Sentence examples for depending on derivatives from inspiring English sources
A more general equation with f depending on derivatives was studied in [24].
In [7, 8], functionals depending on derivatives are also suggested for investigating the asymptotic stability of neutral nonlinear systems.
A more general equation with f depending on derivatives and the boundary conditions with two nonlocal terms was studied by Zhang [4], where f can be singular at t = 0 and/or t = 1 and be allowed to change sign.
By reformulating a typical optimal design problem in a variational format, we are able to prove existence results in a rather general framework allowing non-linear state equations and costs functional depending on derivatives of states.
In our problem the nonlinearity depends on derivatives, and we allow solutions to be unbounded, which is an extra interesting feature.
Firstly, BVP (1.1) depends on derivative, and the boundary conditions are more general.
Problems of the classical calculus of variations with integrand depending on fractional derivatives instead of ordinary derivatives are first introduced by Agrawal [17] in 2002.
The approach also enables treatment of actions depending on higher derivatives of position; I thus derive balances for an elastica which are applicable to moving contact problems.
end{aligned}By the properties of B it follows that (tilde{B}) is a differential operator of order (k+2), depending on horizontal derivatives of (a_{ij}^epsilon ) of order at most (k+1).
There exists a differential operator B of order (k+1), depending on horizontal derivatives of (a_{ij}^epsilon ) of order at most k, such that begin{aligned} X_{i_k}^epsilon cdots X_{i_1}^epsilon left( L_{epsilon,(x_0,t_0)}-L_epsilon right) = sum _{i,j=1}^n left( a_{ij}^epsilon - a_{ij}^epsilon (x_0, t_0) right) X_{i_k}^epsilon cdots X_{i_1}^epsilon X^epsilon _iX^epsilon _j + B. end{aligned}.
Conditions are given to characterize neutral systems with delays for which linear observers exist not depending on the derivatives of the state.
