Optimal control of delay systems with differential and algebraic dynamic constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 2, pp. 285-309.

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.

DOI : https://doi.org/10.1051/cocv:2005008
Classification : 49J53,  49K24,  49K25,  49M25,  90C31,  93C30
Mots clés : optimal control, variational analysis, functional-differential inclusions of neutral type, differential and algebraic dynamic constraints, discrete approximations, generalized differentiation, necessary optimality conditions
@article{COCV_2005__11_2_285_0,
     author = {Mordukhovich, Boris S. and Wang, Lianwen},
     title = {Optimal control of delay systems with differential and algebraic dynamic constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {285--309},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {2},
     year = {2005},
     doi = {10.1051/cocv:2005008},
     zbl = {1081.49017},
     mrnumber = {2141891},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2005008/}
}
Mordukhovich, Boris S.; Wang, Lianwen. Optimal control of delay systems with differential and algebraic dynamic constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 2, pp. 285-309. doi : 10.1051/cocv:2005008. http://www.numdam.org/articles/10.1051/cocv:2005008/

[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). | MR 1101809 | Zbl 0699.65057

[2] E.N. Devdariani and Yu.S. Ledyaev, Maximum principle for implicit control systems. Appl. Math. Optim. 40 (1999) 79-103. | Zbl 0929.49006

[3] A.L. Dontchev and E.M. Farhi, Error estimates for discretized differential inclusions. Computing 41 (1989) 349-358. | Zbl 0667.65067

[4] M. Kisielewicz, Differential Inclusions and Optimal Control. Kluwer, Dordrecht (1991). | MR 1135796 | Zbl 0731.49001

[5] B.S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40 (1976) 960-969. | Zbl 0362.49017

[6] B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control. Nauka, Moscow (1988). | MR 945143 | Zbl 0643.49001

[7] B.S. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340 (1993) 1-35. | Zbl 0791.49018

[8] B.S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882-915. | Zbl 0844.49017

[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. | Zbl 1041.49019

[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. | Zbl 1006.49016

[11] B.S. Mordukhovich and L. Wang, Optimal control of constrained delay-differential inclusions with multivalued initial condition. Control Cybernet. 32 (2003) 585-609. | Zbl 1127.49303

[12] B.S. Mordukhovich and L. Wang, Optimal control of neutral functional-differential inclusions. SIAM J. Control Optim. 43 (2004) 116-136. | MR 2082695 | Zbl 1070.49016

[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.

[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. | Zbl 0891.49013

[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. | Zbl 0878.49012

[17] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1998). | MR 1491362 | Zbl 0888.49001

[18] G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). | MR 1867542 | Zbl 0992.34001

[19] R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). | MR 1756410 | Zbl 0952.49001

[20] J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, New York (1972). | MR 372708 | Zbl 0253.49001