# 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

## Access Full Article

top## Abstract

top## How to cite

topMordukhovich, 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

top- [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] E.N. Devdariani and Yu.S. Ledyaev, Maximum principle for implicit control systems. Appl. Math. Optim. 40 (1999) 79–103. Zbl0929.49006
- [3] A.L. Dontchev and E.M. Farhi, Error estimates for discretized differential inclusions. Computing 41 (1989) 349–358. Zbl0667.65067
- [4] M. Kisielewicz, Differential Inclusions and Optimal Control. Kluwer, Dordrecht (1991). Zbl0731.49001MR1135796
- [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] B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). Zbl0643.49001MR945143
- [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] 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] 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] 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] 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] B.S. Mordukhovich and L. Wang, Optimal control of neutral functional-differential inclusions. SIAM J. Control Optim. 43 (2004) 116-136. Zbl1070.49016MR2082695
- [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] 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] 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] 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] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1998). Zbl0888.49001MR1491362
- [18] G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). Zbl0992.34001MR1867542
- [19] R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). Zbl0952.49001MR1756410
- [20] J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). Zbl0253.49001MR372708

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.