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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 $[{\mathcal L}(y)]_{\min} = \int^{b}_{a}L(t,y^{\sigma}(t),y^{\Delta}(t))\Delta t \rightarrow \min$
$y(a)= \alpha, \;y(b)=\beta$ where $$\alpha, \beta \in R^{n}$$ and $$L : T \times R^{2n}$$ 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.
Reviewer: W.Hejmo (Kraków)

##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations 49K15 Optimality conditions for problems involving ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 39A10 Additive difference equations