Positivity 11 (2007), 319–340
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020319-22, published online April 6, 2007
On K-Positivity Properties of Time-Invariant
Linear Systems Subject to Discrete Point Lags
Manuel De la Sen
Abstract. This paper discusses non-negativity and positivity concepts and
related properties for the state and output trajectory solutions of dynamic
linear time-invariant systems described by functional diﬀerential equations
subject to point time-delays. The various non-negativity and positivity intro-
duced hierarchically from the weakest one to the strongest one while sepa-
rating the corresponding properties when applied to the state space or to the
output space as well as for the zero-initial state or zero-input responses. The
formulation is developed by deﬁning cones for the input, state and output
spaces of the dynamic system.
Mathematics Subject Classiﬁcation (2000). 47A15; 47B65; 34D20; 34D25.
Keywords. Dynamic Systems, Functional Equations, Positivity, Stability,
Positive Systems have an important relevance since the input, state and output
signals in many physical or biological systems are necessarily positive, [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Therefore, important attention has
been paid to such systems in the last decades. For instance, an hydrological sys-
tem composed of a set of lakes in which the input is the inﬂow into the upstream
lake and the output is the outﬂow from the downstream lake is an Externally
Positive System since the output is always positive under a positive input, .
Also, hyperstable single-input single-output systems are Externally Positive since
the Impulse Response Kernel is everywhere positive. This also implies that the
associated transfer functions (provided they are time-invariant) are Positive Real
and their input/output instantaneous power and time-integral energy are positive.
However, hyperstable systems of second and higher orders are not guaranteed to
be Externally Positive since the Impulse Response Kernel Matrix is everywhere
positive deﬁnite but not necessarily positive, . The properties of those systems