Weak maximum principle and accessory problem for control problems on time scales

Nonlinear Analysis, Theory, Methods and Applications

Volumn 70, Issue 9, 2009, Pages 3209-3226

  Hilscher, Roman ^a   Zeidan, Vera ^b  

^a MASARYK UNIVERSITY   (Czech Republic)

^b MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Controllability; Dubois Reymond lemma; Feasible family; First variation; Normality; Optimal control problem; Second variation; Time scale; Weak maximum principle

Indexed keywords

CONTROLLABILITY; MAXIMUM PRINCIPLE;

DUBOIS-REYMOND LEMMA; FEASIBLE FAMILY; FIRST VARIATION; NORMALITY; OPTIMAL CONTROL PROBLEM; SECOND VARIATION; TIME SCALE; WEAK MAXIMUM PRINCIPLE;

OPTIMIZATION;

EID: 61349150026     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2008.04.025     Document Type: Article

References (29)

1. Agarwal R.P., Bohner M., and Wong P.J.Y. Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99 2-3 (1999) 153-166
2. Ahlbrandt C.D., Bohner M., and Ridenhour J. Hamiltonian systems on time scales. J. Math. Anal. Appl. 250 2 (2000) 561-578
3. Ahlbrandt C.D., and Peterson A.C. Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations (1996), Kluwer Academic Publishers, Boston
4. Atici F.M., Biles D.C., and Lebedinsky A. An application of time scales to economics. Math. Comput. Modelling 43 7-8 (2006) 718-726
5. Bernstein D.S. Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory (2005), Princeton University Press, Princeton
6. Bohner M. Calculus of variations on time scales. Dynam. Systems Appl. 13 3-4 (2004) 339-349
7. Bohner M., and Peterson A. Dynamic Equations on Time Scales. An Introduction with Applications (2001), Birkhäuser, Boston
8. Boltyanskii V.G. Optimal Control of Discrete Systems (1978), John Wiley & Sons, New York, Toronto
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)
10. Došlý O., and Hilger S. A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales. In: Agarwal R.P., Bohner M., and O'Regan D. (Eds). Dynamic Equations on Time Scales. J. Comput. Appl. Math. 141 1-2 (2002) 147-158
11. Erbe L., and Hilger S. Sturmian theory on measure chains. Differential Equations Dynam. Systems 1 3 (1993) 223-244
12. Erbe L., and Peterson A. Oscillation criteria for second order matrix dynamic equations on a time scale. In: Agarwal R.P., Bohner M., and O'Regan D. (Eds). Dynamic Equations on Time Scales. J. Comput. Appl. Math. 141 1-2 (2002) 169-185
13. Gelfand I.M., and Fomin S.V. Calculus of Variations (1963), Prentice-Hall, Englewood Cliffs, NJ
14. Hestenes M.R. Calculus of Variations and Optimal Control Theory (1966), John Wiley & Sons, New York
15. Hestenes M.R. Quadratic variational theory. In: Balakrishnan A.V. (Ed). Control Theory and the Calculus of Variations (1969), Academic Press, New York 1-37
16. Hestenes M.R. Quadratic control problems. J. Optim. Theory Appl. 17 1-2 (1975) 1-42
17. Hilscher R. Linear Hamiltonian systems on time scales: Positivity of quadratic functionals. In: Peterson A.C. (Ed). Boundary Value Problems and Related Topics. Math. Comput. Modelling 32 5-6 (2000) 507-527
18. Hilscher R., and Zeidan V. Discrete optimal control: The accessory problem and necessary optimality conditions. J. Math. Anal. Appl. 243 2 (2000) 429-452
19. Hilscher R., and Zeidan V. Second order sufficiency criteria for a discrete optimal control problem. J. Difference Equ. Appl. 8 6 (2002) 573-602
20. Hilscher R., and Zeidan V. Symplectic difference systems: Variable stepsize discretization and discrete quadratic functionals. Linear Algebra Appl. 367 (2003) 67-104
21. 1 solutions with variable endpoints. J. Math. Anal. Appl. 289 1 (2004) 143-166
22. Hilscher R., and Zeidan V. Nonnegativity and positivity of a quadratic functional in the discrete calculus of variations: A survey. J. Difference Equ. Appl. 11 9 (2005) 857-875
23. Hilscher R., and Zeidan V. Legendre, Jacobi, and Riccati type conditions for time scale variational problem with application. Dynam. Systems Appl. 16 3 (2007) 451-480
24. Hilscher R., and Zeidan V. Time scale embedding theorem and coercivity of quadratic functionals. Analysis (Munich) 28 1 (2008) 1-28
25. Jacobson D.H., Martin D.H., Pachter M., and Geveci T. Extensions of Linear-Quadratic Control Theory. Lecture Notes in Control and Information Sciences vol. 27 (1980), Springer-Verlag, Berlin
26. Kelley W.G., and Peterson A.C. Difference Equations: An Introduction with Applications (1991), Academic Press, San Diego
27. Lewis F.L. Optimal Control (1986), John Wiley & Sons, Inc., New York
28. Reid W.T. Ordinary Differential Equations (1971), Wiley, New York
29. Zeidan V., and Zezza P. The conjugate point condition for smooth control sets. J. Math. Anal. Appl. 132 2 (1988) 572-589