×

Variational problems with free ends on bounded and unbounded intervals. (Russian) Zbl 0659.49001

The author derives the boundary conditions for a function minimizing a functional \(K(u)=A(u)-f(u)\), on a set \({\mathfrak M}\subset L^ r_{2,\alpha,\beta}(0,b)\) where \(A(u)=A(u,u)\), \[ A(u,v)=\int^{b}_{0}\sum^{r}_{m=0}a_ m(t)u^{(m)}(t)v^{(m)}(t)dt,\quad r\geq 1,\quad 0<b<\infty; \]
\[ | a_ r(t)| \leq \lambda t^{2\alpha}(b-t)^{2\beta},\quad | a_ m(t)| \leq \lambda,\quad m=0,1,...,r-1,\quad t\in (0,b). \] The condition \(\| u-v\|^ 2_ 2\leq c^ 2A(u-v)\), \(c>0\), for all \(u,v\in {\mathfrak M}\) implies that the functional K(u) is bounded below. The classes \({\mathfrak M}\) fulfilling that condition are investigated. The case of the unbounded interval (1,\(\infty)\) for a variable t is investigated too.
Reviewer: I.Bock

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
49S05 Variational principles of physics
PDFBibTeX XMLCite