Coupled intervals in the discrete calculus of variations: necessity and sufficiency. (English) Zbl 1032.49038

As the authors state in the Introduction, “in this paper a thorough study is presented of the nonnegativity and positivity of the discrete quadratic functional of the form” \[ {\mathcal I}(\eta):=\eta^T_0\Gamma_0\eta_0 +\eta^T_{N+1}\Gamma\eta_{N+1} + \displaystyle \sum_{k=0}^N [\eta^T_{k+1}P_k\eta_{k+1}+2\eta^T_{k+1}Q_k\Delta \eta_k+\Delta\eta^T_kR_k\Delta\eta_k] \] over (variable) endpoints constraints of the form \[ M_0\eta_0=0, \;M\eta_{N+1}=0. \] In order to extend to the “variable endpoints” case some of their previous results [e.g., R. Hilscher and V. Zeidan, J. Differ. Equ. Appl. 8, 573–602 (2002; Zbl 1010.49020)], the authors introduce the new concept of coupled interval which is shown to contain as a particular case the classical concept of conjugate interval in calculus of variations.
As in their previous papers, the authors prove a large number of results that contain different types of equivalent conditions for the nonnegativity and for the positivity of the functional \({\mathcal I}(\eta)\); these results are then used to obtain corresponding necessary and/or sufficient optimality conditions for local minima of some classes of problems in “discrete” calculus of variations.


49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N10 Linear-quadratic optimal control problems
39A12 Discrete version of topics in analysis
39A13 Difference equations, scaling (\(q\)-differences)
93C55 Discrete-time control/observation systems


Zbl 1010.49020
Full Text: DOI


[1] Agarwal, R.P.; Ahlbrandt, C.D.; Bohner, M.; Peterson, A.; Agarwal, R.P.; Bohner, M., Discrete linear Hamiltonian systems: A survey, Discrete and Continuous Hamiltonian Systems, Dynam. systems appl., 8, 3-4, 307-333, (1999) · Zbl 0942.39009
[2] Ahlbrandt, C.D., Discrete variational inequalities, (), 93-107 · Zbl 0869.49007
[3] Ahlbrandt, C.D.; Heifetz, M., Discrete Riccati equations of filtering and control, (), 1-16 · Zbl 0863.93049
[4] Ahlbrandt, C.D.; Hooker, J.W., A variational view of non-oscillation theory for linear difference equations, (), 1-21 · Zbl 0805.93024
[5] Ahlbrandt, C.D.; Peterson, A.C., Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations, (1996), Kluwer Academic Publishers Boston · Zbl 0860.39001
[6] Bohner, M., Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, J. math. anal. appl., 199, 804-826, (1996) · Zbl 0855.39018
[7] Bohner, M., Riccati matrix difference equations and linear Hamiltonian difference systems, Dynam. contin. discrete impuls. systems, 2, 2, 147-159, (1996) · Zbl 0873.39003
[8] Bohner, M.; Došlý, O., Disconjugacy and transformations for symplectic systems, Rocky mountain J. math., 27, 3, 707-743, (1997) · Zbl 0894.39005
[9] Bohner, M.; Došlý, O., Positivity of block tridiagonal matrices, SIAM J. matrix anal., 20, 1, 182-195, (1998) · Zbl 0921.39004
[10] Bohner, M.; Došlý, O.; Kratz, W.; Agarwal, R.P.; Bohner, M., Discrete reid roundabout theorems, Discrete and Continuous Hamiltonian Systems, Dynam. systems appl., 8, 3-4, 345-352, (1999) · Zbl 0946.39001
[11] Došlá, Z.; Došlý, O., Quadratic functionals with general boundary conditions, Appl. math. optim., 36, 243-262, (1997) · Zbl 1057.49502
[12] Došlá, Z.; Zezza, P., Quadratic functionals with a variable singular end point, Comment. math. univ. carolin., 33, 3, 411-425, (1992) · Zbl 0779.49026
[13] Erbe, L.; Yan, P., Disconjugacy for linear Hamiltonian difference systems, J. math. anal. appl., 167, 355-367, (1992) · Zbl 0762.39003
[14] Erbe, L.; Yan, P., Qualitative properties of Hamiltonian difference systems, J. math. anal. appl., 171, 334-345, (1992) · Zbl 0768.39001
[15] Hilscher, R., Disconjugacy of symplectic systems and positivity of block tridiagonal matrices, Rocky mountain J. math., 29, 4, 1301-1319, (1999) · Zbl 0956.39010
[16] 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
[17] Hilscher, R.; Zeidan, V., Second order sufficiency criteria for a discrete optimal control problem, J. differ. equations appl., 8, 6, 573-602, (2002) · Zbl 1010.49020
[18] R. Hilscher, V. Zeidan, Nonnegativity of a discrete quadratic functional in terms of the (strengthened) Legendre and Jacobi conditions, in: R.P. Agarwal (Ed.), Advances in Difference Equations IV, Comput. Math. Appl., to appear · Zbl 1044.49020
[19] Luenberger, D.G., Linear and nonlinear programming, (1984), Addison-Wesley Reading, MA · Zbl 0241.90052
[20] Stefani, G.; Zezza, P., Constrained regular LQ-control problems, SIAM J. control optim., 35, 3, 876-900, (1997) · Zbl 0876.49031
[21] Zeidan, V., Sufficiency criteria via focal points and via coupled points, SIAM J. control optim., 30, 1, 82-98, (1992) · Zbl 0780.49018
[22] Zeidan, V., Sufficient conditions for variational problems with variable endpoints: coupled points, Appl. math. optim., 27, 2, 191-209, (1993) · Zbl 0805.49012
[23] Zeidan, V.; Agarwal, R.P.; Bohner, M., Nonnegativity and positivity of a quadratic functional, Discrete and Continuous Hamiltonian Systems, Dynam. systems appl., 8, 3-4, 571-588, (1999) · Zbl 0936.49022
[24] Zeidan, V.; Zezza, P., Coupled points in the calculus of variations and applications to periodic problems, Trans. amer. math. soc., 315, 1, 323-335, (1989) · Zbl 0677.49020
[25] Zezza, P., Jacobi condition for elliptic forms in Hilbert spaces, J. optim. theory appl., 76, 2, 357-380, (1993) · Zbl 0798.49027
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