal control, Geometry in nonlinear control and differential inclusions. Mathematics institute of Polish academy of sciences. Banach center publications, 32 (1995), 379-393.

[28]

R. Vinter and H. Zheng, The extended Euler-Lagrange condition for nonconvex variational problems, SIAM journal on control and optimization, 35 (1997), 56-77. doi: 10.1137/S0363012995283133.

[29]

R. Vinter and P. Woodford, On the occurrence of intermediate local minimizers that are not strong local minimizers, Systems & Control Letters, 31 (1997), 235-242. doi: 10.1016/S0167-6911(97)00041-8.

[30]

R. Vinter, The Hamiltonian inclusion for nonconvex velocity sets, SIAM Journal on Optimization and Control, 52 (2014), 1237-1250. doi: 10.1137/130917417.

[31]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.

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