In the sequel we use for simplicity the term quasi-definite linear functional (positive definite
linear functional) for linear functionals that are quasi-definite (positive definite
) on the space of polynomials of sufficiently large degree.
It is clear that [C.sub.k], [C.sup.*.sub.k], [C.sub.d], and [C.sup.*.sub.d] are positive definite
and [C.sub.h], [C.sub.hk], and [C.sub.hd] are nonnegative definite.
Given a positive definite
matrix R and three positive constants [c.sub.1], [c.sub.2], [T.sub.f] ([c.sub.1] < [c.sub.2]), the polynomial fuzzy system (3) with control input u(t) is said to be finite-time stabilization with respect to [mathematical expression not reproducible].
Let X be the Hermitian positive definite
solution of (1).
if [rho](A) < 1, C is a fixed positive definite
matrix, matrix H solves the corresponding Lyapunov matrix equation (5), and
Equation (15) is a matrix Lyapunov equation where matrices P and Q are positive definite
If [A.sub.d] is positive definite
and E is negative definite or [A.sub.d] is negative definite and E is positive definite
, then the S-system in (1) is stable.
The Ghost-Fluid Method leads to a symmetric positive definite
linear systems that captures the discontinuity in the normal derivative while smearing out the discontinuity in the tangential direction to the interface.
where S [member of] [R.sup.nxn] is a positive definite
symmetric matrix and T [member of] [R.sup.nxn] is a unity upper triangular matrix.
In classical analysis a complex valued continuous function is said positive definite
If there exists symmetric positive definite
matrix Q, such that the following LMI holds:
For two symmetric matrices of the same dimensions X and Y, X > Y means that X-Y is positive definite
. [R.sup.+] is the set of positive real numbers.