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 & Continuous Dynamical Systems - A, 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, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[2]

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

[3]

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

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

[5]

J. Bismut, Large Deviations and the Malliavin Calculus,, Birkhäuser, (1984).   Google Scholar

[6]

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

[7]

F. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory,, Springer-Verlag, (1998).   Google Scholar

[8]

F. Clarke, Necessary conditions in dynamic optimization,, Mem. Amer. Math. Soc., 173 (2005).  doi: 10.1090/memo/0816.  Google Scholar

[9]

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

[10]

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

[11]

R. Gamkrelidze, Principles of Optimal Control Theory,, Plenum Press, (1978).   Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1007/BF01322309.  Google Scholar

[15]

V. Jurdjevic, Geometric control theory,, Cambridge University Press, (1997).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

J. Lee, Introduction to Smooth Manifolds,, Springer-Verlag, (2013).   Google Scholar

[20]

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

[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).   Google Scholar

[22]

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

[23]

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

[24]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications,, American Mathematical Society, (2002).   Google Scholar

[25]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes,, John Wiley & Sons, (1962).   Google Scholar

[26]

G. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002).   Google Scholar

[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.   Google Scholar

[28]

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

[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.  doi: 10.1016/S0167-6911(97)00041-8.  Google Scholar

[30]

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

[31]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

J. Bismut, Large Deviations and the Malliavin Calculus,, Birkhäuser, (1984).   Google Scholar

[6]

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

[7]

F. Clarke, Yu. Ledyaev, R. Stern and P. Wolenski, Nonsmooth Analysis and Control Theory,, Springer-Verlag, (1998).   Google Scholar

[8]

F. Clarke, Necessary conditions in dynamic optimization,, Mem. Amer. Math. Soc., 173 (2005).  doi: 10.1090/memo/0816.  Google Scholar

[9]

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

[10]

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

[11]

R. Gamkrelidze, Principles of Optimal Control Theory,, Plenum Press, (1978).   Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1007/BF01322309.  Google Scholar

[15]

V. Jurdjevic, Geometric control theory,, Cambridge University Press, (1997).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

J. Lee, Introduction to Smooth Manifolds,, Springer-Verlag, (2013).   Google Scholar

[20]

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

[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).   Google Scholar

[22]

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

[23]

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

[24]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications,, American Mathematical Society, (2002).   Google Scholar

[25]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes,, John Wiley & Sons, (1962).   Google Scholar

[26]

G. Smirnov, Introduction to the Theory of Differential Inclusions,, American Mathematical Society, (2002).   Google Scholar

[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.   Google Scholar

[28]

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

[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.  doi: 10.1016/S0167-6911(97)00041-8.  Google Scholar

[30]

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

[31]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).   Google Scholar

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