×
Hilscher, Roman; Zeidan, Vera

Weak maximum principle and accessory problem for control problems on time scales. (English) Zbl 1157.49030

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, 3209-3226 (2009).
Summary: We derive the first and second variations for a nonlinear time scale optimal control problem with control and state-endpoints equality constraints. Using the first variation, a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the Dubois-Reymond Lemma to the optimal control setting and time scales. A second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions. The control problem is studied under a controllability assumption, and with or without the shift in the state variable. These two forms of the problem are shown to be equivalent.

Cited in 37 Documents

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
39A12 Discrete version of topics in analysis
34K35 Control problems for functional-differential equations
49N25 Impulsive optimal control problems
93B05 Controllability
93C15 Control/observation systems governed by ordinary differential equations
93C55 Discrete-time control/observation systems

Keywords:

time scale; optimal control problem; controllability; normality; weak maximum principle; DuBois-Reymond Lemma; first variation; second variation; feasible family
Cite Review PDF
Full Text: DOI

References:

[1] Agarwal, R. P.; Bohner, M.; Wong, P. J.Y., Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput., 99, 2-3, 153-166 (1999) · Zbl 0938.34015
[2] Ahlbrandt, C. D.; Bohner, M.; Ridenhour, J., Hamiltonian systems on time scales, J. Math. Anal. Appl., 250, 2, 561-578 (2000) · Zbl 0966.39010
[3] Ahlbrandt, C. D.; Peterson, A. C., Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0860.39001
[4] Atici, F. M.; Biles, D. C.; Lebedinsky, A., An application of time scales to economics, Math. Comput. Modelling, 43, 7-8, 718-726 (2006) · Zbl 1187.91125
[5] Bernstein, D. S., Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory (2005), Princeton University Press: Princeton University Press Princeton · Zbl 1075.15001
[6] Bohner, M., Calculus of variations on time scales, Dynam. Systems Appl., 13, 3-4, 339-349 (2004) · Zbl 1069.39019
[7] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales. An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[8] Boltyanskii, V. G., Optimal Control of Discrete Systems (1978), John Wiley & Sons: John Wiley & Sons New York, Toronto · Zbl 0249.49010
[9] H. Brezis, Analyse Fonctionnelle. Théorie et Applications [Functional Analysis. Theory and Applications], Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (in French); H. Brezis, Analyse Fonctionnelle. Théorie et Applications [Functional Analysis. Theory and Applications], Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (in French) · Zbl 0511.46001
[10] Došlý, O.; Hilger, S.; Agarwal, R. P.; Bohner, M.; O’Regan, D., A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Dynamic Equations on Time Scales. Dynamic Equations on Time Scales, J. Comput. Appl. Math., 141, 1-2, 147-158 (2002) · Zbl 1009.34033
[11] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential Equations Dynam. Systems, 1, 3, 223-244 (1993) · Zbl 0868.39007
[12] Erbe, L.; Peterson, A.; Agarwal, R. P.; Bohner, M.; O’Regan, D., Oscillation criteria for second order matrix dynamic equations on a time scale, Dynamic Equations on Time Scales. Dynamic Equations on Time Scales, J. Comput. Appl. Math., 141, 1-2, 169-185 (2002) · Zbl 1017.34030
[13] Gelfand, I. M.; Fomin, S. V., Calculus of Variations (1963), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0127.05402
[14] Hestenes, M. R., Calculus of Variations and Optimal Control Theory (1966), John Wiley & Sons: John Wiley & Sons New York · Zbl 0173.35703
[15] Hestenes, M. R., Quadratic variational theory, (Balakrishnan, A. V., Control Theory and the Calculus of Variations (1969), Academic Press: Academic Press New York), 1-37 · Zbl 0292.49019
[16] Hestenes, M. R., Quadratic control problems, J. Optim. Theory Appl., 17, 1-2, 1-42 (1975) · Zbl 0292.49019
[17] Hilscher, R.; Peterson, A. C., Linear Hamiltonian systems on time scales: Positivity of quadratic functionals, Boundary Value Problems and Related Topics. Boundary Value Problems and Related Topics, Math. Comput. Modelling, 32, 5-6, 507-527 (2000) · Zbl 0982.37054
[18] Hilscher, R.; Zeidan, V., Discrete optimal control: The accessory problem and necessary optimality conditions, J. Math. Anal. Appl., 243, 2, 429-452 (2000) · Zbl 0987.49016
[19] Hilscher, R.; Zeidan, V., Second order sufficiency criteria for a discrete optimal control problem, J. Difference Equ. Appl., 8, 6, 573-602 (2002) · Zbl 1010.49020
[20] Hilscher, R.; Zeidan, V., Symplectic difference systems: Variable stepsize discretization and discrete quadratic functionals, Linear Algebra Appl., 367, 67-104 (2003) · Zbl 1021.39008
[21] Hilscher, R.; Zeidan, V., Calculus of variations on time scales: Weak local piecewise \(C_{rd}^1\) solutions with variable endpoints, J. Math. Anal. Appl., 289, 1, 143-166 (2004) · Zbl 1043.49004
[22] Hilscher, R.; Zeidan, V., Nonnegativity and positivity of a quadratic functional in the discrete calculus of variations: A survey, J. Difference Equ. Appl., 11, 9, 857-875 (2005) · Zbl 1098.49025
[23] Hilscher, R.; Zeidan, V., Legendre, Jacobi, and Riccati type conditions for time scale variational problem with application, Dynam. Systems Appl., 16, 3, 451-480 (2007) · Zbl 1160.39004
[24] Hilscher, R.; Zeidan, V., Time scale embedding theorem and coercivity of quadratic functionals, Analysis (Munich), 28, 1, 1-28 (2008) · Zbl 1136.49017
[25] Jacobson, D. H.; Martin, D. H.; Pachter, M.; Geveci, T., (Extensions of Linear-Quadratic Control Theory. Extensions of Linear-Quadratic Control Theory, Lecture Notes in Control and Information Sciences, vol. 27 (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0435.93001
[26] Kelley, W. G.; Peterson, A. C., Difference Equations: An Introduction with Applications (1991), Academic Press: Academic Press San Diego · Zbl 0733.39001
[27] Lewis, F. L., Optimal Control (1986), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0665.93065
[28] Reid, W. T., Ordinary Differential Equations (1971), Wiley: Wiley New York · Zbl 0212.10901
[29] Zeidan, V.; Zezza, P., The conjugate point condition for smooth control sets, J. Math. Anal. Appl., 132, 2, 572-589 (1988) · Zbl 0646.49011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.