Optimal control of delay systems with differential and algebraic dynamic constraints

Boris S. Mordukhovich; Lianwen Wang

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 2, page 285-309
  • ISSN: 1292-8119

Abstract

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This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between “slow” and “fast” variables. We pursue a twofold goal: to study variational stability for this class of control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems under consideration and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. The authors are not familiar with any results in these directions for such systems even in the delay-free case. In the first part of the paper we establish the value convergence of discrete approximations as well as the strong convergence of optimal arcs in the classical Sobolev space W 1 , 2 . Then using discrete approximations as a vehicle, we derive necessary optimality conditions for the initial continuous-time systems in both Euler-Lagrange and hamiltonian forms via basic generalized differential constructions of variational analysis.

How to cite

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Mordukhovich, Boris S., and Wang, Lianwen. "Optimal control of delay systems with differential and algebraic dynamic constraints." ESAIM: Control, Optimisation and Calculus of Variations 11.2 (2005): 285-309. <http://eudml.org/doc/244836>.

@article{Mordukhovich2005,
abstract = {This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between “slow” and “fast” variables. We pursue a twofold goal: to study variational stability for this class of control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems under consideration and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. The authors are not familiar with any results in these directions for such systems even in the delay-free case. In the first part of the paper we establish the value convergence of discrete approximations as well as the strong convergence of optimal arcs in the classical Sobolev space $W^\{1,2\}$. Then using discrete approximations as a vehicle, we derive necessary optimality conditions for the initial continuous-time systems in both Euler-Lagrange and hamiltonian forms via basic generalized differential constructions of variational analysis.},
author = {Mordukhovich, Boris S., Wang, Lianwen},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal control; variational analysis; functional-differential inclusions of neutral type; differential and algebraic dynamic constraints; discrete approximations; generalized differentiation; necessary optimality conditions},
language = {eng},
number = {2},
pages = {285-309},
publisher = {EDP-Sciences},
title = {Optimal control of delay systems with differential and algebraic dynamic constraints},
url = {http://eudml.org/doc/244836},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Mordukhovich, Boris S.
AU - Wang, Lianwen
TI - Optimal control of delay systems with differential and algebraic dynamic constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 2
SP - 285
EP - 309
AB - This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between “slow” and “fast” variables. We pursue a twofold goal: to study variational stability for this class of control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems under consideration and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. The authors are not familiar with any results in these directions for such systems even in the delay-free case. In the first part of the paper we establish the value convergence of discrete approximations as well as the strong convergence of optimal arcs in the classical Sobolev space $W^{1,2}$. Then using discrete approximations as a vehicle, we derive necessary optimality conditions for the initial continuous-time systems in both Euler-Lagrange and hamiltonian forms via basic generalized differential constructions of variational analysis.
LA - eng
KW - optimal control; variational analysis; functional-differential inclusions of neutral type; differential and algebraic dynamic constraints; discrete approximations; generalized differentiation; necessary optimality conditions
UR - http://eudml.org/doc/244836
ER -

References

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  1. [1] K.E. Brennan, S.L. Campbell and L.R. Pretzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations. North-Holland, New York (1989). Zbl0699.65057MR1101809
  2. [2] E.N. Devdariani and Yu.S. Ledyaev, Maximum principle for implicit control systems. Appl. Math. Optim. 40 (1999) 79–103. Zbl0929.49006
  3. [3] A.L. Dontchev and E.M. Farhi, Error estimates for discretized differential inclusions. Computing 41 (1989) 349–358. Zbl0667.65067
  4. [4] M. Kisielewicz, Differential Inclusions and Optimal Control. Kluwer, Dordrecht (1991). Zbl0731.49001MR1135796
  5. [5] B.S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976) 960–969. Zbl0362.49017
  6. [6] B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). Zbl0643.49001MR945143
  7. [7] B.S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340 (1993) 1–35. Zbl0791.49018
  8. [8] B.S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882–915. Zbl0844.49017
  9. [9] B.S. Mordukhovich, J.S. Treiman and Q.J. Zhu, An extended extremal principle with applications to multiobjective optimization. SIAM J. Optim. 14 (2003) 359–379. Zbl1041.49019
  10. [10] B.S. Mordukhovich and R. Trubnik, Stability of discrete approximation and necessary optimality conditions for delay-differential inclusions. Ann. Oper. Res. 101 (2001) 149–170. Zbl1006.49016
  11. [11] B.S. Mordukhovich and L. Wang, Optimal control of constrained delay-differential inclusions with multivalued initial condition. Control Cybernet. 32 (2003) 585–609. Zbl1127.49303
  12. [12] B.S. Mordukhovich and L. Wang, Optimal control of neutral functional-differential inclusions. SIAM J. Control Optim. 43 (2004) 116-136. Zbl1070.49016MR2082695
  13. [13] B.S. Mordukhovich and L. Wang, Optimal control of differential-algebraic inclusions, in Optimal Control, Stabilization, and Nonsmooth Analysis, M. de Queiroz et al., Eds., Lectures Notes in Control and Information Sciences, Springer-Verlag, Heidelberg 301 (2004) 73–83. Zbl1160.49024
  14. [14] M.D.R. de Pinho and R.B. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations. J. Math. Anal. Appl. 212 (1997) 493–516. Zbl0891.49013
  15. [15] C. Pantelides, D. Gritsis, K.P. Morison and R.W.H. Sargent, The mathematical modelling of transient systems using differential-algebraic equations. Comput. Chem. Engrg. 12 (1988) 449–454. 
  16. [16] R.T. Rockafellar, Equivalent subgradient versions of Hamiltonian and Euler–Lagrange conditions in variational analysis. SIAM J. Control Optim. 34 (1996) 1300–1314. Zbl0878.49012
  17. [17] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1998). Zbl0888.49001MR1491362
  18. [18] G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). Zbl0992.34001MR1867542
  19. [19] R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). Zbl0952.49001MR1756410
  20. [20] J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). Zbl0253.49001MR372708

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