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Quadratic functionals for second order matrix equations on time scales. (English) Zbl 0938.49001

This paper is devoted to the variational problem of the functional

${\left[ℒ\left(y\right)\right]}_{min}={\int }_{a}^{b}L\left(t,{y}^{\sigma }\left(t\right),{y}^{{\Delta }}\left(t\right)\right){\Delta }t\to min$
$y\left(a\right)=\alpha ,\phantom{\rule{4pt}{0ex}}y\left(b\right)=\beta$

where $\alpha ,\beta \in {R}^{n}$ and $L:T×{R}^{2n}$ is of ${C}^{2}$ class. Examination of the above problem is based upon well-known results dealing with quadratic functional problems, i.e., Jacobi’s condition, discrete version of Jacobi’s condition and Jacobi’s condition on time scales. The paper may be useful in analysis of an optimal problem with square criterion of quality.

##### MSC:
 49J15 Optimal control problems with ODE (existence) 49K15 Optimal control problems with ODE (optimality conditions) 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 39A10 Additive difference equations