Massera's lemma

Massera’s lemma is used in the construction of a converse Lyapunov function of the following form (also known as the integral construction)

${\displaystyle V(\zeta )=\int _{0}^{\infty }G(|\varphi (t,\zeta )|)dt}$

for an asymptotically stable dynamical system whose stable trajectory starting from ${\displaystyle \zeta {\text{ is }}\varphi (t,\zeta )}$

The lemma states:

Let ${\displaystyle g:[0,\infty )\rightarrow R}$ be a positive, continuous, strictly decreasing function with ${\displaystyle g(t)\rightarrow 0}$ as ${\displaystyle t\rightarrow \infty }$. Let ${\displaystyle h:[0,\infty )\rightarrow R}$ be a positive, continuous, nondecreasing function. Then there exists a function ${\displaystyle G:[0,\infty )\rightarrow [0,\infty )}$ such that

• ${\displaystyle G}$ and its derivative ${\displaystyle G'}$ are class-K functions defined for all t ≥ 0
• There exist positive constants k₁, k₂, such that for any continuous function u satisfying 0 ≤ u(t) ≤ g(t) for all t ≥ 0,

${\displaystyle \int _{0}^{\infty }G(u(t))\,dt\leq k_{1};\quad \int _{0}^{\infty }G'(u(t))h(t)\,dt\leq k_{2}.}$

Massera's lemma for single variable functions was extended to the multivariable case by Vu and Liberzon.[2]

Let ${\displaystyle g:[0,\infty )\rightarrow R}$ be a positive, continuous, strictly decreasing function with ${\displaystyle g(t)\rightarrow 0}$ as ${\displaystyle t\rightarrow \infty }$. Let ${\displaystyle h:[0,\infty )\rightarrow R}$ be a positive, continuous, nondecreasing function. Then there exists a differentiable function ${\displaystyle G:[0,\infty )\rightarrow [0,\infty )}$ such that

• ${\displaystyle G}$ and its derivative ${\displaystyle G'}$ are class-K functions on ${\displaystyle [0,\infty )}$.
• For every positive integer ${\displaystyle l}$, there exist positive constants k₁, k₂, such that for any continuous function ${\displaystyle u:\mathbb {R} ^{l}\rightarrow [0,\infty )}$ satisfying

${\displaystyle 0\leq u(t_{1},\ldots ,t_{l})\leq g(t_{1}+\cdots +t_{l})}$ for all ${\displaystyle t_{i}\geq 0}$, ${\displaystyle i=1,\ldots ,l}$

we have

${\displaystyle \int _{0}^{\infty }\cdots \int _{0}^{\infty }G(u(s_{1},\ldots ,s_{l}))ds_{1}\ldots ds_{l}<k_{1}}$

${\displaystyle \int _{0}^{\infty }\cdots \int _{0}^{\infty }G'(u(s_{1},\ldots ,s_{l}))\times h(s_{1}+\cdots +s_{l})ds_{1}\ldots ds_{l}<k_{2}}$

• Massera, José Luis (1949), "On Liapounoff's conditions of stability", Annals of Mathematics, Second Series, 50 (3): 705–721, doi:10.2307/1969558, JSTOR 1969558, MR 0035354
• Massera, José Luis (1956), "Contributions to stability theory", Annals of Mathematics, Second Series, 64 (1): 182–206, doi:10.2307/1969955, JSTOR 1969955, MR 0079179
• Massera, José Luis; Schäffer, Juan Jorge (1966), Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Boston, MA: Academic Press, MR 0212324