## 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.

### MSC:

 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
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### References:

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