First- and second-order necessary conditions for control problems with constraints

Authors:
Zsolt Páles and Vera Zeidan

Journal:
Trans. Amer. Math. Soc. **346** (1994), 421-453

MSC:
Primary 49K15; Secondary 49J52

DOI:
https://doi.org/10.1090/S0002-9947-1994-1270667-9

MathSciNet review:
1270667

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Abstract: Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form , where is a closed convex set of a Banach space with nonempty interior. The inequality constraints depend on a parameter belonging to a compact metric space . The equality constraints are split into two sets of equations and , where the first equation is an abstract control equation, and is assumed to have a full rank property in . The objective function is where is a compact metric space, is upper semicontinuous in and Lipschitz in . The results are in terms of a function that disappears when the parameter spaces and are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.

**[1]**V. M. Alekseev, S. V. Fomin, and V. M. Tihomirov,*Optimal control*, "Nauka", Moscow, 1979. (Russian) MR**566022 (81g:49001)****[2]**J. P. Aubin,*Applied functional analysis*, Wiley-Interscience, New York, 1978. MR**549483 (81a:46083)****[3]**J. P. Aubin and H. Frankowska,*Set-valued analysis, systems and control*:*Foundations and applications*, Vol. 2, Birkhäuser-Verlag, Boston, Basel, and Berlin, 1990. MR**1048347 (91d:49001)****[4]**A. Ben-Tal,*Second order theory of extremum problems*, Extremal Methods and System Analysis (A. V. Fiacco and K. Kortanek, eds.), Springer-Verlag, Berlin, 1980, pp. 336-356. MR**563871 (83c:90143)****[5]**A. Ben-Tal and J. Zowe,*A unified theory of first and second order conditions for extremum problems in topological vector spaces*, Math. Programming Study**19**(1982), 39-76. MR**669725 (84d:90090)****[6]**F. H. Clarke,*Optimization and nonsmooth analysis*, Canad. Math. Soc. Series of Monographs and Advanced Texts, Wiley, New York, 1983. MR**709590 (85m:49002)****[7]**R. Cominetti,*Metric regularity, tangent sets, and second-order optimality conditions*, Appl. Math. Optim.**21**(1990), 265-287. MR**1036588 (91g:90174)****[8]**A. Ya. Dubovitskii and A. A. Milyutin,*Extremum problems with constraints*, Dokl. Akad. Nauk SSSR**149**(1963), 759-762 = Soviet Math. Dokl.**4**(1963), 452-455.**[9]**-,*Second variations in extremal problems with constraints*, Dokl. Akad. Nauk SSSR**160**(1965), 18-21. MR**0218943 (36:2027)****[10]**B. Fuchssteiner and W. Lusky,*Convex cones*, North-Holland Math. Studies 56, North-Holland, Amsterdam, New York, and Oxford, 1981. MR**640719 (83m:46018)****[11]**I. V. Girsanov,*Lectures on mathematical theory of extremum problems*, Lecture Notes in Economics and Mathematical Systems 67, Springer-Verlag, Berlin, Heidelberg, and New York, 1972. MR**0464021 (57:3958)****[12]**A. D. Ioffe,*On some recent developments in the theory of second order optimality conditions*, Optimization, (S. Dolecki, ed.), Lectures Notes in Math., vol. 1405, Springer-Verlag, New York and Berlin, 1989, pp. 55-68. MR**1036544 (90m:90300)****[13]**-,*Variational analysis of a composite function*:*a formula for the lower second order epi-derivative*, J. Math. Anal. Appl.**160**(1991), 379-405. MR**1126124 (92m:46061)****[14]**A. D. Ioffe and V. M. Tihomirov,*Theory of extremal problems*, North-Holland, Amsterdam, 1979. MR**528295 (80d:49001b)****[15]**H. Kawasaki,*An envelope like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems*, Math. Programming**41**(1988), 73-96. MR**941318 (89d:90191)****[16]**-,*The upper and second order directional derivatives for a sup-type function*, Math. Programming**41**(1988), 327-339. MR**955209 (90a:90171)****[17]**-,*Second order necessary optimality conditions for minimizing a sup type function*, Math. Programming**49**(1991), 213-229. MR**1087454 (92b:49048)****[18]**-,*Second order necessary and sufficient optimality conditions for minimizing a sup type function*, Appl. Math. Optim.**26**(1992), 195-220. MR**1166212 (93d:49032)****[19]**E. S. Levitin, A. A. Milyutin, and N. P. Osmolovskii,*Higher order conditions for a local minimum in problems with constraints*, Uspehi Mat. Nauk**33**(1978), 85-148. MR**526013 (80f:49001)****[20]**H. Maurer and J. Zowe,*First and second order necessary and sufficient conditions for infinite dimensional programming problems*, Math. Programming**16**(1979), 98-110. MR**517762 (81e:90093)****[21]**N. P. Osmolovskii,*Second order conditions for weak local minimum in an optimal control problem*(*necessity, sufficiency*), Soviet Math. Dokl.**16**(1975), 1480-1484.**[22]**Zs. Páles and V. M. Zeidan,*Nonsmooth optimum problems with constraints*, SIAM J. Control Optim. (to appear). MR**1288258 (95g:49026)****[23]**M. Schechter,*Principles of functional analysis*, Academic Press, New York, 1971. MR**0445263 (56:3607)****[24]**G. Stefani and P. Zezza,*Optimal control with mixed state-control constraints*, preprint.**[25]**V. Zeidan and P. Zezza,*The conjugate point condition for smooth control sets*, J. Math. Anal. Appl.**132**(1988), 572-589. MR**943530 (89j:49014)****[26]**E. Zeidler,*Nonlinear functional analysis and its applications*, Vol. I, Springer-Verlag, Berlin, Heidelberg, and New York, 1984.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1270667-9

Keywords:
Nonsmooth functions,
second-order necessary conditions,
mixed state and/or control equality constraints,
state and/or control inequality constraints with parameter,
abstract control equation,
optimal controls

Article copyright:
© Copyright 1994
American Mathematical Society